Sample Size for AP Statistics (page 2)

Practice problems for these concepts can be found at:

• Confidence Intervals and Introduction to Inference Multiple Choice Practice Problems for AP Statistics
• Confidence Intervals and Introduction to Inference Free Response Practice Problems for AP Statistics
• Confidence Intervals and Introduction to Inference Review Problems for AP Statistics
• Confidence Intervals and Introduction to Inference Rapid Review for AP Statistics

It is always desirable to select as large a sample as possible when doing research because larger samples are less variable than small samples. However, it is often expensive or difficult to draw larger samples so that we try to find the optimum sample size: large enough to accomplish our goals, small enough that we can afford it or manage it. We will look at techniques in this section for selecting sample sizes in the case of a large sample test for a single population mean and for a single population proportion.

Sample Size for Estimating a Population Mean (Large Sample)

The large sample confidence interval for a population mean is given by ± z* . The margin of error is given by z* . Let M be the desired maximum margin of error. Then, M ≤ z* . Solving for n, we have . Using this "recipe," we can calculate the minimum n needed for a fixed confidence level and a fixed maximum margin of error.

One obvious problem with using this expression as a way to figure n is that we will not know σ, so we need to estimate it in some way. Sometimes you will know the approximate range of the data (call it R) you will be dealing with (for example, you are going to study children from 6 months of age to 18 months of age). If so, and assuming that your data will be approximately normally distributed, σ could be estimated using R/4 or R/6. This is based on the fact that most data in a normal distribution are within two or three standard deviations of , making for a range of four or six standard deviations.

Another way would be to utilize some historical knowledge about the standard deviation for each type of data we are examining, as shown in the following example:

example: A machine for inflating tires, when properly calibrated, inflates tires to 32 lbs, but it is known that the machine varies with a standard deviation of about 0.8 lbs. How large a sample is needed in order be 99% confident that the mean inflation pressure is within a margin of error of M = 0.10 lbs?

solution:

= 424.49. Since n must be an integer and n ≥ 424.49, choose n = 425. You would need a sample of at least 425 tires.

In this course you will not need to find a sample size for constructing a confidence interval involving t. This is because you need to know the sample size before you can determine t* since there is a different t distribution for each different number of degrees of freedom. For example, for a 95% confidence interval for the mean of a normal distribution, you know that z* = 1.96 no matter what sample size you are dealing with, but you can't determine t* without already knowing n.