EXAMPLE 7.21
Planning a survey of college students. In Example 7.1 (page 411), we calculated a 95% confidence interval for the mean hours per week a college student watches traditional television. The margin of error based on an SRS of n = 8 students was 12.42 hours. Suppose that a new study is being planned and the goal is to have a margin of error of five hours. How many students need to be sampled?
The sample standard deviation in Example 7.1 is s = 14.854 hours. To be conservative, we’ll guess that the population standard deviation is 17.5 hours.
1. To compute an initial n, we replace t* with z*. This results in
Round up to get n = 48.
2. We now check to see if this sample size satisfies the requirement when we switch back to t*. For n = 48, we have n − 1 = 47 degrees of freedom and t* = 2.011. Using this value, the expected margin of error is
This is larger than m = 5, so the requirement is not satisfied.
3. The following table summarizes these calculations for some larger values of n.
| n | |
|---|---|
| 49 | 5.03 |
| 50 | 4.97 |
| 51 | 4.92 |
The requirement is first satisfied when n = 50. Thus, we need to sample at least n = 50 students for the expected margin of error to be no more than five hours.
Figure 7.17 shows the Minitab input window used to do these calculations. Because the default confidence level is 95%, only the desired margin of error m and the estimate for s need to be entered.