Tutorial Reference Minitab 16 Excel 3y5a26

Minitab 16

Excel 2010

Reference Guide Prepared for MATH201/MATH202 Bryan Crissinger University of Delaware Department of Mathematical Sciences

Contents

Introduction ................................................................................................................................................ 5 Troubleshooting (Minitab) .......................................................................................................................... 6 Troubleshooting (Excel)............................................................................................................................. 7 Getting More Help ..................................................................................................................................... 8

Descriptive Statistics Frequency Table (Minitab) ........................................................................................................................ 9 Bar Chart (Minitab) .................................................................................................................................. 11 Bar Chart (Excel) ..................................................................................................................................... 13 Pie Chart (Minitab) ................................................................................................................................... 14 Pie Chart (Excel) ..................................................................................................................................... 16 Creating New Variables (Minitab) ............................................................................................................ 17 Creating New Variables (Excel) .............................................................................................................. 20 Descriptive Statistics (Minitab) ................................................................................................................ 21 Descriptive Statistics (Excel) ................................................................................................................... 24 Dotplot (Minitab) ...................................................................................................................................... 26 Histogram (Minitab) ................................................................................................................................. 28 Histogram (Excel) .................................................................................................................................... 33 Boxplot (Minitab) ...................................................................................................................................... 35 Stem-and-Leaf Plot (Minitab) .................................................................................................................. 37 Normal Probability Plot (Minitab) ............................................................................................................. 39 Graphical Summary (Minitab) .................................................................................................................. 41 Scatterplot (Minitab) ................................................................................................................................ 43 Scatterplot (Excel) ................................................................................................................................... 46

Probability Distributions Binomial Distribution (Minitab) ................................................................................................................. 47 Binomial Distribution (Excel) ................................................................................................................... 50 Poisson Distribution (Minitab) .................................................................................................................. 52 Poisson Distribution (Excel) .................................................................................................................... 55 Normal Distribution (Minitab) ................................................................................................................... 57 Normal Distribution (Excel) ...................................................................................................................... 62 t Distribution (Minitab).............................................................................................................................. 64 t Distribution (Excel) ................................................................................................................................ 68 Chi-Square Distribution (Minitab) ............................................................................................................ 71 Chi-Square Distribution (Excel) ............................................................................................................... 74 2

F Distribution (Minitab) ............................................................................................................................ 77 F Distribution (Excel) ............................................................................................................................... 80

One-Sample Inference z-interval and z-test for 𝝁 (Minitab).......................................................................................................... 82 t-test and t-interval for 𝝁 (Minitab) ........................................................................................................... 84 t-interval for 𝝁 (Excel) .............................................................................................................................. 86 Interval and Test for 𝒑 (Minitab) .............................................................................................................. 88 Interval and Test for 𝝈𝟐 (Minitab)............................................................................................................. 90

Two-Sample Inference t-Interval and t-Test for 𝝁𝟏 − 𝝁𝟐 Using Independent Samples (Minitab) ................................................. 92 t-Test for 𝝁𝟏 − 𝝁𝟐 Using Independent Samples (Excel) .......................................................................... 95 t-Interval and t-Test for 𝝁𝟏 − 𝝁𝟐 Using Paired Samples (Minitab) ........................................................... 98 t-Test for 𝝁𝟏 − 𝝁𝟐 Using Paired Samples (Excel) .................................................................................. 100 z-Interval and z-Test for 𝒑𝟏 − 𝒑𝟐 Using Independent Samples (Minitab) .............................................. 102 Test and Interval for 𝝈𝟐𝟏 /𝝈𝟐𝟐 Using Independent Samples (Minitab) ...................................................... 105 F-Test for 𝝈𝟐𝟏 /𝝈𝟐𝟐 Using Independent Samples (Excel) .......................................................................... 108

ANOVA One-Way ANOVA (Minitab) ................................................................................................................... 111 One-Way ANOVA (Excel)...................................................................................................................... 115 Two-Way ANOVA (Minitab) ................................................................................................................... 117 Two-Way ANOVA (Excel)...................................................................................................................... 121 Interaction Plot (Minitab)........................................................................................................................ 125

Contingency Tables Chi-Square Test for One-Way Table (Minitab) ...................................................................................... 126 Chi-Square Test for Two-Way Table (Minitab) ...................................................................................... 129

Regression Regression (Minitab) ............................................................................................................................. 132 Regression (Excel) ................................................................................................................................ 137

Control Charts ̅ Chart (Minitab) .................................................................................................................................... 140 𝒙 R Chart (Minitab) ................................................................................................................................... 143 P Chart (Minitab) ................................................................................................................................... 146 3

Time Series Time Series Plot (Minitab) ..................................................................................................................... 148 Moving Averages (Minitab) .................................................................................................................... 149 Single Exponential Smoothing (Minitab) ............................................................................................... 151 Trend Analysis (Minitab) ........................................................................................................................ 153 Trend Analysis (Excel)........................................................................................................................... 156 Seasonal Regression Models ................................................................................................................ 158

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Introduction Minitab Modern statistical practice always involves the use of software to do data analysis. We will use Minitab (version 16) or Excel (2010 or 2013) to do many statistical analyses and it will be beneficial for you to use such a package to do homework problems that require the use of software, especially where hand computations are burdensome. 101 Ewing is not open to students other than during lab times, so if you need to use Minitab, there are two options: 

Computer labs on campus where Minitab software is installed: 305 Pearson Hall, 111 and 113 MacDowell Hall, Smith Hall Computing Site, B&E Lab in Purnell



Minitab at www.onthehub.com/minitab. This site allows you try Minitab free for 30 days, rent Minitab for 6 months ($30), or buy a copy ($100). Minitab is currently available for Windows only.

Excel While Excel is designed as spreadsheet software, it can perform basic data analysis functions as well. Since Excel is used extensively in business, it’s worthwhile to learn about its statistical capabilities (and limitations). To enable the data analysis features in Excel, you must make sure two add-ins are activated. Excel 2007: Click on the Office Button in the upper left corner of the window Excel 2010 or 2013: Choose File > Options and then select Excel Options. Choose Add-Ins from the menu at the left. You should see the following active application add-ins: Analysis ToolPak Analysis ToolPak – VBA If not, make sure Excel Add-ins are selected to manage at the bottom of the window and click Go. Check the two Analysis ToolPak add-ins and click Ok. You may have to restart Excel for the change to take effect. These two features activate the Data Analysis option under the Data tab at the top of your Excel window.

Textbook Datasets You can directly input to Minitab and Excel most of the datasets referenced in your textbook, rather than entering the data by hand. The files are located on the CD that comes with your textbook.

5

Troubleshooting (Minitab) In Minitab Express, when I try to copy and paste a data set into the worksheet, everything ends up in one column.

Minitab Express may recognize the columns for data copied from Excel better than from other sources. Try copying/pasting into an Excel spreadsheet first. If the same thing happens in Excel, check out the Troubleshooting (Excel) section below to separate the data into columns in Excel. Once you do that, copy from Excel and paste into your Minitab Express worksheet.

I'm using File > Open Project… to open a data file but the data file doesn't show up in the dialog box.

If the data file is a Minitab worksheet, you must use File > Open Worksheet… instead of File > Open Project….

Minitab shows a column contains text data, e.g. C3-T, but I only have numbers in the column.

It's possible you may have typed or copied a non-numeric character in one of the cells. Try using Data > Change Data Type > Text to Numeric… and store the numeric columns in a new column.

In a dialog box, a column I need to use is not shown in the list of available columns on the left.

Try clicking in the box where you want to use the column first. If that doesn't work, it could be that Minitab is expecting a numeric data column and the column you're trying to use contains text data. Text data columns are indicated with a T suffix in the column heading, e.g. C3-T.

Paired t-test or regression: I get an error message that my two columns must be of the same length.

Both columns must have the same number of values (can include missing values).

2-Sample t-test: I get an error that there must be exactly two distinct subgroups.

You may be using stacked data format where there are more than 2 distinct values in the grouping variable column (subscripts).

6

Troubleshooting (Excel)

When I try to copy and paste a data set into the worksheet, everything ends up in one column.

Copy the data as usual. Select the cell in the top left corner of the space where you want to paste the data. Right-click on this cell, select Paste Special…, and choose Text as the source. If that doesn't work, highlight the column you pasted, choose Data > Text to Columns, and choose Delimited as the file type. Select delimiters from the list until the columns are shown separated properly. If that fails, try importing the data to Minitab first, copy the Minitab worksheet, and paste into Excel.

There are no statistics displayed when I request descriptive statistics.

You must check the Summary statistics box in the dialog box.

A histogram is not displayed when I request a histogram.

You must check the Chart Output box in the dialog box.

I get this error message: "Input range contains non-numeric data."

If an input range contains a column label, check the Labels or Labels in First Row box in the dialog box. Otherwise, there may be non-numeric characters in the column (usually indicated by left-justified numbers).

Paired t-test or regression: I get an error message that my two columns must have the same number of rows.

Excel doesn't handle missing data for either a paired analysis of means or regression. All columns used must contain only non-missing values and be of the same length.

7

Getting More Help This guide is designed to be a quick-reference tool, not an exhaustive reference. Minitab

Excel

Help Buttons You can access documentation for specific dialog boxes by clicking on the Help button in the dialog box.

? Button Click on the ? in the upper right of the window to access Microsoft's help for Excel.

Help > Help Access help by topic as well as use the index and search features.

Note: The Excel/DDXL or Excel/XLSTAT parts of the Using Technology sections at the ends of the chapters will not apply unless you install the DDXL add-in (on the CD accompanying the textbook). We do not use this add-in as it is not generally available with Excel.

Help > Methods and Formulas This feature shows the methods and formulas used in the procedures you specify. Textbook: Using Technology Sections You'll find documentation on using many of the tools listed in this guide and others in the Using Technology sections at the end of each chapter.

8

Frequency Table (Minitab) The data should be entered in the Minitab worksheet with one row per observation. In this example we have data on several students in a class. The variables are Student ID and Class. Choose Stat > Tables > Tally Individual Variables.

In the dialog box, select the categorical variable you want to summarize with a frequency table. The default frequency table includes only category counts (frequencies) but you can also request percents (similar to relative frequencies) in this dialog box, as shown here.

9

Frequency Table (Minitab) The frequency table will be displayed in the Session window. Note that misspelled words are considered a different category.

10

Bar Chart (Minitab) The data should be entered in the Minitab worksheet with one row per observation. In this example we have data on several students in a class. The variables are Student ID and Class. Choose Graph > Bar Chart.

Choose the Simple bar chart option and specify Class as the categorical variable you want to chart. You can customize the chart by using the six buttons in the dialog box.

11

Bar Chart (Minitab) The chart will display in a separate graph window. Chart of Class 6 5

Count

4 3 2 1 0

Frehsman

Freshman

Junior Class

Senior

Sophomore

12

Bar Chart (Excel) Enter the frequency table in the worksheet; Excel won’t do this automatically for you. Highlight the cells containing the frequency table and choose Insert > Column for a vertical bar chart or Insert > Bar for a horizontal bar chart.

13

Pie Chart (Minitab) The data should be entered in the Minitab worksheet with one row per observation. In this example we have data on several students in a class. The variables are Student ID and Class. Choose Graph > Pie Chart.

The default pie chart requires data in the format as shown above with one row per observation (Chart counts of unique values). Specify Class as the categorical variable you want to chart. You can customize the chart by using the six buttons in the dialog box. Use the Labels button and the Slice Labels tab to label the pie slices with the category percentages.

14

Pie Chart (Minitab)

The pie chart will be shown in a separate graph window. Pie Chart of Class

7.1%

14.3%

C ategory Frehsman Freshman Junior Senior Sophomore

42.9%

21.4%

14.3%

15

Pie Chart (Excel) Enter the frequency table in the worksheet; Excel won’t do this automatically for you. Highlight the cells containing the frequency table and choose Insert > Pie.

An easy way to add the percentages to the chart is to select the first Chart Layout after you've created the pie chart. Click on the chart title to edit it.

Class Rank 2 16%

4 36%

4 19%

6 29%

16

Creating New Variables (Minitab) You can create new columns using the data in existing columns quickly and easily. Option 1: Calc > Row Statistics…

Here we want to compute an average price for each of the stocks in the worksheet using the four existing prices. Input variables: select the four columns of prices Store result in: specify a name for the new column where Minitab will store the average price for each stock

17

Creating New Variables (Minitab) Minitab computes the average prices and stores them in the worksheet in the next available column.

Option 2: Calc > Calculator…

18

Creating New Variables (Minitab) Store result in variable: specify a name for the new column where Minitab will store the average price for each stock Expression: write the expression for the calculation you want to do This option gives you more flexibility in the kinds of calculations you can do.

19

Creating New Variables (Excel) In Excel, you create new columns by writing formulas in the cells. Begin the formula with an equal sign.

Then copy the cell containing the formula and paste it to all the cells below it. An easy way to do that is to drag the lower right corner of the cell straight down. The cursor will change from a white cross to a black cross.

Since we did not use any absolute cell references (e.g. $B$2), the references automatically change to the appropriate rows.

20

Descriptive Statistics (Minitab) Option 1: Use Stat > Basic Statistics > Display Descriptive Statistics… to obtain many numeric descriptive statistics for columns of numeric data. Here we have data on the number of soft drinks consumed per week for a sample of males and a sample of females. The data are shown in the worksheet in two different formats: Unstacked Format (C1 and C2): Each sample of data has its own column. Stacked Format (C3 and C4): Both samples are stacked in C3 with a column of gender indicators in C4. In practice, we typically have grouped data in only one format.

21

Descriptive Statistics (Minitab) Variables: Specify the column(s) containing the data you want to summarize. If you want to summarize data separately for different groups, specify the analysis as shown below depending on whether you have unstacked or stacked data. Unstacked

Stacked

The output will display in the session window. Note the slight differences in the output for the two data formats. Many tools in Minitab can accommodate data in either format.

22

Descriptive Statistics (Minitab) Option 2: Use Calc > Column Statistics… to obtain a single statistic for a numeric data column.

Specify the Statistic you want. Input variable: Specify the column containing the data you want to summarize. By default, the output will display in the Session window unless you specify an optional storage column.

23

Descriptive Statistics (Excel) Use Data > Data Analysis > Descriptive Statistics. Here we have data in unstacked format.

Input Range: Highlight the cells containing the data. Grouped By: Columns This tells Excel we have unstacked data. Labels in First Row: Check this box if the Input Range contains column labels. If not, don't check the box. Checking the box tells Excel to ignore what's in the first row of the Input Range. Summary Statistics: You must check this box for the output to display.

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Descriptive Statistics (Excel) The output will be displayed in a separate worksheet. Make the columns containing the labels (here A and C) wider so that the entire labels are displayed.

