Determining The Sample Size (continuous Data) 182z4u

Sample Size Determination Before you can calculate a sample size, you need to determine a few things about the target population and the sample you need: 1. Population Size — how many total people fit your demographic? For instance, if you want to know about mothers living in the US, your population size would be the total number of mothers living in the US. Don’t worry if you are unsure about this number. It is common for the population to be unknown or approximated. 2. Margin of Error (Confidence Interval) — No sample will be perfect, so you need to decide how much error to allow. The confidence interval determines how much higher or lower than the population mean you are willing to let your sample mean fall. If you’ve ever seen a political poll on the news, you’ve seen a confidence interval. It will look something like this: “68% of voters said yes to Proposition Z, with a margin of error of +/- 5%.” 3. Confidence Level — How confident do you want to be that the actual mean falls within your confidence interval? The most common confidence intervals are 90% confident, 95% confident, and 99% confident. When X is normally distributed, the range of values between X Bar ±1.96σ is called the 95% confidence interval for µ. The two boundaries of the interval, X Bar−1.96σ and X Bar +1.96σ are called the 95% confidence limits. That is, there is a 95% chance that the following statement will we true: X Bar −1.96σ ≤ µ ≤ X Bar +1.96σ 4. Standard of Deviation — How much variance do you expect in your responses? Since we haven’t actually istered our survey yet, the safe decision is to use .5 – this is the most forgiving number and ensures that your sample will be large enough. Okay, now that we have these values defined, we can calculate our needed sample size.

Your confidence level corresponds to a Z-score. This is a constant value needed for this equation. Here are the z-scores for the most common confidence levels: 

90% – Z Score = 1.64

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95% – Z Score = 1.96

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99% – Z Score = 2.57

Determining the Sample Size Continuous Data Formula for determining sample size is given below:

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Z is the value from the table of probabilities of the standard normal distribution for the desired confidence level (e.g., Z = 1.96 for 95% confidence)

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E is the margin of error that the investigator specifies as important from a clinical or practical standpoint.

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σ is the standard deviation of the outcome of interest.

Example 1: We would like to start an ISP and need to estimate the average Internet usage of households in one week for our business plan and model. How many households must we randomly select to be 95 percent sure that the sample mean is within 1 minute of the population mean . Assume that a previous survey of household usage has shown = 6.95 minutes. Solution we are solving for the sample size . A 95% degree confidence corresponds to

= 0.05. Each of the shaded tails in the

following figure has an area of = 0.025. The region to the left of and to the right of = 0 is 0.5 – 0.025, or 0.475. In the table of the standard normal ( ) distribution, an area of 0.475 corresponds to a value of 1.96. The critical value is therefore

= 1.96.

The margin of error = 1 and the standard deviation for sample size, we can calculate :

= 6.95. Using the formula

So we will need to sample at least 186 (rounded up) randomly selected households. With this sample we will be 95 percent confident that the sample mean within 1 minute of the true population of Internet usage.

This formula can be used when you know

will be

and want to determine the sample size

necessary to establish, with a confidence of , the mean value to within You can still use this formula if you don’t know your population standard deviation and you have a small sample size. Although it is unlikely that you know when the population mean is not known, you may be able to determine from a similar process or from a pilot test/simulation.

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Example 2: An investigator wants to estimate the mean systolic blood pressure in children with congenital heart disease who are between the ages of 3 and 5. How many children should be enrolled in the study? The investigator plans on using a 95% confidence interval (so Z=1.96) and wants a margin of error of 5 units. The standard deviation of systolic blood pressure is unknown, but the investigators conduct a literature search and find that the standard deviation of systolic blood pressures in children with other cardiac defects is between 15 and 20. To estimate the sample size, we consider the larger standard deviation in order to obtain the most conservative (largest) sample size.

In order to ensure that the 95% confidence interval estimate of the mean systolic blood pressure in children between the ages of 3 and 5 with congenital heart disease is within 5 units of the true mean, a sample of size 62 is needed. [Note: We

always round up; the sample size formulas always generate the minimum number of subjects needed to ensure the specified precision.] We assumed a standard deviation of 15, the sample size would have been n=35. Because the estimates of the standard deviation were derived from studies of children with other cardiac defects, it would be advisable to use the larger standard deviation and plan for a study with 62 children. Selecting the smaller sample size could potentially produce a confidence interval estimate with a larger margin of error. Example 3: We want to estimate the mean systolic blood pressure of Malaysian females. The standard deviation is around 20 mmHg and we wish to estimate the true mean to within 5mmHg with 95% confidence. What is the required sample size? Answer We are given σ = 20, Δ = 5 and z = 1.96. n=(1.96×205)2=61.47 women