To do these exercises, go to www.macmillanhighered.com/fapp10e.

Question 7.105

4. The idea of an 80% confidence interval is that the interval captures the true parameter value in 80% of all samples. That’s not high enough confidence for practical use, but 80% hits and 20% misses make it easy to see how a confidence interval behaves in repeated samples from the same population. Go to the Confidence Interval applet.

  1. Use the slider to set the confidence level to 80%. Click “Sample” to choose an SRS and calculate the confidence interval. Do this 10 times to simulate 10 SRSs with 10 confidence intervals. How many of the 10 intervals captured the true mean? How many missed?
  2. You see that we can’t predict whether the next sample will hit or miss. The confidence level, however, tells us what percentage of responses will hit in the long run. Reset the applet and click “Sample 25” to get the confidence intervals from 25 SRSs. How many hit? (You can read the number of hits and misses under the “Sample 25” button.)
  3. Keep clicking “Sample 25” and record the percent of hits among 100, 200, 300, 400, and 500 SRSs. Even 500 samples is not truly “the long run,” but we expect the percentage of hits in 500 samples to be fairly close to the confidence level of 80%.