25

Dotplot (Minitab) Use Graph > Dotplot…

The dialog box gives you several input options depending on the format of the data. One Y, Simple: Use this for one-sample analysis, i.e. a single column of numeric data. One Y, With Groups: Use for stacked data. Multiple Y's, Simple: Use for unstacked data. Multiple Y's, With Groups: Use for stacked data having multiple response/comparison variables.

While you have the option to stack the dots in the two dotplots with grouped data, this is not recommended as it makes determining features such as shape difficult to determine.

26

Dotplot (Minitab) One Y, Simple: Displays the distribution of number of soft drinks combined for males and females. Dotplot of Number

2

4

6 Number

8

10

12

Multiple Y's, Simple: Uses the unstacked data to display the distribution of number of soft drinks separately for males and females. Dotplot of NumMales, NumFemales

NumMales NumFemales

2

4

6 Data

8

10

12

One Y, With Groups: Same comparison analysis using the stacked data.

Gender

Dotplot of Number

Female

Male

27

2

4

6 Number

8

10

12

Histogram (Minitab) Use Graph > Histogram…

The dialog box gives you several input options depending on the format of the data. Simple: Use this for one-sample analysis, i.e. a single column of numeric data. With Outline and Groups: Use for stacked data.

You also have the option of having Minitab draw the best fitting normal distribution either on the histogram.

28

Histogram (Minitab) Simple: Displays the distribution of number of soft drinks combined for males and females. Histogram of Number 16 14

Frequency

12 10 8 6 4 2 0

1.5

3.0

4.5

6.0 7.5 Number

9.0

10.5

12.0

Group Comparisons An alternative to using the With Outline and Groups option to create multiple histograms for grouped data (as the picture can get messy with all the outlines overlaid) is to use the Simple option and the Multiple Graphs… button.

Stacked Data Graph Variables: Specify the single column of numeric data, i.e. the response/comparison variable.

29

Histogram (Minitab) Click on Multiple Graphs… and request that the multiple histograms be shown In separate s of the same graph. Also check the box so that the X axis scales and bins will be the same for the histograms.

By Variable tab: Specify Gender as the "by variable", i.e. the grouping variable.

Histogram of Number

It is important that the x-axis scales be identical to allow for an accurate comparison of the features of the distributions.

1.5

Female

12

Frequency

8 6 4 2

1.5

3.0

4.5

6.0

7.5

9.0 10.5 12.0

Number variable: Gender

30

4.5

6.0

7.5

Male

10

0

3.0

9.0 10.5 12.0

Histogram (Minitab) Unstacked Data Graph Variables: Specify both numeric data columns.

Click on Multiple Graphs… and request that the multiple histograms be shown In separate s of the same graph. Also check the box so that the X axis scales and bins will be the same for the histograms.

Note that the scales of the frequency axes are different; these could be made identical by checking the Same Y in the Multiple Graphs… dialog.

Histogram of NumMales, NumFemales 1.5

NumMales

12

5

Frequency

4 8 3 6 2

4

1

2

1.5

3.0

4.5

31

6.0

7.5

4.5

6.0

7.5

9.0

NumFemales

10

0

3.0

9.0

10.5 12.0

0

10.5 12.0

Histogram (Minitab) You can change the default interval (bin) definitions by right-clicking on the bars of the histogram once it's created and selecting Edit Bars… Under the Binning tab change Interval Type to Cutpoint. Define the Cutpoint positions with a list of the endpoints of the intervals. You can do this long-hand: 0 1 2 3 4 5 6 7 8 9 10 11 12 or by using Minitab short-hand as shown here. 0:12/1 requests intervals starting at 0 and ending at 12 each with width of 1.

Histogram of Number 10

Frequency

8

6

4

2

0

0

2

4

6 Number

8

10

12

Note: Minitab's interval/bin definitions use the [ , ) convention. For example, in the above histogram, the interval [1, 2) includes the 4 subjects who drink 1 soft drink per week. Note: You can also use the Midpoint Interval Type and change the Number of intervals to the desired number without having to specify the individual endpoints.

32

Histogram (Excel) In addition to having the data entered in the worksheet, you must enter a list of the endpoints of the intervals (bins) for the histogram somewhere in the worksheet.

Use Data > Data Analysis > Histogram. Input Range: Highlight the column of cells containing the data. Bin Range: Highlight the column of cells containing the interval endpoints. Labels: Check this box if both the Input Range and Bin Range contain column labels. If not, don't check the box. Checking the box tells Excel to ignore what's in the first row of these ranges. Chart Output: You must check this box for the histogram to display.

33

Histogram (Excel) Once the histogram is created, select the bars portion of the graph and right-click. Choose Format Data Series… and change the Gap Width to 0% (No Gap).

Excel outputs both the histogram and a frequency table showing the counts in each interval (bin). The Bin Endpoints in the frequency table are the upper endpoints of each interval. For example, there are 5 observations in the interval (6, 8], 3 observations in the interval (8, 10], etc.

In order to construct multiple histograms, you must do a separate analysis for each group. Be sure to use the same Bin Range for each.

Note: Excel's interval/bin definitions use the ( , ] convention. For example, in the above histogram, the interval (4, 6] includes the 7 subjects who drink either 5 or 6 soft drinks per week.

34

Boxplot (Minitab) Use Graph > Boxplot… For this example we'll use data on length of time in practice (years) for a sample of physicians.

The dialog box gives you several input options depending on the format of the data. One Y, Simple: Use this for one-sample analysis, i.e. a single column of numeric data. One Y, With Groups: Use for stacked data. Multiple Y's, Simple: Use for unstacked data. Multiple Y's, With Groups: Use for stacked data having multiple response/comparison variables.

35

Boxplot (Minitab) Here we show an analysis of length of time in practice (YRSPRAC) by specialty (SPEC). Since these data are stacked (single numeric column of times and a categorical column of specialty indicators), we'll use the One Y, With Groups input option. Graph variables: Specify the single column of numeric data, i.e. the response/comparison variable. Categorical variables for grouping: Specify the categorical grouping variable.

You can display the boxplot horizontally instead of vertically (the default) by using the Scale… button and checking the box to Transpose value and category scales.

Boxplot of YRSPRAC

MED

SPEC

The output shows an outlier at 40 years in practice in the Surgery specialty sample. Minitab classifies any observations outside the inner fences as outliers and shows them as asterisks in the boxplot.

SURG

The whiskers extend to the most extreme observations just inside the inner fences.

0

10

20 YRSPRAC

30

Hover the mouse pointer over the boxplot to see some descriptive statistics and numeric features of the boxplot.

36

40

Stem-and-Leaf Plot (Minitab) Use Graph > Stem-and-Leaf…

Graph variables: Specify a column of numeric data. By variable: Specify the categorical grouping variable. Note: The by variable needs to be coded with numbers in the worksheet for you to be able to use it here. In our data example, we'd first have to create a new column with numeric codes first: for example 1 = SURG 2 = MED. You can do this automatically using Data > Code > Text to Numeric…

37

Stem-and-Leaf Plot (Minitab) This particular graph is displayed to the Session Window.

The leftmost column in the stem-and-leaf plot shown is called the depths. The numbers are cumulative counts of numbers of leaves in each row staring from each extreme and increa to the row containing the median. The depth for the row containing the median is a simple count of the number of leaves in that row and is indicated in parentheses. In this example, there are 16 leaves in the first row, 21 in the second row for a total of 37, 2 leaves in the last row, 1 in the next to last row for a total of 3, etc. The median is in the third row. There are 23 leaves in that row.

38

Normal Probability Plot (Minitab) Use Graph > Probability Plot… > Single Using the Multiple option overlays several normal probability plots on the same set of axes which can look rather jumbled.

Graph variables: Specify a column of numeric data.

You can request multiple normal probability plots for grouped data in a way similar to that for histograms. See the documentation for Histogram (Minitab) for details.

39

Normal Probability Plot (Minitab)

Probability Plot of YRSPRAC Normal - 95% CI

99.9

Mean StDev N AD P-Value

99 95 90

Percent

The extent to which the pattern of points deviates from a straight line is an indication as to the lack of fit of a normal model for the data. Minitab also provides confidence bands and the Anderson-Darling test for normality:

80 70 60 50 40 30 20 10

H0: data come from a normal model Ha: data do not come from a normal model

5 1 0.1

40

-20

-10

0

10 20 YRSPRAC

30

40

50

14.60 9.161 112 0.954 0.015

Graphical Summary (Minitab) This tool displays many important numeric statistics alongside several graphical summaries for numeric variables. Use Stat > Basic Statistics > Graphical Summary…

The data shown are eruption data for several eruptions of the Old Faithful geyser in Yellowstone National Park, Wyoming. Here we summarize the actual time until the next eruption in minutes (ATM). Graph variables: Specify a column of numeric data. By variables (optional): Specify an optional categorical variable for grouping if you want separate analyses for different groups. Confidence level: Specify a confidence level for confidence intervals for the population mean, median, and standard deviation.

41

Graphical Summary (Minitab) In addition to many basic descriptive statistics, Minitab shows the results of a formal test for normality (see Normal Probability Plot) and several confidence intervals. The histogram is editable, just like the output of Graph > Histogram so that, for example, you can change the interval/bin definitions by right-clicking on one of the bars (see Histogram). Summary for ATM A nderson-D arling N ormality Test

50

60

70

80

90

A -S quared P -V alue <

1.65 0.005

M ean S tD ev V ariance S kew ness Kurtosis N

76.352 16.494 272.044 -0.07322 -1.40627 54

M inimum 1st Q uartile M edian 3rd Q uartile M aximum

100

49.000 60.000 82.000 91.000 107.000

95% C onfidence Interv al for M ean 71.850

80.854

95% C onfidence Interv al for M edian 65.000

85.643

95% C onfidence Interv al for S tD ev 9 5 % C onfidence Inter vals

13.865

Mean Median 65

70

75

80

85

42

20.362

Scatterplot (Minitab) There are several ways to obtain a scatterplot of (x,y) pairs of numeric data. The data must be entered in the worksheet with two columns for the numeric data and where the rows represent each observation. Here we have 24 orange juice samples, 6 from each of 4 brands, the pectin content, and measure of sweetness for each. Basic scatterplots: Use Graph > Scatterplot…

You can also add regression lines and use different plotting symbols for different groups. We show two examples here.

Simple: Specify the columns containing the y-axis and x-axis coordinates. Scatterplot of SweetIndex vs Pectin 6.0 5.9

SweetIndex

5.8 5.7 5.6 5.5 5.4 5.3 5.2 5.1 200

43

250

300 Pectin

350

400

Scatterplot (Minitab) With Regression and Groups: Specify the columns containing the y-axis and x-axis coordinates. Categorical variables for grouping: Specify the column containing the group indicators. Scatterplot of SweetIndex vs Pectin Brand A B C D

6.0 5.9

SweetIndex

5.8 5.7 5.6 5.5 5.4 5.3 5.2 5.1 200

250

300 Pectin

350

400

Scatterplot with some regression output: Use Stat > Regression > Fitted Line Plot… Response (Y): Specify the column containing the y-axis coordinates. Predictor (X): Specify the column containing the x-axis coordinates. Type of Regression Model: Specify the relationship between x and y; the default is Linear; you may also specify a Quadratic or Cubic model.

44

Scatterplot (Minitab) The graph output shows the regression line overlaid on the scatterplot as well as the estimated regresstion equation and some basic regression statistics. The session window shows these statistics as well as the regression ANOVA table.

Regression Analysis: SweetIndex versus Pectin

Fitted Line Plot

SweetIndex = 6.252 - 0.002311 Pectin

The regression equation is SweetIndex = 6.252 - 0.002311 Pectin

S R-Sq R-Sq(adj)

6.0 5.9

S = 0.214998

R-Sq = 22.9%

R-Sq(adj) = 19.4%

Analysis of Variance

SweetIndex

5.8 5.7 5.6 5.5 5.4 5.3

Source Regression Error Total

DF 1 22 23

SS 0.30140 1.01693 1.31833

MS 0.301402 0.046224

F 6.52

P 0.018

5.2 5.1 200

45

250

300 Pectin

350

400

0.214998 22.9% 19.4%

Scatterplot (Excel) The data must be entered in the worksheet with two columns for the numeric data and where the rows represent each observation. The column containing the x-axis coordinates must be first. Here we have 24 orange juice samples, the pectin content, and measure of sweetness for each. Highlight both columns of data and then choose Insert > Scatter. Axis labels should be added: with the graph window active, choose Layout > Axis Titles and select a format for the Horizontal and Vertical axes. You can also delete the legend and delete or change the chart title.

SweetIndex

A regression line can be overlaid on the scatterplot by rightclicking on one of the points and selecting Add Trendline… 6.1 6 5.9 5.8 5.7 5.6 5.5 5.4 5.3 5.2 5.1 0

100

200

300

400

Pectin

46

500

Binomial Distribution (Minitab) Use Graph > Probability Distribution Plot… > View Probability.

Distribution tab: Select the Binomial distribution from the dropdown box. Number of trials: Specify 𝑛. Event probability: Specify 𝑝.

47

Binomial Distribution (Minitab) Shaded Area tab: The following dialogs show how to specify several kinds of probability calculations. To input values of 𝑥 and compute probabilities, specify that you want to Define Shaded Area By X Value. Below are several examples where the random variable 𝑋 has a Binomial distribution with 𝑛 = 10 and 𝑝 = .3. 𝑃(𝑋 ≥ 4) = .3504 Distribution Plot

Binomial, n=10, p=0.3 0.30 0.25

Probability

0.20 0.15 0.10 0.05 0.00

0.3504

0

4 X

𝑃(𝑋 ≤ 4) = .8497 Distribution Plot

Binomial, n=10, p=0.3 0.30 0.25

0.8497

Probability

0.20 0.15 0.10 0.05 0.00

48

4 X

8

Binomial Distribution (Minitab) 𝑃(4 ≤ 𝑋 ≤ 6) = .3398 Distribution Plot

Binomial, n=10, p=0.3 0.30 0.25

Probability

0.20 0.15 0.3398

0.10 0.05 0.00

0

4 X

6

8

Note: To get individual probabilities of the form 𝑃(𝑋 = 𝑘), use the Middle option and specify the same values for x1 and x2. 𝑃(𝑋 = 4) = .2001 Distribution Plot

Binomial, n=10, p=0.3 0.30 0.25 0.2001

Probability

0.20 0.15 0.10 0.05 0.00

49

0

4 X

8

Binomial Distribution (Excel) In a blank cell, type an equal sign to insert a function and select More Functions… from the drop-down list of functions.

Select the BINOMDIST function and complete the dialog box. Number_s: Specify the value of 𝑘. Trials: Specify 𝑛. Probability_s: Specify 𝑝. Cumulative: Specify TRUE to compute 𝑃(𝑋 ≤ 𝑘) or FALSE to compute 𝑃(𝑋 = 𝑘). In this example we compute 𝑃(𝑋 ≤ 4) = .849731667 for 𝑛 = 10 and 𝑝 = .3. The result is shown in the dialog box as soon as you specify all four inputs.

