Applet Exercises

Applet Exercises

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

Question 7.102

1. Use the Simple Random Sample applet to choose the sample of songs in Example 4 (on page 299). Here’s how:

  • Assign labels 01 to 27 by entering 27 in the “Population 1 to” box and clicking “Reset.”
  • Then enter 4 in the “Select a sample of size” box and click “Sample.”

Which songs from the list in Example 4 (page 299) make up your sample? Click “Reset” and choose another sample. Which songs did you choose this time? You see that random sampling gives different samples each time—what matters is that all songs have the same chance to be chosen.

Question 7.103

2. You can use the Simple Random Sample applet to choose treatment groups at random for a randomized comparative experiment. In outlining the design of the experiment in Exercise 35 (page 333) to compare the growth of cataracts in rats who were given tea extract to rats who were given a placebo, you should randomly choose the subjects (rats) for the first treatment (tea extract).

  1. Use the applet to choose an SRS of 7 out of 14 to receive the first treatment. Which subjects make up this group?
  2. The applet allows you to assign subjects randomly to more than two groups. Suppose you had a total of 36 rats and you wanted to assign a different treatment to each of four 9-rat groups. After you choose the first group, the “Population Hopper” contains the 27 subjects that were not chosen, in scrambled order. Click “Sample” again to choose 9 of these remaining subjects to receive the second treatment. Do this once more to choose the third group. The 9 subjects that remain in the “Population Hopper” form the fourth group. Which of the 36 subjects will receive each of the four treatments?

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Question 7.104

3. Suppose that 60% of the population bought a lottery ticket in the last 12 months. (This is the setting for Exercise 51.) You can use the Probability applet to simulate the behavior of random samples of size 50 from this population. You want to take many samples from this population to observe how the sample proportion that plays the lottery varies from sample to sample. By moving the sliders, specify the “Probability of Heads” setting in the applet as 0.6 and the number of tosses as 50. This simulates an SRS of size 50 from a large population. Each head in the sample is a person who plays the lottery, and each tail is a person who does not play. By alternating between “Toss” and “Reset,” you can take many samples quickly.

  1. Take 25 samples, recording the proportion in each sample that plays the lottery. (The applet gives this proportion at the top left of its display.) Make a histogram of the 25 sample proportions.
  2. Another population contains only 20% of people who play the lottery. Take 25 samples of size 50 from this population, record the number in each sample that plays, and make a histogram of the 25 sample proportions. How do the centers of your two histograms reflect the differing truths about the two populations?

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%.