50

Binomial Distribution (Excel)

Here we compute 𝑃(𝑋 = 4) = .200120949 by specifying FALSE for Cumulative.

51

Poisson Distribution (Minitab) Use Graph > Probability Distribution Plot… > View Probability.

Distribution tab: Select the Poisson distribution from the dropdown box. Mean: Specify 𝜆.

52

Poisson Distribution (Minitab) Shaded Area tab: The following dialogs show how to specify several kinds of probability calculations. To input values of 𝑥 and compute probabilities, specify that you want to Define Shaded Area By X Value. Below are several examples where the random variable 𝑋 has a Poisson distribution with 𝜆 = 3.8. 𝑃(𝑋 ≥ 6) = .1844 Distribution Plot Poisson, Mean=3.8

0.20

Probability

0.15

0.10

0.05 0.1844 0.00

0

X

6

𝑃(𝑋 ≤ 3) = .4735 Distribution Plot Poisson, Mean=3.8

0.20 0.4735

Probability

0.15

0.10

0.05

0.00

53

3

X

11

Poisson Distribution (Minitab) 𝑃(2 ≤ 𝑋 ≤ 7) = .8525 Distribution Plot Poisson, Mean=3.8

0.20

0.8525

Probability

0.15

0.10

0.05

0.00

0

2

X

7

11

Note: To get individual probabilities of the form 𝑃(𝑋 = 𝑘), use the Middle option and specify the same values for x1 and x2. 𝑃(𝑋 = 4) = .1944 Distribution Plot Poisson, Mean=3.8 0.1944

0.20

Probability

0.15

0.10

0.05

0.00

54

0

4

X

11

Poisson Distribution (Excel) In a blank cell, type an equal sign to insert a function and select More Functions… from the drop-down list of functions.

Select the POISSON function and complete the dialog box. X: Specify the value of 𝑘. Mean: Specify 𝜆. Cumulative: Specify TRUE to compute 𝑃(𝑋 ≤ 𝑘) or FALSE to compute 𝑃(𝑋 = 𝑘). In this example we compute 𝑃(𝑋 ≤ 4) = .667843601 for 𝜆 = 3.8. The result is shown in the dialog box as soon as you specify all three inputs.

55

Poisson Distribution (Excel) Here we compute 𝑃(𝑋 = 4) = .194358757 by specifying FALSE for Cumulative.

56

Normal Distribution (Minitab) Use Graph > Probability Distribution Plot… > View Probability.

Distribution tab: Select the Normal distribution from the drop-down box. Mean: Specify 𝜇. Standard deviation: Specify 𝜎. Note: The default normal distribution is the standard normal.

57

Normal Distribution (Minitab) Shaded Area tab: The following dialogs show how to specify several kinds of probability calculations. To input values of 𝑥 and compute probabilities, specify that you want to Define Shaded Area By X Value. Below are several examples where the random variable 𝑋 has a standard normal distribution. 𝑃(𝑋 ≥ 1.28) = .1003 Distribution Plot

Normal, Mean=0, StDev=1 0.4

Density

0.3

0.2

0.1 0.1003 0.0

0 X

1.28

𝑃(𝑋 ≤ 1.28) = .8997 Distribution Plot

Normal, Mean=0, StDev=1 0.4

Density

0.3

0.8997

0.2

0.1

0.0

58

0 X

1.28

Normal Distribution (Minitab) 𝑃(|𝑋| > 1.96) = 𝑃(𝑋 < −1.96) + 𝑃(𝑋 > 1.96) = .0250 + .0250 = .0500 Distribution Plot

Normal, Mean=0, StDev=1 0.4

Density

0.3

0.2

0.1 0.02500 0.0

0.02500 -1.96

0 X

1.96

𝑃(−2 < 𝑋 < 1) = .8186 Distribution Plot

Normal, Mean=0, StDev=1 0.4 0.8186

Density

0.3

0.2

0.1

0.0

59

-2

0 X

1

Normal Distribution (Minitab) Note: For continuous distributions, 𝑃(𝑋 = 𝑘) = 0 since there is no area under the curve at a point. 𝑃(𝑋 = 1.55) = 0 Distribution Plot

Normal, Mean=0, StDev=1 0.4

Density

0.3

0.2 0 0.1

0.0

0 X

1.55

To input probability values and compute 𝑥 values, specify that you want to Define Shaded Area By Probability. Find 𝑥0 such that 𝑃(𝑋 > 𝑥0 ) = .90. 𝑥0 = −1.282 Distribution Plot

Normal, Mean=0, StDev=1 0.4

Density

0.3

0.9

0.2

0.1

0.0

60

-1.282

0 X

Normal Distribution (Minitab) Find 𝑥0 such that 𝑃(𝑋 < 𝑥0 ) = .75.

𝑥0 = .6745 Distribution Plot

Normal, Mean=0, StDev=1 0.4

Density

0.3

0.75

0.2

0.1

0.0

Find 𝑥0 such that 𝑃(|𝑋| > 𝑥0 ) = .01.

0 X

0.6745

𝑥0 = 2.576 Distribution Plot

Normal, Mean=0, StDev=1 0.4

Density

0.3

0.2

0.1

0.0

61

0.005

0.005 -2.576

0 X

2.576

Normal Distribution (Excel) In a blank cell, type an equal sign to insert a function and select More Functions… from the drop-down list of functions.

Select the NORMDIST function and complete the dialog box. X: Specify the value of 𝑘. Mean: Specify 𝜇. Standard_dev: Specify 𝜎. Cumulative: Specify TRUE to compute 𝑃(𝑋 ≤ 𝑘). Specifying FALSE will compute 𝑓(𝑘), the height of the normal curve at 𝑋 = 𝑘. In this example we compute 𝑃(𝑋 ≤ −.55) = .291159687 for the standard normal distribution. The result is shown in the dialog box as soon as you specify all four inputs.

62

Normal Distribution (Excel) To input probability values and compute 𝑥 values, use the NORMINV function. Find 𝑥0 such that 𝑃(𝑋 ≤ 𝑥0 ) = .95.

𝑥0 = 1.644853627

Probability: Specify the cumulative probability, i.e. the area under the curve to the left of 𝑥0 . Mean: Specify 𝜇. Standard_dev: Specify 𝜎.

63

t Distribution (Minitab) Use Graph > Probability Distribution Plot… > View Probability.

Distribution tab: Select the t distribution from the drop-down box. Degrees of freedom: Specify the degrees of freedom.

64

t Distribution (Minitab) Shaded Area tab: The following dialogs show how to specify several kinds of probability calculations. To input values of 𝑥 (𝑡 values) and compute probabilities, specify that you want to Define Shaded Area By X Value. Below are several examples where the random variable 𝑋 has a t distribution with 12 degrees of freedom. 𝑃(𝑡 > 2.179) = .02499 Distribution Plot T, df=12

0.4

Density

0.3

0.2

0.1

0.02499 0.0

0 X

2.179

𝑃(𝑡 ≤ 3.055) = .9950 Distribution Plot T, df=12

0.4

0.9950

Density

0.3

0.2

0.1

0.0

65

0 X

3.055

t Distribution (Minitab) 𝑃(|𝑡| > 1.356) = .1000 Distribution Plot T, df=12

0.4

Density

0.3

0.2

0.1 0.1000 0.0

0.1000 -1.356

0 X

1.356

𝑃(1.5 ≤ 𝑡 ≤ 2.5) = .06577 Distribution Plot T, df=12

0.4

Density

0.3

0.2

0.1 0.06577 0.0

66

0 X

1.5

2.5

t Distribution (Minitab) To input probability values and compute 𝑥 values, specify that you want to Define Shaded Area By Probability. Find 𝑡0 such that 𝑃(𝑡 ≥ 𝑡0 ) = .05.

𝑡0 = 1.782 Distribution Plot T, df=12

0.4

Density

0.3

0.2

0.1 0.05 0.0

Find 𝑡0 such that 𝑃(|𝑡| > 𝑡0 ) = .01.

0 X

1.782

𝑡0 = 3.055 Distribution Plot T, df=12

0.4

Density

0.3

0.2

0.1

0.0

67

0.005 -3.055

0.005 0 X

3.055

t Distribution (Excel) In a blank cell, type an equal sign to insert a function and select More Functions… from the drop-down list of functions. There are several functions you can use depending on what you want to do. In the following examples we'll use a t-distribution with 6 degrees of freedom.

Finding lower-tailed probabilities:

Select the T.DIST function and complete the dialog box. X: Specify the 𝑡 value. Deg_freedom: Specify the degrees of freedom. Cumulative: Specify TRUE to compute 𝑃(𝑡 ≤ −1.76). Specifying FALSE will compute 𝑓(−1.76), the height of the t curve at −1.76. In this example we compute 𝑃(𝑡 ≤ −1.76) = .064447607. The result is shown in the dialog box as soon as you specify all three inputs.

68

t Distribution (Excel) Finding upper-tailed probabilities:

Select the T.DIST.RT function and complete the dialog box. X: Specify the 𝑡 value. Deg_freedom: Specify the degrees of freedom.

In this example we compute 𝑃(𝑡 > 2.73) = .017093009. The result is shown in the dialog box as soon as you specify both inputs.

Finding two-tailed probabilities: Select the T.DIST.2T function and complete the dialog box. X: Specify the 𝑡 value. This must be a positive value. Deg_freedom: Specify the degrees of freedom.

In this example we compute 𝑃(|𝑡| > 1.56) = .169778183. The result is shown in the dialog box as soon as you specify both inputs.

69

t Distribution (Excel) To input probability values and compute 𝑡 values, use these functions: Find 𝑡0 such that 𝑃(𝑡 ≤ 𝑡0 ) = .95.

𝑡0 = 1.943180281

Select the T.INV function and complete the dialog box. Probability: Specify the cumulative probability, i.e. the area under the curve to the left of 𝑡0 . Deg_freedom: Specify the degrees of freedom.

Find 𝑡0 such that 𝑃(|𝑡| > 𝑡0 ) = .05. 𝑡0 = 2.446911851 Probability: Specify the total tail probability, i.e. the area under the curve to the left of −𝑡0 plus the area to the right of 𝑡0 . Deg_freedom: Specify the degrees of freedom.

70

Chi-Square Distribution (Minitab) Use Graph > Probability Distribution Plot… > View Probability.

Distribution tab: Select the Chi-Square distribution from the drop-down box. Degrees of freedom: Specify the degrees of freedom.

71

Chi-Square Distribution (Minitab) Shaded Area tab: The following dialogs show how to specify several kinds of probability calculations. To input values of 𝑥 2 and compute probabilities, specify that you want to Define Shaded Area By X Value. Below are several examples where the random variable 𝑋 2 has a Chi-Square distribution with 9 degrees of freedom. 𝑃(𝑋 2 > 1.735) = .995 Distribution Plot Chi-Square, df=9

0.10

Density

0.08

0.06 0.9950

0.04

0.02

0.00

0 1.735

X

𝑃(𝑋 2 ≤ 1.735) = .005 Distribution Plot Chi-Square, df=9

0.10

Density

0.08

0.06

0.04

0.02 0.005001 0.00 0 1.735

72

X

Chi-Square Distribution (Minitab) To input probability values and compute 𝑥 2 values, specify that you want to Define Shaded Area By Probability. Find 𝑥02 such that 𝑃(𝑋 2 > 𝑥02 ) = .05.

𝑥02 = 16.92 Distribution Plot Chi-Square, df=9

0.10

Density

0.08

0.06

0.04

0.02 0.05 0.00

0

X

Find 𝑥12 and 𝑥22 such that 𝑃(𝑋 2 < 𝑥12 𝑜𝑟 𝑋 2 > 𝑥22 ) = .05.

16.92

𝑥12 = 2.700 𝑥22 = 19.02 Distribution Plot Chi-Square, df=9

0.10

Density

0.08

0.06

0.04

0.02

0.00

73

0.025 0.025 0

2.700

X

19.02

Chi-Square Distribution (Excel) In a blank cell, type an equal sign to insert a function and select More Functions… from the drop-down list of functions.

There are several functions you can use depending on what you want to do. In the following examples we'll use a Chi-Square distribution with 9 degrees of freedom.

Finding lower-tailed probabilities:

Select the CHISQ.DIST function and complete the dialog box. X: Specify the 𝑥 2 value. Deg_freedom: Specify the degrees of freedom. Cumulative: Specify TRUE to compute 𝑃(𝑋 2 ≤ 1.6837). Specifying FALSE will compute 𝑓(14.6837), the height of the chi-square curve at 14.6837. In this example we compute 𝑃(𝑋 2 ≤ 14.6837) = .900001297. The result is shown in the dialog box as soon as you specify all three inputs.

74

Chi-Square Distribution (Excel) Finding upper-tailed probabilities:

Select the CHISQ.DIST.RT function and complete the dialog box. X: Specify the 𝑥 2 value. Deg_freedom: Specify the degrees of freedom.

In this example we compute 𝑃(𝑋 2 > 3.325) = .950005451. The result is shown in the dialog box as soon as you specify both inputs.

To input probability values and compute 𝑥 2 values, use these functions: Find 𝑥02 such that 𝑃(𝑋 2 ≤ 𝑥02 ) = .90.

𝑥02 = 14.68365657

Select the CHISQ.INV function and complete the dialog box. Probability: Specify the cumulative probability, i.e. the area under the curve to the left of 𝑥02 . Deg_freedom: Specify the degrees of freedom.

75

Chi-Square Distribution (Excel) Find 𝑥02 such that 𝑃(𝑋 2 > 𝑥02 ) = .05.

𝑥02 = 16.9189776

Select the CHISQ.INV.RT function and complete the dialog box. Probability: Specify the upper-tailed probability, i.e. the area under the curve to the right of 𝑥02 . Deg_freedom: Specify the degrees of freedom.

76

F Distribution (Minitab) Use Graph > Probability Distribution Plot… > View Probability.

Distribution tab: Select the F distribution from the drop-down box. Numerator df: Specify the numerator degrees of freedom. Denominator df: Specify the denominator degrees of freedom.

77

F Distribution (Minitab) Shaded Area tab: The following dialogs show how to specify several kinds of probability calculations. To input values of 𝑓 and compute probabilities, specify that you want to Define Shaded Area By X Value. Below are several examples where the random variable 𝐹 has an F distribution with 4 numerator degrees of freedom and 10 denominator degrees of freedom. 𝑃(𝐹 > 3.48) = .04993 Distribution Plot F, df1=4, df2=10

0.7 0.6

Density

0.5 0.4 0.3 0.2 0.1 0.0

78

0.04993 0

X

3.48

F Distribution (Minitab) To input probability values and compute 𝑓 values, specify that you want to Define Shaded Area By Probability. Find 𝑓0 such that 𝑃(𝐹 > 𝑓0 ) = .025.

𝑓0 = 4.468 Distribution Plot F, df1=4, df2=10

0.7 0.6

Density

0.5 0.4 0.3 0.2 0.1 0.0

79

0.025 0

X

4.468

F Distribution (Excel) In a blank cell, type an equal sign to insert a function and select More Functions… from the drop-down list of functions.

There are several functions you can use depending on what you want to do. In the following examples we'll use an F distribution with 4 numerator degrees of freedom and 10 denominator degrees of freedom.

Finding upper-tailed probabilities:

Select the F.DIST.RT function and complete the dialog box. X: Specify the 𝑓 value. Deg_freedom1: Specify the numerator degrees of freedom. Deg_freedom2: Specify the denominator degrees of freedom. In this example we compute 𝑃(𝐹 > 5.99) = .010023913. The result is shown in the dialog box as soon as you specify all three inputs.

80

F Distribution (Excel) To input probability values and compute 𝑓 values, use these functions: Find 𝑓0 such that 𝑃(𝐹 > 𝑓0 ) = .10.

𝑓0 = 2.605336431

Select the F.INV.RT function and complete the dialog box. Probability: Specify the upper-tailed probability, i.e. the area under the curve to the right of 𝑓0 . Deg_freedom1: Specify the numerator degrees of freedom. Deg_freedom2: Specify the denominator degrees of freedom.

81

z-interval and z-test for 𝝁 (Minitab) Use Stat > Basic Statistics > 1-Sample Z… As an example, we use body temperature data collected on a sample of healthy adults.

We'll assume that 𝜎 = .75. Samples in columns: Specify the column containing the raw data. Summarized data: Use this option if you have only the sample size (𝑛) and the sample mean (𝑥̅ ) available. Standard deviation: Enter 𝜎. If you want to do a z-test with 𝜎 unknown (i.e. for large sample sizes), find the sample standard deviation (𝑠) first and enter it here. If you want to perform a hypotheses test, check the box and enter 𝜇0 as the Hypothesized mean. By default, Minitab calculates only a confidence interval for 𝜇. 82

z-interval and z-test for 𝝁 (Minitab) To change the interval/test defaults, use Options… Confidence level: Specify a different confidence level for the interval. The default is 95%. Alternative: Specify a different direction for 𝐻𝑎 . The default is not equal (≠) for a two-tailed test. Leave this as not equal to obtain the usual "two-tailed" confidence interval. Changing this option will provide one-sided confidence intervals. The output shows the hypotheses, 𝐻0 : 𝜇 = 98.6 𝐻𝑎 : 𝜇 ≠ 98.6 indicates that the z-test statistic and confidence interval are computed using 𝜎 = .75, and displays the endpoints of the confidence interval and the hypothesis test results (z-test statistic and pvalue), in addition to some descriptive statistics.

83

t-test and t-interval for 𝝁 (Minitab) Use Stat > Basic Statistics > 1-Sample t… As an example, we use earnings per share figures (EPS) collected on a sample of publically-traded companies.

Samples in columns: Specify the column containing the raw data. Summarized data: Use this option if you only have the sample size (𝑛), the sample mean (𝑥̅ ), and the sample standard deviation (𝑠) available. If you want to perform a hypotheses test, check the box and enter 𝜇0 as the Hypothesized mean. By default, Minitab calculates only a confidence interval for 𝜇.

84

t-test and t-interval for 𝝁 (Minitab) To change the interval/test defaults, use Options… Confidence level: Specify a different confidence level for the interval. The default is 95%. Alternative: Specify a different direction for 𝐻𝑎 . The default is not equal (≠) for a two-tailed test. Leave this as not equal to obtain the usual "two-tailed" confidence interval. Changing this option will provide one-sided confidence intervals. The output shows the hypotheses, 𝐻0 : 𝜇 = 4 𝐻𝑎 : 𝜇 < 4 displays the hypothesis test results (t-test statistic and pvalue), and the upper bound of a one-sided confidence interval (since the alternative hypothesis is specified as one-tailed), in addition to some descriptive statistics.

85

t-interval for 𝝁 (Excel) Excel does not output results of a hypothesis test for 𝜇 but you can use it to construct a confidence interval using the t procedure. Enter the raw data in a column of the worksheet. Use Data > Data Analysis > Descriptive Statistics.

Input Range: Highlight the cells containing the data. Grouped By: Columns If the Input Range contains more than 1 column of data, Excel does a separate analysis for each one. Labels in First Row: Check this box if the Input Range contains column labels. If not, don't check the box. Checking the box tells Excel to ignore what's in the first row of the Input Range. Summary Statistics: Check this box for the descriptive statistics to display. Confidence Level for Mean: Check this box and specify the desired confidence level (95% is the default).

86

t-interval for 𝝁 (Excel) In addition to the univariate descriptive statistics, Excel displays the margin of error for the confidence interval calculated as 𝑡𝛼⁄2

𝑠 √𝑛

.

The confidence interval is then 𝑥̅ ± 𝑡𝛼⁄2

𝑠

.

√𝑛

87

Interval and Test for 𝒑 (Minitab) Use Stat > Basic Statistics > 1 Proportion… As an example, we use the results of a set of coin tosses where 9 were heads and 21 were tails.

Samples in columns: Specify the column containing the raw data, i.e. a column of "success"/"failure" indicators. Summarized data: Use this input option if you only know the Number of events ("successes") and Number of trials (𝑛). If you want to perform a hypotheses test, check the box and enter 𝑝0 as the Hypothesized proportion. By default, Minitab calculates only a confidence interval for 𝑝.

88

Interval and Test for 𝒑 (Minitab) To change the interval/test defaults, use Options… Confidence level: Specify a different confidence level for the interval. The default is 95%. Alternative: Specify a different direction for 𝐻𝑎 . The default is not equal (≠) for a two-tailed test. Leave this as not equal to obtain the usual "two-tailed" confidence interval. Changing this option will provide one-sided confidence intervals. Use test and interval based on normal distribution: Check this box to use the zinterval and z-test procedures if the sample size requirements are met. If not, leave the box unchecked to use more robust procedures based on the binomial distribution. The output shows the hypotheses, 𝐻0 : 𝑝 = .5 𝐻𝑎 : 𝑝 ≠ .5 displays the endpoints of the confidence interval and the hypothesis test results (z-test statistic and pvalue), in addition to the sample counts and sample proportion of successes.

89

Interval and Test for 𝝈𝟐 (Minitab) Use Stat > Basic Statistics > 1 Variance… As an example, we use earnings per share figures (EPS) collected on a sample of publically-traded companies.

Data: Choose the input option. Samples in columns: Specify this option if the raw data are in a column of the worksheet. Sample standard deviation/variance: Specify this option if you have only summary statistics available for the sample data. If you want to perform a hypotheses test, check the box and enter 𝜎02 as the Value if you choose Hypothesized variance. Enter 𝜎0 as the Value if you choose Hypothesized standard deviation. By default, Minitab calculates only a confidence interval for 𝜎 and 𝜎 2

90

Interval and Test for 𝝈𝟐 (Minitab) To change the interval/test defaults, use Options… Confidence level: Specify a different confidence level for the interval. The default is 95%. Alternative: Specify a different direction for 𝐻𝑎 . The default is not equal (≠) for a two-tailed test. Leave this as not equal to obtain the usual "twotailed" confidence interval. Changing this option will provide one-sided confidence intervals. Test and CI for One Variance: EPS

The output shows the hypotheses, 𝐻0 : 𝜎 2 = 9 𝐻𝑎 : 𝜎 2 ≠ 9 displays the endpoints of the confidence intervals for both 𝜎 and 𝜎 2 , and the hypothesis test results, in addition to summary statistics of the sample data. Note that two different methods are used depending on the assumption you want to make about the shape of the population distribution from which the data are sampled.

Method Null hypothesis Alternative hypothesis

Sigma-squared = 9 Sigma-squared not = 9

The chi-square method is only for the normal distribution. The Bonett method is for any continuous distribution. Statistics Variable EPS

N 5

StDev 1.41

Variance 1.99

95% Confidence Intervals Variable EPS

Method Chi-Square Bonett

CI for StDev (0.85, 4.06) (0.86, 3.80)

Method Chi-Square Bonett

Test Statistic 0.89 —

CI for Variance (0.72, 16.46) (0.75, 14.42)

Tests Variable EPS

DF 4 —

P-Value 0.147 0.103

Since the Chi-square procedures are sensitive to the normal distribution assumption, it's a good idea to the data are closely normally distributed (e.g. with a normal probability plot) before using them.

91

t-Interval and t-Test for 𝝁𝟏 − 𝝁𝟐 Using Independent Samples (Minitab) Use Stat > Basic Statistics > 2-Sample t… Our example is data collected on the length of cuts (in feet) of columns from two different saws (A and B). We're interested in whether these data show enough evidence that one saw is cutting the columns shorter, on average, than the other saw. The data are shown in the worksheet in both unstacked format (C1 and C2) and stacked format (C3 and C4-T).

Samples in one column: Use this option for stacked data. Specify the column containing the numeric response/comparison variable (Samples) and the column containing the grouping variable (Subscripts). Note: When you use this option, Minitab uses the group which comes first alphanumerically in the Subscripts column as the first term in the difference in means. Here it would be 𝜇𝐴 − 𝜇𝐵 . 92

t-Interval and t-Test for 𝝁𝟏 − 𝝁𝟐 Using Independent Samples (Minitab) Samples in different columns: Use this option for unstacked data. Specify the both columns of numeric values. Note: When you use this option, the group whose data is specified as the First column is used as the first term in the difference in means. Here it would be 𝜇𝐵 − 𝜇𝐴 . Summarized data: Use this option if you have only the sample sizes and descriptive statistics available for each sample. Assume equal variances: Check this box if you want to assume equal population variances, i.e. 𝜎12 = 𝜎22 . The default is to not assume equal variances. The differences in the tests are shown below. Assumption 𝜎12 = 𝜎22

𝜎12 ≠ 𝜎22

Test Statistic 𝑡=

(𝑥̅ 1 −𝑥̅ 2 )−𝐷0

𝑡=

(𝑥̅ 1 −𝑥̅2 )−𝐷0

2( 1 + 1 ) √𝑠𝑝 𝑛1 𝑛2

2

Degrees of Freedom where 𝑠𝑝2 =

(𝑛1 −1 )𝑠12 +(𝑛2 −1 )𝑠22 𝑛1 +𝑛2 −2

𝑛1 + 𝑛2 − 2 2

2 2

𝑠 𝑠 ( 1+ 2)

2

𝑠 𝑠 √ 1+ 2

𝑛1 𝑛2

⌊

𝑛1 𝑛2 2 2 2 𝑠2 𝑠 ( 1) ( 2) 𝑛1 𝑛2 + 𝑛1 −1 𝑛2 −1

⌋

To change the interval/test defaults, use Options… Confidence level: Specify a different confidence level for the interval. The default is 95%. Test difference: Specify the value of 𝐷0 . The default is 0. Alternative: Specify a different direction for 𝐻𝑎 . The default is not equal (≠) for a twotailed test. Leave this as not equal to obtain the usual "two-tailed" confidence interval. Changing this option will provide one-sided confidence intervals.

93

t-Interval and t-Test for 𝝁𝟏 − 𝝁𝟐 Using Independent Samples (Minitab) The output shows descriptive statistics for both groups. In addition, the difference in means on which we're making inference is shown (𝜇𝐴 − 𝜇𝐵 ) as well as the point estimate for the difference (𝑥̅1 − 𝑥̅2 ) based on the data. The two last lines display the results of the inference procedures: the endpoints of the confidence interval for 𝜇𝐴 − 𝜇𝐵 and the results of the test of 𝐻0 : 𝜇𝐴 − 𝜇𝐵 = 0 𝐻𝑎 : 𝜇𝐴 − 𝜇𝐵 ≠ 0 . Two-Sample T-Test and CI: Length, Saw Two-sample T for Length Saw A B

N 9 9

Mean 8.0489 8.0700

StDev 0.0372 0.0224

SE Mean 0.012 0.0075

Difference = mu (A) - mu (B) Estimate for difference: -0.0211 95% CI for difference: (-0.0524, 0.0102) T-Test of difference = 0 (vs not =): T-Value = -1.46

94

P-Value = 0.168

DF = 13

t-Test for 𝝁𝟏 − 𝝁𝟐 Using Independent Samples (Excel) Use Data > Data Analysis > t-Test: Two-Sample Assuming Equal Variances or Data > Data Analysis > t-Test: Two-Sample Assuming Unequal Variances depending on whether you want to assume equal population variances (𝜎12 = 𝜎22 ) or not. The differences in the tests are shown below. Assumption 𝜎12 = 𝜎22

𝜎12 ≠ 𝜎22

Test Statistic 𝑡=

𝑡=

(𝑥̅ 1 −𝑥̅ 2 )−𝐷0 2( 1 + 1 ) √𝑠𝑝 𝑛1 𝑛2

Degrees of Freedom where 𝑠𝑝2 =

(𝑛1 −1 )𝑠12 +(𝑛2 −1 )𝑠22 𝑛1 +𝑛2 −2

𝑛1 + 𝑛2 − 2 2

2 2

𝑠 𝑠 ( 1+ 2)

(𝑥̅ 1 −𝑥̅2 )−𝐷0 𝑠2 𝑠2 √ 1+ 2 𝑛1 𝑛2

⌊

𝑛1 𝑛2 2 2 2 𝑠2 𝑠 ( 1) ( 2) 𝑛1 𝑛2 + 𝑛1 −1 𝑛2 −1

⌋

Our example is data collected on the length of cuts (in feet) of columns from two different saws (A and B). We're interested in whether these data show enough evidence that one saw is cutting the columns shorter, on average, than the other saw. The data are shown in the worksheet in unstacked format.

95

t-Test for 𝝁𝟏 − 𝝁𝟐 Using Independent Samples (Excel) Variable 1 Range: Highlight the cells containing the numeric responses for the first group. Variable 2 Range: Highlight the cells containing the numeric responses for the second group. Note: The group whose data is specified as the Variable 1 Range is used as the first term in the difference in means. Here it would be 𝜇𝐴 − 𝜇𝐵 . Hypothesized Mean Difference: Specify 𝐷0 . Labels: Check this box if the Input Ranges contain column labels. If not, don't check the box. Checking the box tells Excel to ignore what's in the first row of the Input Ranges. Alpha: Specify 𝛼 for the test. The output contains descriptive statistics for both groups and results of the hypothesis test of 𝐻0 : 𝜇𝐴 − 𝜇𝐵 = 0 . Both one-tail and two-tail p-values and critical values are reported.

t-Test: Two-Sample Assuming Unequal Variances

Mean Variance Observations Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail

96

A B 8.048889 8.07 0.001386 0.0005 9 9 0 13 -1.45831 0.084245 1.770933 0.16849 2.160369

t-Test for 𝝁𝟏 − 𝝁𝟐 Using Independent Samples (Excel) IMPORTANT NOTE: Excel does not report one-tailed p-values and critical values correctly, in general. The one-tailed “p-value” reported is actually the area under the tcurve in the upper or lower tail, depending on whether the value of the test statistic “t Stat” is positive or negative, respectively. Excel mistakenly assumes that “t Stat” will be positive if you have an upper-tailed test and negative if you have a lower-tailed test, which usually happens, but not always. It is poor statistical practice to base the direction of the hypotheses on the data. Also, the one-tailed “t Critical” reported is actually the absolute value of the critical value, so it could have the wrong sign depending on what the alternative hypothesis is. Two-tailed p-values are reported correctly but there is a second critical value in the lower tail having the same absolute value with a negative sign. The following chart shows the correct rejection regions and p-values for the corresponding tests. The highlighted values are the correct values. Alternative Hypothesis 𝐻𝑎 : 𝜇𝐴 − 𝜇𝐵 < 0 𝐻𝑎 : 𝜇𝐴 − 𝜇𝐵 > 0 𝐻𝑎 : 𝜇𝐴 − 𝜇𝐵 ≠ 0

Rejection Region 𝑡 < −1.770933 𝑡 > 1.770933 |𝑡| > 2.160369

p-value 𝑃(𝑇 < −1.45831) = .084245 𝑃(𝑇 > −1.45831) = 1 − .084245 = .915755 𝑃(|𝑇| > |−1.45831|) = .16849

97

t-Interval and t-Test for 𝝁𝟏 − 𝝁𝟐 Using Paired Samples (Minitab) Use Stat > Basic Statistics > Paired t… The example data are distances (yards) a golf ball was driven off a tee for a sample of golfers. Each golfer hit two balls, one of brand A and one of brand B. The data are shown in unstacked format.

Samples in columns: Use this option for unstacked data. Specify the both columns of numeric values. Note: When you use this option, the group whose data is specified as the First sample is used as the first term in the difference in means. Here it would be 𝜇𝐴 − 𝜇𝐵 . Summarized data (differences): Use this option if you have only the sample sizes and descriptive statistics available for the paired differences.

98

t-Interval and t-Test for 𝝁𝟏 − 𝝁𝟐 Using Paired Samples (Minitab) To change the interval/test defaults, use Options… Confidence level: Specify a different confidence level for the interval. The default is 95%. Test difference: Specify the value of 𝐷0 . The default is 0. Alternative: Specify a different direction for 𝐻𝑎 . The default is not equal (≠) for a twotailed test. Leave this as not equal to obtain the usual "two-tailed" confidence interval. Changing this option will provide one-sided confidence intervals. The output shows descriptive statistics for both groups and the paired differences. The two last lines display the results of the inference procedures: the endpoints of the confidence interval for 𝜇𝐴 − 𝜇𝐵 and the results of the test of 𝐻0 : 𝜇𝐴 − 𝜇𝐵 = 0 𝐻𝑎 : 𝜇𝐴 − 𝜇𝐵 ≠ 0 .

Paired T-Test and CI: A, B Paired T for A - B A B Difference

N 17 17 17

Mean 256.18 251.41 4.76

StDev 18.70 14.93 9.09

SE Mean 4.54 3.62 2.20

95% CI for mean difference: (0.09, 9.44) T-Test of mean difference = 0 (vs not = 0): T-Value = 2.16

P-Value = 0.046

Note: An alternative is to use Stat > Basic Statistics > 1-Sample Z… or Stat > Basic Statistics > 1-Sample t… and use the paired differences as the input data.

99

t-Test for 𝝁𝟏 − 𝝁𝟐 Using Paired Samples (Excel) Use Data > Data Analysis > t-test: Paired Two-Sample for Means. The example data are distances (yards) a golf ball was driven off a tee for a sample of golfers. Each golfer hit two balls, one of brand A and one of brand B. The data are shown in unstacked format.

Variable 1 Range: Highlight the cells containing the numeric responses for the first group. Variable 2 Range: Highlight the cells containing the numeric responses for the second group. Note: The group whose data is specified as the Variable 1 Range is used as the first term in the difference in means. Here it would be 𝜇𝐴 − 𝜇𝐵 . Hypothesized Mean Difference: Specify 𝐷0 . Labels: Check this box if the Input Ranges contain column labels. If not, don't check the box. Checking the box tells Excel to ignore what's in the first row of the Input Ranges. Alpha: Specify 𝛼 for the test.

100

t-Test for 𝝁𝟏 − 𝝁𝟐 Using Paired Samples (Excel) The output contains descriptive statistics for both groups and the Pearson correlation coefficient (𝑟) between the two samples. Also shown are the results of the hypothesis test of 𝐻0 : 𝜇𝐴 − 𝜇𝐵 = 0 . Both one-tail and two-tail p-values and critical values are reported.

t-Test: Paired Two Sample for Means

Mean Variance Observations Pearson Correlation Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail

A B 256.1765 251.4118 349.7794 223.0074 17 17 0.877617 0 16 2.162019 0.023056 1.745884 0.046112 2.119905

IMPORTANT NOTE: Excel does not report one-tailed p-values and critical values correctly, in general. The one-tailed “p-value” reported is actually the area under the t-curve in the upper or lower tail, depending on whether the value of the test statistic “t Stat” is positive or negative, respectively. Excel mistakenly assumes that “t Stat” will be positive if you have an upper-tailed test and negative if you have a lowertailed test, which usually happens, but not always. It is poor statistical practice to base the direction of the hypotheses on the data. Also, the one-tailed “t Critical” reported is actually the absolute value of the critical value, so it could have the wrong sign depending on what the alternative hypothesis is. Two-tailed p-values are reported correctly but there is a second critical value in the lower tail having the same absolute value with a negative sign. The following chart shows the correct rejection regions and p-values for the corresponding tests. The highlighted values are the correct values. Alternative Hypothesis 𝐻𝑎 : 𝜇𝐴 − 𝜇𝐵 < 0 𝐻𝑎 : 𝜇𝐴 − 𝜇𝐵 > 0 𝐻𝑎 : 𝜇𝐴 − 𝜇𝐵 ≠ 0

Rejection Region 𝑡 < −1.745884 𝑡 > 1.745884 |𝑡| > 2.119905

p-value 𝑃(𝑇 < 2.162019) = 1 − .023056 = .976944 𝑃(𝑇 > 2.162019) = .023056 𝑃(|𝑇| > |2.162019|) = .046112

Note: An alternative is to use Data > Data Analysis > Descriptive Statistics and use the paired differences as the input data. This approach will only output the margin of error for a paired t-interval for 𝜇1 − 𝜇2 . 101

z-Interval and z-Test for 𝒑𝟏 − 𝒑𝟐 Using Independent Samples (Minitab) Use Stat > Basic Statistics > 2 Proportions… The example data are indicators of whether or not a washing machine needed repairs in the first five years of operation for a sample of the company's machines and a sample of a competitor's machines. The data are shown in both unstacked format (C1-T and C2T) and stacked format (C3-T and C4-T). 13 of the 100 company's machines and 26 of the 100 competitor's machines needed repairs.

Samples in one column: Use this option for stacked data. Specify the column containing the categorical response/comparison variable (Samples) and the column containing the grouping variable (Subscripts). Note: When you use this option, Minitab uses the group which comes first alphanumerically in the Subscripts column as the first term in the difference in proportions. Here it would be 𝑝𝐶𝑜𝑚𝑝𝑎𝑛𝑦 − 𝑝𝐶𝑜𝑚𝑝𝑒𝑡𝑖𝑡𝑜𝑟 .

102

z-Interval and z-Test for 𝒑𝟏 − 𝒑𝟐 Using Independent Samples (Minitab) Samples in different columns: Use this option for unstacked data. Specify the both columns of categorical responses. Note: When you use this option, the group whose data is specified as the First column is used as the first term in the difference in proportions. Here it would be 𝑝𝐶𝑜𝑚𝑝𝑎𝑛𝑦 − 𝑝𝐶𝑜𝑚𝑝𝑒𝑡𝑖𝑡𝑜𝑟 . Summarized data: Use this option if you have only the sample sizes (Trials) and numbers of "successes" (Events) for each sample. To change the interval/test defaults, use Options… Confidence level: Specify a different confidence level for the interval. The default is 95%. Test difference: Specify the value of 𝐷0 . The default is 0. Alternative: Specify a different direction for 𝐻𝑎 . The default is not equal (≠) for a twotailed test. Leave this as not equal to obtain the usual "two-tailed" confidence interval. Changing this option will provide one-sided confidence intervals. Use pooled estimate of p for test: Check this box if you want Minitab to compute the z-test statistic as 𝑧=

(𝑝̂1 −𝑝̂2 )−0 1 1 √𝑝̂(1−𝑝̂)(𝑛 +𝑛 ) 1 2

𝑥 +𝑥

where 𝑝̂ = 𝑛1+𝑛2 is the pooled estimate of 𝑝 and 𝑥1 and 𝑥2 are the 1

2

numbers of "successes" (Events) in each group. This is only appropriate under the assumption that 𝑝1 = 𝑝2 = 𝑝, i.e. 𝑝1 − 𝑝2 = 0, so only check this box if you specify 𝐷0 = 0. If you do not check the box, Minitab computes the z-test statistic as 𝑧=

(𝑝̂1 −𝑝̂2 )−𝐷0 ̂ (1−𝑝 ̂1) 𝑝 ̂ (1−𝑝 ̂2) 𝑝 √ 1 + 2 𝑛1 𝑛2

.

Checking or not checking the box has no effect on the confidence interval for 𝑝1 − 𝑝2 .

103

z-Interval and z-Test for 𝒑𝟏 − 𝒑𝟐 Using Independent Samples (Minitab) The output shows the numbers of events (X), the sample sizes (N), and sample proportions (Sample p) for both samples. In addition, the difference in proportions on which we're making inference is shown (𝑝1 − 𝑝2) as well as the point estimate for the difference (𝑝̂1 − 𝑝̂2) based on the data. The next two lines display the results of the z-procedures: the endpoints of the confidence interval for 𝑝1 − 𝑝2 and the results of the test of 𝐻0 : 𝑝1 − 𝑝2 = 0 𝐻𝑎 : 𝑝1 − 𝑝2 ≠ 0 . The last line shows the p-value for Fisher's exact test of hypotheses. Use this test when the large samples condition (required for the validity of the z-procedures) fails.

Test and CI for Two Proportions Sample 1 2

X 13 26

N 100 100

Sample p 0.130000 0.260000

Difference = p (1) - p (2) Estimate for difference: -0.13 95% CI for difference: (-0.238331, -0.0216688) Test for difference = 0 (vs not = 0): Z = -2.32 Fisher's exact test: P-Value = 0.031

104

P-Value = 0.020

Test and Interval for 𝝈𝟐𝟏 /𝝈𝟐𝟐 Using Independent Samples (Minitab) Use Stat > Basic Statistics > 2 Variances… Our example is data collected on the length of cuts (in feet) of columns from two different saws (A and B). We're interested in whether these data show enough evidence that the saws differ on the variation in the lengths of cuts. The data are shown in the worksheet in both unstacked format (C1 and C2) and stacked format (C3 and C4T).

Data: Specify the input option. Samples in one column: Use this option for stacked data format. Specify the numeric response/comparison variable (Samples) and the categorical grouping variable (Subscripts). Note: When you use this option, Minitab uses the group which comes first alphanumerically in the Subscripts column as the numerator term in the ratio of variances. Here it would be 𝜎𝐴2 /𝜎𝐵2 . 105

Test and Interval for 𝝈𝟐𝟏 /𝝈𝟐𝟐 Using Independent Samples (Minitab) Samples in different columns: Use this option for unstacked data. Specify both columns of numeric responses. Note: When you use this option, the group whose data is specified as the First column is used as the numerator term in the ratio of variances. Here it would be 𝜎𝐴2 /𝜎𝐵2 . Sample standard deviations/variances: Use these options if you have only descriptive statistics available for both groups. The group whose summaries are specified as the First sample is used as the numerator term in the ratio of variances.

To change the interval/test defaults, use Options… Confidence level: Specify a different confidence level for the interval. The default is 95%. Hypothesized ratio: Specify the Value of the null hypothesized ratio of standard deviations (default) or variances. The default Value is 1, i.e. 𝐻0 :

𝜎1 𝜎2

= 1 or 𝐻0 :

𝜎12 𝜎22

= 1.

Alternative: Specify a different direction for 𝐻𝑎 . The default is not equal (≠) for a twotailed test. Leave this as not equal to obtain the usual "two-tailed" confidence interval. Changing this option will provide one-sided confidence intervals. 106

Test and Interval for 𝝈𝟐𝟏 /𝝈𝟐𝟐 Using Independent Samples (Minitab) The output shows the hypotheses, 𝐻0 :

2 𝜎𝐴

=1

2 𝜎𝐵

𝐻𝑎 :

2 𝜎𝐴

2 𝜎𝐵

≠1

displays descriptive statistics for each group, and point estimates of the population ratios of standard deviations and variances: point estimate of 𝜎12 𝜎22

𝜎12 𝜎22

𝑠1 𝑠2

𝜎

𝑠2

as the point estimate of 𝜎1 and 𝑠12 as the 2

2

. Next are the endpoints of the confidence intervals for both

𝜎1 𝜎2

and

, and the hypothesis test results.

Note that two different methods are used depending on the assumption you want to make about the shapes of the two population distributions from which the data are sampled. Since the F procedures are sensitive to the normal distribution assumption, it's a good idea to that both samples of data are closely normally distributed (e.g. with a normal probability plot) before using them. Test and CI for Two Variances: Length vs Saw Method Null hypothesis Alternative hypothesis Significance level

Variance(A) / Variance(B) = 1 Variance(A) / Variance(B) not = 1 Alpha = 0.05

Statistics Saw A B

N 9 9

StDev 0.037 0.022

Variance 0.001 0.001

Ratio of standard deviations = 1.665 Ratio of variances = 2.772 95% Confidence Intervals Distribution of Data Normal Continuous

CI for StDev Ratio (0.791, 3.506) (0.446, 3.441)

CI for Variance Ratio (0.625, 12.290) (0.199, 11.844)

Tests Method F Test (normal) Levene's Test (any continuous)

DF1 8 1

DF2 8 16

Test Statistic 2.77 0.87 107

P-Value 0.171 0.365

F-Test for 𝝈𝟐𝟏 /𝝈𝟐𝟐 Using Independent Samples (Excel) Use Data > Data Analysis > F-Test Two-Sample for Variances. Our example is data collected on the length of cuts (in feet) of columns from two different saws (A and B). We're interested in whether these data show enough evidence that the saws differ on the variation in the lengths of cuts. The data are shown in the worksheet in unstacked format.

Variable 1 Range: Highlight the cells containing the numeric responses for the first group. Variable 2 Range: Highlight the cells containing the numeric responses for the second group. Note: The group whose data is specified as the Variable 1 Range is used as the numerator term in the ratio of variances. Here it would be 𝜎𝐴2 /𝜎𝐵2 . Labels: Check this box if the Input Ranges contain column labels. If not, don't check the box. Checking the box tells Excel to ignore what's in the first row of the Input Ranges. Alpha: Specify 𝛼 for the test.

108

F-Test for 𝝈𝟐𝟏 /𝝈𝟐𝟐 Using Independent Samples (Excel) The output contains descriptive statistics for both groups and the results of the hypothesis test of 𝐻0 :

2 𝜎𝐴

2 𝜎𝐵

=1

Only "one-tail p-values" and "one-tail critical values" are reported.

F-Test Two-Sample for Variances

Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail

A 8.048889 0.001386 9 8 2.772222 0.085334 3.438101

B 8.07 0.0005 9 8

IMPORTANT NOTE: Excel does not report one-tailed p-values and critical values correctly, in general. The one-tailed “pvalue” reported is actually the area under the F-curve in the upper or lower tail, depending on the value of the test statistic “F.” Excel mistakenly assumes that “F” will be relatively large if you have an upper-tailed test and relatively small if you have a lower-tailed test, which usually happens, but not always. The one-tailed “F Critical” reported also corresponds to the direction Excel chooses for the alternative. It is poor statistical practice to base the direction of the hypotheses on the data.

To conduct a two-tailed test, 𝐻0 : 𝐻𝑎 :

2 𝜎𝐴

2 𝜎𝐵 2 𝜎𝐴

2 𝜎𝐵

=1 ≠1

use the Excel function F.TEST. In a blank cell, type an equal sign to insert a function and select More Functions… from the drop-down list of functions.

109

F-Test for 𝝈𝟐𝟏 /𝝈𝟐𝟐 Using Independent Samples (Excel) Select the F.TEST function and complete the dialog box. Array 1: Highlight the cells containing the numeric responses for the first group. You may highlight the column labels if you choose; Excel will ignore any non-numeric data. Array 2: Highlight the cells containing the numeric responses for the second group. You may highlight the column labels if you choose; Excel will ignore any nonnumeric data. Note: The group whose data is specified as the Array 1 range is used as the numerator term in the ratio of variances. Here it would be 𝜎𝐴2 /𝜎𝐵2 . The two-tailed p-value (.170667608 here) is shown in the dialog box as soon as you specify both arrays.

110

One-Way ANOVA (Minitab) Stacked Data Use Stat > ANOVA > One-Way… All response data must be in a single column with a second column indicating the levels of the factor. In the worksheet at the right, we have random samples of 10 scores from each of 3 different training programs.

Response: Specify the response variable. Factor: Specify the column containing the factor levels. Use the Comparisons… button to specify a multiple comparisons analysis. Here we request Tukey's adjustment for multiple comparisons so that the family (experimentwise) error rate remains at .05 (5%).

111

One-Way ANOVA (Minitab) The output shows the ANOVA table, individual confidence intervals for the population means (NOT adjusted for multiple comparisons), and the multiple comparisons analysis (if requested). One-way ANOVA: Score versus Program Source Program Error Total

DF 2 27 29

S = 9.823

Level 1 2 3

N 10 10 10

SS 62.1 2605.3 2667.4

MS 31.0 96.5

R-Sq = 2.33%

Mean 71.200 70.300 67.800

StDev 10.261 9.627 9.566

F 0.32

P 0.728

R-Sq(adj) = 0.00% Individual 95% CIs For Mean Based on Pooled StDev -------+---------+---------+---------+-(-----------*------------) (------------*-----------) (------------*-----------) -------+---------+---------+---------+-65.0 70.0 75.0 80.0

Pooled StDev = 9.823 Grouping Information Using Tukey Method Program 1 2 3

N 10 10 10

Mean 71.200 70.300 67.800

Grouping A A A

Means that do not share a letter are significantly different. Tukey 95% Simultaneous Confidence Intervals All Pairwise Comparisons among Levels of Program Individual confidence level = 98.04% Program = 1 subtracted from: Program 2 3

Lower -11.803 -14.303

Center -0.900 -3.400

Upper 10.003 7.503

+---------+---------+---------+--------(---------------*--------------) (--------------*---------------) +---------+---------+---------+---------14.0 -7.0 0.0 7.0

Program = 2 subtracted from: Program 3

Lower -13.403

Center -2.500

Upper 8.403

+---------+---------+---------+--------(--------------*---------------) +---------+---------+---------+---------14.0 -7.0 0.0 7.0

112

One-Way ANOVA (Minitab) Unstacked Data Use Stat > ANOVA > One-Way(Unstacked)… Response data must be in separate columns, one for each sample. In the worksheet at the right, we have random samples of 10 scores from each of 3 different training programs.

Responses (in separate columns): Specify the columns containing the response data. Use the Comparisons… button to specify a multiple comparisons analysis. Here we request Tukey's adjustment for multiple comparisons so that the family (experimentwise) error rate remains at .05 (5%).

113

One-Way ANOVA (Minitab) The output shows the ANOVA table, individual confidence intervals for the population means (NOT adjusted for multiple comparisons), and the multiple comparisons analysis (if requested).

One-way ANOVA: Score_1, Score_2, Score_3 Source Factor Error Total

DF 2 27 29

S = 9.823

Level Score_1 Score_2 Score_3

N 10 10 10

SS 62.1 2605.3 2667.4

MS 31.0 96.5

R-Sq = 2.33%

Mean 71.200 70.300 67.800

F 0.32

P 0.728

R-Sq(adj) = 0.00%

StDev 10.261 9.627 9.566

Individual 95% CIs For Mean Based on Pooled StDev -------+---------+---------+---------+-(-----------*------------) (------------*-----------) (------------*-----------) -------+---------+---------+---------+-65.0 70.0 75.0 80.0

Pooled StDev = 9.823 Grouping Information Using Tukey Method Score_1 Score_2 Score_3

N 10 10 10

Mean 71.200 70.300 67.800

Grouping A A A

Means that do not share a letter are significantly different. Tukey 95% Simultaneous Confidence Intervals All Pairwise Comparisons Individual confidence level = 98.04% Score_1 subtracted from: Score_2 Score_3

Lower -11.803 -14.303

Center -0.900 -3.400

Upper 10.003 7.503

+---------+---------+---------+--------(---------------*--------------) (--------------*---------------) +---------+---------+---------+---------14.0 -7.0 0.0 7.0

Score_2 subtracted from: Score_3

Lower -13.403

Center -2.500

Upper 8.403

+---------+---------+---------+--------(--------------*---------------) +---------+---------+---------+---------14.0 -7.0 0.0 7.0

114

One-Way ANOVA (Excel) Enter the data in the worksheet in unstacked format, i.e. the responses in separate columns, one for each sample. In the worksheet at the right, we have random samples of 10 scores from each of 3 different training programs. Use Data > Data Analysis > ANOVA: Single Factor.

Input Range: Specify the input data as the range of cells containing response columns. Grouped By: Select columns. (You can also input the response data in separate rows, one for each sample, but this is not typically done. In that case, select Rows here.) Labels in First Row: Check this box if the Input Ranges contain column labels. If not, don't check the box. Checking the box tells Excel to ignore what's in the first row of the Input Range. Alpha: The level of significance for critical values. The default is .05.

Note: Excel does not do multiple comparisons.

115

One-Way ANOVA (Excel) The output shows summary statistics for each sample and the ANOVA table which includes the critical value for the F-test statistic.

Anova: Single Factor SUMMARY Groups Score_1 Score_2 Score_3

Count Sum 10 712 10 703 10 678

Average Variance 71.2 105.2889 70.3 92.67778 67.8 91.51111

ANOVA Source of Variation Between Groups Within Groups

SS 62.06667 2605.3

df MS F P-value F crit 2 31.03333 0.321614 0.727717 3.354131 27 96.49259

Total

2667.367

29

116

Two-Way ANOVA (Minitab) Data must be entered in stacked format with one column for the numeric responses and one column for each factor. In this example, we have factor A at 3 levels and factor B at 2 levels. There are two ways to do the analysis in Minitab. One way is to use Stat > ANOVA > Two-Way…

Response: Specify the column containing the responses. Row factor / Column factor: Specify columns containing the levels of each factor. The output will be the ANOVA table. Two-way ANOVA: Score versus FactorA, FactorB Source FactorA FactorB Interaction Error Total S = 6.285

DF 2 1 2 6 11

SS 118.5 108.0 46.5 237.0 510.0

MS 59.25 108.00 23.25 39.50

R-Sq = 53.53%

F 1.50 2.73 0.59

P 0.296 0.149 0.584

R-Sq(adj) = 14.80%

Note that this tool does not allow for multiple comparisons. To do multiple comparisons in a two-way ANOVA, use the General Linear Model tool: Stat > ANOVA > General Linear Model

117

Two-Way ANOVA (Minitab) Responses: Specify the column containing the responses. Model: Specify the sources of variation you want to for. Enter both columns containing the factor levels. To for an interaction between two factors, specify an interaction term using an asterisk, as shown. You should only specify an interaction term when there are multiple observations per treatment.

The output will show the ANOVA table as well as a list of the factors and levels. General Linear Model: Score versus FactorA, FactorB Factor FactorA FactorB

Type fixed fixed

Levels 3 2

Values 1, 2, 3 1, 2

Analysis of Variance for Score, using Adjusted SS for Tests Source FactorA FactorB FactorA*FactorB Error Total S = 6.28490

DF 2 1 2 6 11

Seq SS 118.50 108.00 46.50 237.00 510.00

R-Sq = 53.53%

Adj SS 118.50 108.00 46.50 237.00

Adj MS 59.25 108.00 23.25 39.50

F 1.50 2.73 0.59

P 0.296 0.149 0.584

R-Sq(adj) = 14.80%

Use the Comparisons… button to request a multiple comparisons analysis. Row or Column Means Comparisons : Specify the column containing the levels of the row or column factor, as shown here. Method: Select a multiple comparisons method. Check the Confidence interval approach for displaying the results.

118

Two-Way ANOVA (Minitab) The output shows a confidence interval (adjusted for multiple comparisons) for the difference, 𝜇𝑖 − 𝜇𝑗 , between each pair of row means (𝜇1 , 𝜇2 , … , 𝜇𝑎 ) or column means (𝜇1 , 𝜇2 , … , 𝜇𝑏 ). Here there are 3 levels of Factor A giving 3 row means (𝜇1 , 𝜇2 , 𝜇3 ) and (32) = 3 pairs. Tukey 95.0% Simultaneous Confidence Intervals Response Variable Score All Pairwise Comparisons among Levels of FactorA FactorA = 1 subtracted from: FactorA 2 3

Lower -18.89 -11.39

FactorA = 2 FactorA 3

Center -5.250 2.250

Upper 8.388 15.888

------+---------+---------+---------+ (-----------*----------) (----------*----------) ------+---------+---------+---------+ -12 0 12 24

subtracted from:

Lower -6.138

Center 7.500

Upper 21.14

------+---------+---------+---------+ (----------*-----------) ------+---------+---------+---------+ -12 0 12 24

Example: A 95% confidence interval for 𝜇2 − 𝜇1 is (−18.89 , 8.388).

Treatment/Cell Means Comparisons : Specify the interaction term using an asterisk, as shown here. Method: Select a multiple comparisons method. Check the Confidence interval approach for displaying the results.

119

Two-Way ANOVA (Minitab) The output shows a confidence interval (adjusted for multiple comparisons) for the difference, 𝜇𝑖𝑗 − 𝜇𝑘𝑙 , between each pair of treatment means (𝜇11 , 𝜇12 , … , 𝜇𝑎𝑏 ). Here there are 6 treatment means (𝜇11 , 𝜇12 , 𝜇21 , 𝜇22 , 𝜇31 , 𝜇32 ) and (62) = 15 pairs. Tukey 95.0% Simultaneous Confidence Intervals Response Variable Score All Pairwise Comparisons among Levels of FactorA*FactorB FactorA = 1 FactorB = 1 subtracted from: FactorA 1 2 2 3 3

FactorB 2 1 2 1 2

FactorA = 1 FactorB = 2 FactorA 2 2 3 3

FactorA 2 3 3

FactorA 3 3

FactorA 3

------+---------+---------+---------+ (---------*---------) (---------*---------) (---------*---------) (---------*---------) (---------*---------) ------+---------+---------+---------+ -25 0 25 50

Lower -39.02 -29.52 -27.02 -26.52

Center -14.00 -4.50 -2.00 -1.50

Upper 11.02 20.52 23.02 23.52

------+---------+---------+---------+ (---------*---------) (---------*---------) (---------*---------) (---------*---------) ------+---------+---------+---------+ -25 0 25 50

Lower -15.52 -13.02 -12.52

Center 9.500 12.000 12.500

Upper 34.52 37.02 37.52

------+---------+---------+---------+ (---------*---------) (---------*---------) (---------*---------) ------+---------+---------+---------+ -25 0 25 50

Upper 27.52 28.02

------+---------+---------+---------+ (---------*---------) (---------*---------) ------+---------+---------+---------+ -25 0 25 50

Upper 25.52

------+---------+---------+---------+ (---------*---------) ------+---------+---------+---------+ -25 0 25 50

subtracted from:

FactorB 1 2

FactorA = 3 FactorB = 1

Upper 33.02 19.02 28.52 31.02 31.52

subtracted from:

FactorB 2 1 2

FactorA = 2 FactorB = 2

Center 8.000 -6.000 3.500 6.000 6.500

subtracted from:

FactorB 1 2 1 2

FactorA = 2 FactorB = 1

Lower -17.02 -31.02 -21.52 -19.02 -18.52

Lower -22.52 -22.02

Center 2.500 3.000

subtracted from:

FactorB 2

Lower -24.52

Center 0.5000

Example: A 95% confidence interval for 𝜇32 − 𝜇12 is (−26.52 , 23.52).

120

Two-Way ANOVA (Excel) Enter the data in the worksheet in a format that resembles the design (rows for factor A and columns for factor B). In this example, we have factor A at 3 levels and factor B at 2 levels. There are 2 observations in each treatment.

If there is more than one observation per treatment, use Data > Data Analysis > ANOVA: Two-Factor With Replication.

Input Range: Specify the input data as the range of cells containing data table. Include the labels in the range. Rows per sample: Indicate how many observations were made in each treatment. Alpha: The level of significance for critical values. The default is .05.

Note: Excel does not do multiple comparisons.

121

Two-Way ANOVA (Excel) The output gives a table of descriptive statistics (count, sum, average, variance) of the responses in each treatment as well as across each row and column. The ANOVA table is displayed at the bottom.

Anova: Two-Factor With Replication SUMMARY

1

2 Total

1

Count Sum Average Variance

2 140 70 2

2 156 78 72

4 296 74 46

2 128 64 2

2 4 147 275 73.5 68.75 40.5 44.25

2

Count Sum Average Variance 3

Count Sum Average Variance

2 2 4 152 153 305 76 76.5 76.25 8 112.5 40.25 Total

Count Sum Average Variance

ANOVA Source of Variation Sample Columns Interaction Within Total

6 420 70 31.2

6 456 76 49.2

SS 118.5 108 46.5 237

df

510

2 1 2 6

MS F P-value F crit 59.25 1.5 0.296296 5.143253 108 2.734177 0.149307 5.987378 23.25 0.588608 0.584233 5.143253 39.5

11 122

Two-Way ANOVA (Excel) If there is only one observation per cell (as in a block design), use Data > Data Analysis > ANOVA: Two-Factor Without Replication. In this example of a block design, we have a factor at 3 levels and a blocking variable at 5 levels.

Input Range: Specify the input data as the range of cells containing data table. Labels: Check this box if you highlighted the row/column labels in the input range. Alpha: The level of significance for critical values. The default is .05. Note: Excel does not do multiple comparisons.

123

Two-Way ANOVA (Excel) The output shows descriptive statistics of the responses across each row and column. The ANOVA table is displayed at the bottom. Anova: Two-Factor Without Replication SUMMARY

Count

Sum 164.0231 86.89977 167.0332 101.9388 218.3436

Average 54.67437 28.96659 55.67772 33.9796 72.78119

Variance 139.9776 133.6837 1312.929 1603.765 158.9724

1 2 3 4 5

3 3 3 3 3

1 2 3

5 285.2015 57.04029 383.9797 5 235.5977 47.11954 778.1437 5 217.4393 43.48785 1341.297

ANOVA Source of Variation Rows Columns Error

SS 3807.157 492.1318 6206.523

Total

10505.81

df

MS F P-value F crit 4 951.7892 1.226824 0.372009 3.837853 2 246.0659 0.317171 0.736959 4.45897 8 775.8153 14

124

Interaction Plot (Minitab) Data must be entered in stacked format with one column for the numeric responses and one column for each factor. In this example, we have factor A at 3 levels and factor B at 2 levels. Use Stat > ANOVA > Interactions Plot…

Responses: Specify the column containing the responses. Factors: Specify the columns containing the levels of the two factors. Display full interaction plot matrix: Check this box to construct the interaction plot two ways: Factor A levels as x-axis and Factor B levels as different plotting symbols; Factor B levels as x-axis and Factor A levels as different plotting symbols. Interaction Plot for Score Data Means 1

2

80 75

FactorA

FactorA 1 2 3

70 65

80

FactorB 1 2

75 FactorB

70 65 1

2

3

125

Chi-Square Test for One-Way Table (Minitab) Use Stat > Tables > Chi-Square Goodness-of-Fit Test (One Variable)… If the data are summarized as frequencies of the categories, use the first input option:

Observed counts: Specify the column containing the frequencies. Category names (optional): Specify the column containing the category names. If you have raw categorical data in a single column with one row per observation, use the second input option:

Categorical data: Specify the column containing the categorical data.

126

Chi-Square Test for One-Way Table (Minitab) Under Test, specify what the null hypothesis of the test looks like: Equal proportions: Use this option if the null hypothesis specifies equal proportions: 1

𝐻0 : 𝑝1 = 𝑝2 = ⋯ = 𝑝𝑘 = 𝑘

Specific proportions: Use this option if the null hypothesis specifies different proportions. Specify a column in the worksheet that contains the proportions specified in the null hypothesis. Here we use 𝐻0 : 𝑝𝐵𝑙𝑢𝑒 = .3, 𝑝𝑅𝑒𝑑 = .4, 𝑝𝑌𝑒𝑙𝑙𝑜𝑤 = .3. Important Note: If you use the Observed counts input option, the specific proportions must be in the same order as the order the categories appear in the frequency table. If you use the Categorical data input option, the specific proportions must be correspond to the alphabetical order of the categories in the data. Here, since we're using the Categorical data input option, the null hypothesized proportion for Blue comes first.

127

Chi-Square Test for One-Way Table (Minitab) The output shows the frequency table with the observed counts, null hypothesized proportions, expected counts, and chi-square contributions, square test statistic value and p-value. Chi-Square Goodness-of-Fit Test for Categorical Variable: Color Category Blue Red Yellow N 44

N* 0

Observed 13 11 20

DF 2

Chi-Sq 5.98106

Test Proportion 0.3 0.4 0.3

Expected 13.2 17.6 13.2

Contribution to Chi-Sq 0.00303 2.47500 3.50303

P-Value 0.050

128

(𝑛𝑖 −𝐸𝑖 )2 𝐸𝑖

, as well as the chi-

Chi-Square Test for Two-Way Table (Minitab) There are two menu options, depending on how the data are shown in the worksheet. Raw/stacked data format: two columns holding the category values of each variable Here we have 3 random samples of 50 Plain, Peanut, and Dark Chocolate M&Ms and observe the color of each. Use Stat > Tables > Cross Tabulation and Chi-Square.

For rows: Specify the column containing the categories you'd like to use as the rows of the 2-way contingency table. For columns: Specify the column containing the categories you'd like to use as the columns of the 2-way contingency table. Use the Chi-Square… button to specify the ChiSquare analysis. You can also display the expected counts.

129

Chi-Square Test for Two-Way Table (Minitab) The output shows the contingency table and two versions of the chi-square test. Our textbook uses the Pearson Chi-Square test. Tabulated statistics: Type, Color Rows: Type

Dark Peanut Plain All

Columns: Color

Blue

Brown

Green

Orange

Red

Yellow

All

9 9 11 29

4 5 8 17

5 6 7 18

7 8 10 25

14 13 9 36

11 9 5 25

50 50 50 150

Cell Contents:

Count

Pearson Chi-Square = 6.105, DF = 10, P-Value = 0.806 Likelihood Ratio Chi-Square = 6.241, DF = 10, P-Value = 0.795

Summarized count data: worksheet holds the contingency table counts Here we have the results of 500 father/son pairs classified by both the father's and the son's occupation. The father's occupation is shown on the rows and the son's on the columns. Use Stat > Tables > Chi-Square Test (Two-Way Table in Worksheet)…

Columns containing the table: Specify the columns of the worksheet containing the cell counts.

130

Chi-Square Test for Two-Way Table (Minitab) The output shows the contingency table with expected counts and the individual contributions to the chi-square statistic, along with the results of the chi-square test. Chi-Square Test: Prof/Bus, Skilled, Unskilled, Farmer Expected counts are printed below observed counts Chi-Square contributions are printed below expected counts Prof/Bus 55 34.20 12.650

Skilled 38 41.40 0.279

Unskilled 7 16.00 5.063

Farmer 0 8.40 8.400

Total 100

2

79 59.85 6.127

71 72.45 0.029

25 28.00 0.321

0 14.70 14.700

175

3

22 49.59 15.350

75 60.03 3.733

38 23.20 9.441

10 12.18 0.390

145

4

15 27.36 5.584

23 33.12 3.092

10 12.80 0.613

32 6.72 95.101

80

Total

171

207

80

42

500

1

Chi-Sq = 180.874, DF = 9, P-Value = 0.000

131

Regression (Minitab) To perform regression analysis, data must be entered in the worksheet with one row per observation and one column for each variable term in the regression model. Here we have 24 orange juice samples, 6 from each of 4 brands, the pectin content, and measure of sweetness for each. We use 3 dummy/indicator variables to indicate the 4 levels of brand.

Use Stat > Regression > Regression… Response: Specify the column containing the y-variable data. Predictors: Specify the column(s) containing the data corresponding to each term in the model. Here we fit a simple linear model of the form 𝑦 = 𝛽0 + 𝛽1 𝑃𝑒𝑐𝑡𝑖𝑛 + 𝜖

132

Regression (Minitab) Use the Graphs… button to request residual plots. Residuals for Plots: Specify the type of residuals. Regular Standardized Deleted

𝑦𝑖 − 𝑦̂𝑖 𝑦𝑖 −𝑦̂𝑖 𝑠 𝑦𝑖 −𝑦̂𝑖∗ 𝑠∗

where 𝑦̂𝑖∗ and 𝑠 ∗ are calculated based on the model fit to the data where observation 𝑖 is omitted. Residual Plots: Request plots individually or a Four-in-one display. Residuals versus the variables: Request residuals (on the y-axis) plotted against any variable in the worksheet.

Use the Options… button to request additional analyses such as Variance inflation factors (VIF) for each term in the model and the Durbin-Watson statistic to test for first-order autocorrelation. Predication intervals for new observations: Use this option to request confidence intervals and prediction intervals for a particular set of x's. Specify the values of the x variables in the same order as their appear in the model, i.e. in the same order as you entered the in the Predictors box. Here we predict Sweetness for 250 ppm of Pectin.

133

Regression (Minitab) The output of this analysis is displayed in both the session window and several graph windows. Regression Analysis: SweetIndex versus Pectin The regression equation is SweetIndex = 6.25 - 0.00231 Pectin Predictor Constant Pectin

Coef 6.2521 -0.0023106

S = 0.214998

SE Coef 0.2366 0.0009049

R-Sq = 22.9%

T 26.42 -2.55

P 0.000 0.018

R-Sq(adj) = 19.4%

Analysis of Variance Source Regression Residual Error Total

DF 1 22 23

SS 0.30140 1.01693 1.31833

MS 0.30140 0.04622

F 6.52

P 0.018

Unusual Observations Obs 1 11 16

Pectin 220 410 383

SweetIndex 5.2000 5.4000 5.3000

Fit 5.7437 5.3047 5.3671

SE Fit 0.0552 0.1453 0.1222

Residual -0.5437 0.0953 -0.0671

St Resid -2.62R 0.60 X -0.38 X

R denotes an observation with a large standardized residual. X denotes an observation whose X value gives it large leverage. Predicted Values for New Observations New Obs 1

Fit 5.6744

SE Fit 0.0443

95% CI (5.5825, 5.7664)

95% PI (5.2192, 6.1297)

Values of Predictors for New Observations New Obs 1

Pectin 250

Residual Plots for SweetIndex

Residuals Versus Pectin

Normal Probability Plot

(response is SweetIndex)

Percent

0.3

0.4

90

0.2

50

0.1

10

0.0

1

0.0 -0.2 -0.4

-0.50

-0.25

-0.1

0.00 Residual

0.25

0.50

5.3

5.4

Histogram

-0.2

-0.5 200

250

300 Pectin

350

400

0.2

6 4 2 0

134

5.7

Versus Order

Residual

-0.4

5.5 5.6 Fitted Value

0.4

8

-0.3 Frequency

Residual

0.2

Versus Fits

99

Residual

0.4

0.0 -0.2 -0.4

-0.4

-0.2 0.0 Residual

0.2

2

4

6

8 10 12 14 16 18 20 22 24 Observation Order

Regression (Minitab) To fit separate, parallel lines to the data, one for each brand, we fit the multiple regression model 𝑦 = 𝛽0 + 𝛽1 𝑃𝑒𝑐𝑡𝑖𝑛 + 𝛽2 𝐵𝑟𝑎𝑛𝑑𝐴 + 𝛽3 𝐵𝑟𝑎𝑛𝑑𝐵 + 𝛽4 𝐵𝑟𝑎𝑛𝑑𝐶 + 𝜖 Use Stat > Regression > Regression… and specify 4 predictor . We use the Options… button to request a confidence interval and prediction interval for a pectin content of 250 ppm from Brand B.

135

Regression (Minitab) Regression Analysis: SweetIndex versus Pectin, Brand_A, Brand_B, Brand_C The regression equation is SweetIndex = 6.19 - 0.00193 Pectin - 0.024 Brand_A - 0.023 Brand_B - 0.108 Brand_C Predictor Constant Pectin Brand_A Brand_B Brand_C

Coef 6.1924 -0.001927 -0.0238 -0.0233 -0.1081

S = 0.227640

SE Coef 0.2894 0.001151 0.1322 0.1362 0.1465

R-Sq = 25.3%

T 21.40 -1.68 -0.18 -0.17 -0.74

P 0.000 0.110 0.859 0.866 0.470

R-Sq(adj) = 9.6%

Analysis of Variance Source Regression Residual Error Total Source Pectin Brand_A Brand_B Brand_C

DF 1 1 1 1

DF 4 19 23

SS 0.33375 0.98458 1.31833

MS 0.08344 0.05182

F 1.61

P 0.213

Seq SS 0.30140 0.00009 0.00407 0.02819

Unusual Observations Obs 1 11

Pectin 220 410

SweetIndex 5.2000 5.4000

Fit 5.7446 5.3789

SE Fit 0.0932 0.1866

Residual -0.5446 0.0211

St Resid -2.62R 0.16 X

R denotes an observation with a large standardized residual. X denotes an observation whose X value gives it large leverage. Predicted Values for New Observations New Obs 1

Fit 5.6873

SE Fit 0.0956

95% CI (5.4873, 5.8873)

95% PI (5.1705, 6.2040)

Values of Predictors for New Observations New Obs 1

Pectin 250

Brand_A 0.000000

Brand_B 1.00

Brand_C 0.000000

136

Regression (Excel) To perform regression analysis, data must be entered in the worksheet with one row per observation and one column for each variable term in the regression model. All columns containing data for the predictor in the model must be in one contiguous block of columns. Here we have 24 orange juice samples, 6 from each of 4 brands, the pectin content, and measure of sweetness for each. We use 3 dummy/indicator variables to indicate the 4 levels of brand.

Use Data > Data Analysis > Regression. Input Y Range: Specify the column containing the y-variable data. Input X Range: Specify the column or block of columns containing the data corresponding to each term in the model. Here we fit a multiple regression model of the form 𝑦 = 𝛽0 + 𝛽1 𝑃𝑒𝑐𝑡𝑖𝑛 + 𝛽2 𝐵𝑟𝑎𝑛𝑑𝐴 + 𝛽3 𝐵𝑟𝑎𝑛𝑑𝐵 + 𝛽4 𝐵𝑟𝑎𝑛𝑑𝐶 + 𝜖 . 137

Regression (Excel) Labels: Check this box if the input ranges include the column names. Residual Plots: Request a residual plot of residuals vs. each predictor variable. Normal Probability Plots: Request a normal probability plot of the response (y) data. This is not really the plot we want to look at; we'd like a normal probability plot of the residuals (see Minitab). The output shows some regression summary statistics, the ANOVA table, and table of parameter estimates. The interval estimates are confidence intervals for the 𝛽 parameters.

SUMMARY OUTPUT

Regression Statistics Multiple R

0.503153

R Square

0.253163

Adjusted R Square

0.095935

Standard Error

0.22764

Observations

24

ANOVA df Regression

SS

MS

4

0.333754

0.083438

Residual

19

0.98458

0.05182

Total

23

1.318333

Coefficients

Standard Error

Intercept

6.19237

Pectin Brand_A

F 1.61016

Significance F 0.2127

Lower 95.0%

Upper 95.0%

6.797988

5.586752

6.797988

-0.00434

0.000481

-0.00434

0.000481

-0.30045

0.25291

-0.30045

0.25291

0.866215

-0.3084

0.261873

-0.3084

0.261873

0.469758

-0.4148

0.198622

-0.4148

0.198622

t Stat

P-value

0.289351

21.4009

9.27E-15

5.586752

-0.00193

0.001151

-1.67519

0.110277

-0.02377

0.132192

-0.17982

0.859196

Brand_B

-0.02326

0.136232

-0.17076

Brand_C

-0.10809

0.146541

-0.73762

138

Lower 95%

Upper 95%

Regression (Excel)

Pectin Residual Plot

Brand_A Residual Plot

0.4 0.3

0.4

0.2

0.2

0 -0.1 0

200

400

Residuals

Residuals

0.1 600

-0.2 -0.3 -0.4

0 0

0.5

1

1.5

-0.2 -0.4

-0.5 -0.6

-0.6

Pectin

Brand_B Residual Plot

Brand_C Residual Plot

0.4

0.4 0.2

0 0

0.5

1

Residuals

Residuals

0.2

1.5

-0.2 -0.4

0 -0.2

0

0.5

-0.4

-0.6

-0.6

Brand_B

Normal Probability Plot 6.5

SweetIndex

Brand_A

6 5.5 5 0

20

40

60

80

100

Sample Percentile

139

120

Brand_C

1

1.5

̅ Chart (Minitab) 𝒙 Use Stat > Control Charts > Variables Charts for Subgroups > X-bar…

There are two ways to input the data: stacked and unstacked. Stacked Format All observations for a chart are in one column: Data for each sample are stacked in a single column. Here we show data for 10 samples of size 4 in a single column. Subgroup sizes: Enter the sample size for each sample (𝑛). Use Xbar Options… to further specify the analysis: Use the Estimate tab to specify the method for estimating the process standard deviation (𝜎). Our text uses the Rbar (𝑅̅ ) method.

140

̅ Chart (Minitab) 𝒙 Use the S Limits tab to request control limits and zone boundaries at 1-, 2-, and 3-standard deviations from the centerline. Enter the multiples separated by spaces. Note: "Standard deviation" here refers to 𝜎 𝜎𝑥̅ = 𝑛. If the Rbar method of estimating √

𝜎 is used, 𝜎𝑥̅ ≈

̅ 𝑅 𝑑2

√𝑛

.

Use the Tests tab to request all relevant tests for special causes. Note: The default settings of K correspond to the tests in our text.

Xbar Chart of X 13 +3SL=12.658 12

Sample Mean

The output is an 𝑥̅ chart. If any of the tests for special causes reveal unusual patterns/points, they will be noted in the session window and flagged on the chart. Here, none of the 8 tests reveal unusual patterns.

+2SL=11.732

11

+1SL=10.806 _ _ X=9.880

10 9

-1SL=8.954

8

-2SL=8.028

7

-3SL=7.102 1

141

2

3

4

5 6 Sample

7

8

9

10

̅ Chart (Minitab) 𝒙 Unstacked Format

Observations for a subgroup are in one row of columns: Data for each sample are in a row of the worksheet and each row shows a different sample. Here we show data for 10 samples of size 3. Specify the columns containing the data for each sample. Using the same Xbar options as above gives the following chart. Xbar Chart of X1, ..., X3 130 +3SL=125.21

Sample Mean

120

+2SL=117.82 +1SL=110.43

110

_ _ X=103.04 100 -1SL=95.65 90

-2SL=88.26 -3SL=80.86

80 1

2

3

4

5 6 Sample

7

8

9

10

142

R Chart (Minitab) Use Stat > Control Charts > Variables Charts for Subgroups > R…

There are two ways to input the data: stacked and unstacked. Stacked Format All observations for a chart are in one column: Data for each sample are stacked in a single column. Here we show data for 10 samples of size 4 in a single column. Subgroup sizes: Enter the sample size for each sample (𝑛). Use R Options… to further specify the analysis: Use the Estimate tab to specify the method for estimating the process standard deviation (𝜎). Our text uses the Rbar (𝑅̅ ) method.

143

R Chart (Minitab) Use the S Limits tab to request control limits and zone boundaries at 1-, 2-, and 3standard deviations from the centerline. Enter the multiples separated by spaces. Note: "Standard deviation" here refers to the standard deviation of the sample range, 𝜎𝑅 .

Use the Tests tab to request all relevant tests for special causes. Note: The default settings of K correspond to the tests in our text.

R Chart of X 9

+3SL=8.699

8 +2SL=7.070

7 Sample Range

The output is an R chart. If any of the tests for special causes reveal unusual patterns/points, they will be noted in the session window and flagged on the chart. Here, none of the 4 tests reveal unusual patterns.

6 +1SL=5.442

5

_ R=3.813

4 3

-1SL=2.185

2 1

-2SL=0.556 -3SL=0

0 1

144

2

3

4

5 6 Sample

7

8

9

10

R Chart (Minitab) Unstacked Format

Observations for a subgroup are in one row of columns: Data for each sample are in a row of the worksheet and each row shows a different sample. Here we show data for 10 samples of size 3. Specify the columns containing the data for each sample. Using the same R options as above gives the following chart. R Chart of X1, ..., X3 60 +3SL=55.79 50

Sample Range

+2SL=44.42 40 +1SL=33.04

30

_ R=21.67

20

-1SL=10.30

10 0

-3SL=0 -2SL=0 1

2

3

4

5 6 Sample

7

8

9

10

145

P Chart (Minitab) The data must be arranged in the worksheet with a column containing the number of "successes" for each sample. Here we have 15 samples of 20 items, each of which we classify as defective or not. The number defective for each sample is shown. Use Stat > Control Charts > Attributes Chart > P…

Variables: Specify the column containing the number of successes for each sample. Subgroup sizes: Enter the sample size for each sample (𝑛).

Use P Chart Options… to further spec ify the analysis: Use the S Limits tab to request control limits and zone boundaries at 1-, 2-, and 3standard deviations from the centerline. Enter the multiples separated by spaces. Note: "Standard deviation" here refers to 𝑝(1−𝑝)

𝜎𝑝̂ = √

𝑛

. 𝑝 is estimated by 𝑝̅ , the

overall proportion of successes.

146

P Chart (Minitab) Use the Tests tab to request all relevant tests for special causes. Note: The default settings of K correspond to the tests in our text.

The output is a P chart. If any of the tests for special causes reveal unusual patterns/points, they will be noted in the session window and flagged on the chart. Here, none of the 4 tests reveal unusual patterns. P Chart of Num Defective 0.30 +3SL=0.2754 0.25 +2SL=0.2125

Proportion

0.20 0.15

+1SL=0.1496

0.10

_ P=0.0867

0.05 -1SL=0.0238 0.00

-3SL=0 -2SL=0 1

2

3

4

5

6

7 8 9 Sample

10 11 12 13 14 15

147

Time Series Plot (Minitab) To plot a time series obtained at regular intervals of time, record the time series values in the worksheet in a single column. Here we have the quarterly price (dollars) of a stock for 4 years. Use Stat > Time Series > Time Series Plot and choose Simple.

Series: Specify the column containing the time series values. The default x-axis is a time index that begins at 1 with the first time point (here, first quarter of 2009).

Time Series Plot of Price 90

Price

80

70

60

50

40 2

148

4

6

8 Index

10

12

14

16

Moving Averages (Minitab) Record the time series values in the worksheet in a single column. Here we have the quarterly price (dollars) of a stock for 4 years. Use Stat > Time Series > Moving Average…

Variable: Specify the column containing the time series values. MA length: Specify the number of time points to include in the moving average. Center the moving averages: Check this box to compute the moving average at time 𝑖, (𝑀𝐴𝑖 ), using the window centered at the time point. The default is to use a window ending at time 𝑖. Use the Storage… button to store the moving averages in the worksheet.

149

Moving Averages (Minitab) Use the Graphs… button to plot the smoothed vs. the actual values. The default is to plot the predicted (previous smoothed value) vs. the actual values.

The output shows the original time series (actual values), the smoothed series using moving averages, and three accuracy measures. Moving Average Plot for Price Variable A ctual Smoothed

90

Mov ing A v erage Length 3

Price

80

A ccuracy Measures MA PE 8.5700 MA D 6.1674 MSD 53.6898

70

60

50

40 2

4

6

8 10 Index

12

14

16

150

Single Exponential Smoothing (Minitab) Record the time series values in the worksheet in a single column. Here we have the quarterly price (dollars) of a stock for 4 years. Use Stat > Time Series > Single Exp Smoothing…

Variable: Specify the column containing the time series values. Weight to Use in Smoothing: Select the Use radio button and specify the weight, 𝑤. The default is to use an optimizing algorithm to determine the weight. Under Options… set the initial smoothed value as the the first actual value (K = 1). This matches the technique our text uses.

Use the Storage… button to store the smoothed values in the worksheet.

151

Single Exponential Smoothing (Minitab) Use the Graphs… button to plot the smoothed vs. the actual values. The default is to plot the predicted (previous smoothed value) vs. the actual values.

The output shows the original time series (actual values), the smoothed series using single exponential smoothing, and three accuracy measures. Note: Minitab refers to the smoothing weight, 𝑤, as "Alpha." Smoothing Plot for Price Single Exponential Method

Variable A ctual Smoothed

Price

90 80

Smoothing Constant A lpha 0.7

70

A ccuracy Measures MA PE 10.8279 MA D 7.5114 MSD 94.7440

60 50 40 2

4

6

8 10 Index

12

14

16

152

Trend Analysis (Minitab) Record the time series values in the worksheet in a single column. Here we have the quarterly price (dollars) of a stock for 4 years. Use Stat > Time Series > Trend Analysis…

Variable: Specify the column containing the time series values. Model Type: Select the type of trend model you want to fit. Here we select a linear trend model (linear regression model) of the form 𝑌𝑡 = 𝛽0 + 𝛽1 𝑡 + 𝜖 where 𝑡 = 1, 2, 3, … , 𝑛. Minitab creates this column for you so that you don't need to specify a predictor column. Use Generate forecasts to compute predicted values of the time series for a specified number of time periods past the end of the observed series. Here we request a forecasted price for first quarter 2013.

153

Trend Analysis (Minitab) The graph shows the estimated trend model, the original time series (actual values), the predicted values using the estimated trend model (fits), and the forecasted values, along with three accuracy measures. These summaries are also shown in the session window where the values of the forecasts are also displayed. Trend Analysis for Price Data Length NMissing

Trend Analysis Plot for Price Linear Trend Model Yt = 43.53 + 2.93*t

Price 16 0

100 90

A ccuracy Measures MA PE 8.8131 MA D 5.7672 MSD 42.1620

80 Price

Fitted Trend Equation Yt = 43.53 + 2.93*t

Variable A ctual Fits Forecasts

70 60

Accuracy Measures

50

MAPE MAD MSD

40

8.8131 5.7672 42.1620

2

Forecasts Period 17

Forecast 93.3646

Note: You can also use Stat > Regression > Regression… to obtain the same estimated trend model. You must first add a time index column having the values 1, 2, 3, … , 𝑛 to the worksheet and then specify that column as the Predictor column.

154

4

6

8

10 Index

12

14

16

Trend Analysis (Minitab) Use the Options… button to request forecasts by specifying the value of 𝑡 for the desired time period. The output will be formatted differently but will show the same estimated model and forecasts. Note: Be careful when using p-values and confidence intervals in a time series context as the assumption of independent errors may be violated. Regression Analysis: Price versus t The regression equation is Price = 43.5 + 2.93 t Predictor Constant t

Coef 43.532 2.9313

S = 6.94155

SE Coef 3.640 0.3765

R-Sq = 81.2%

T 11.96 7.79

P 0.000 0.000

R-Sq(adj) = 79.9%

Analysis of Variance Source Regression Residual Error Total

DF 1 14 15

SS 2921.5 674.6 3596.1

MS 2921.5 48.2

F 60.63

P 0.000

Unusual Observations Obs 12

t 12.0

Price 91.96

Fit 78.71

SE Fit 2.18

Residual 13.26

St Resid 2.01R

R denotes an observation with a large standardized residual. Predicted Values for New Observations New Obs 1

Fit 93.36

SE Fit 3.64

95% CI (85.56, 101.17)

95% PI (76.55, 110.18)

Values of Predictors for New Observations New Obs 1

t 17.0

155

Trend Analysis (Excel) Record the time series values in the worksheet in a single column. Here we have the quarterly price (dollars) of a stock for 4 years. Add a time index column having the values 1, 2, 3, … , 𝑛 to the worksheet. Use Data > Data Analysis > Regression to fit the trend model to the data. Here we fit the linear trend model (linear regression model) of the form 𝑌𝑡 = 𝛽0 + 𝛽1 𝑡 + 𝜖

Input Y Range: Specify the column containing the time series values. Input X Range: Specify the column containing the time index column. Labels: Check this box if the input ranges contain the column labels.

156

Trend Analysis (Excel) From the regression output, you can obtain the estimated trend model and use it to generate forecasts. Note: Be careful when using p-values and confidence intervals in a time series context as the assumption of independent errors may be violated.

SUMMARY OUTPUT

Regression Statistics Multiple R

0.901338

R Square

0.81241

Adjusted R Square

0.799011

Standard Error

6.941553

Observations

16

ANOVA df Regression

SS

MS

F 60.63097

1

2921.513

2921.513

Residual

14

674.5922

48.18516

Total

15

3596.105

Coefficients

Standard Error

43.532 2.931328

Intercept t

Significance F 1.87E-06

t Stat

P-value

3.640181

11.95875

9.78E-09

35.72459

0.376459

7.786589

1.87E-06

2.123905

157

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

51.33942

35.72459

51.33942

3.738751

2.123905

3.738751

Seasonal Regression Models Record the time series values in the worksheet in a single column. Here we have the quarterly price (dollars) of a stock for 4 years. Include columns for 𝑙 − 1 dummy/indicator variables to represent 𝑙 seasons. Here we use 3 dummy variables to represent the 4 quarters. If you want to incorporate a trend component in the model, add a time index column to the worksheet.

Refer to Regression (Minitab) or Regression (Excel) in this guide to fit the desired model to the data: Components Seasonal only Seasonal + Trend

Model 𝑌𝑡 = 𝛽0 + 𝛽1 𝑥1 + 𝛽2 𝑥2 + 𝛽3 𝑥3 + 𝜖 𝑌𝑡 = 𝛽0 + 𝛽1 𝑥1 + 𝛽2 𝑥2 + 𝛽3 𝑥3 + 𝛽4 𝑡 + 𝜖

158