Chapter 8 CHECK-IN QUESTIONS

  1. 8.1

    1. 236.
    2. The proportion p is the percent of all people aged 18 to 21 years old who said they use Instagram.
    3. X is 158. It is the number of 18- to 21-year-olds from the sample who use Instagram.
    4. p^=158/236.

  2. 8.3

    1. 0.0306.
    2. (0.6695 ± 0.0600).
    1. (60.95%, 72.95%).
    2. We are 95% confident that between 60.95% and 92.95% of 18- to 21-year-olds use Instagram.

  3. 8.7 P-value=0.0074. There is evidence that your product is better than the competitor’s product.

  4. 8.9

    1. p^=0.35, z=−1.34, P-value=0.1802. The results are exactly the same.
    2. (14.1%, 55.9%).

  5. 8.11 Answers will vary.

  6. 8.13 Power=32.5%.

  7. 8.15

    1. Let D=p1−p2. μD=0.3−0.6=−0.3. σD=0.0501.

  8. 8.17 p^1−p^2=−0.1443. The 95% CI=(−0.2772, −0.0114).

  9. 8.19 H0: p1=p2 versus Ha: p1≠p2. z=−1.97. P-value=0.0488. There is evidence that there is a difference between the two age groups.

  10. 8.21

    1. The margins of error are 0.24, 0.2245, 0.196.
    2. Yes, all of them are much larger than the required 0.1. You should increase the sample sizes.

CHAPTER 8 EXERCISES

  1. 8.1

    1. n=4272, X=897.
    2. p^=897/4272.
    3. The sample proportion is an unbiased estimator for the population proportion.

  2. 8.3 p^=0.21, SEp^=0.0062, m=0.0122. (b) Yes, n is sufficiently large. (c) (0.1978, 0.2222). (d) We are 95% confident that the true population proportion of adults who use smart watches and fitness trackers is between 19.78% and 22.22%.

  3. 8.5

    1. n=200, X=180.
    2. p^=180/200.
    3. The sample proportion is an unbiased estimator for the population proportion.

  4. 8.7

    1. H0: p=0.5 versus Ha: p≠0.5.
    2. Yes, 82 is a large enough sample size.
    3. z=1.77, P-value=0.0768. There is not sufficient evidence that children are not equally likely to choose a regular-grain snack or the whole-grain alternative.

  5. 8.9 n=1594, using p*=897/4272.

  6. 8.11 p^=0.6548, 95% CI=(0.6146, 0.6680).

  7. 8.13

    1. X=934.5, which rounds to 935.
    2. (0.8711, 0.9089).
    3. (87.1%, 90.9%).
    4. For example, parents might be conscious of violence because of recent events in the news.

  8. 8.15

    1. Values of p^ outside of (0.5661, 0.8339) will result in rejecting Ho.
    2. For n=90, values outside of (0.6053, 0.7947) will result in rejecting Ho.
    3. Larger sample sizes narrows the distribution and the confidence interval.

  9. 8.17

    1. 67,179 students.
    2. (0.41682, 0.42318).

  10. 8.19 p^=0.43, (0.4043, 0.4557).

  11. 8.21

    1. m=0.001321.
    2. Other sources of error are much more significant than sampling error.

  12. 8.23

    1. p^=0.3275, (0.3008, 0.3541).
    2. Speakers and listeners probably perceive sermon length differently.

  13. 8.25

    1. p^=0.5067. z=1.34, P-value=0.1802.
    2. (0.497, 0.516).

  14. 8.27 n=9604.

  15. 8.29 The sample sizes are 97, 171, 225, 257, 267, 257, 225, 171, 97. Use n=267.

  16. 8.31 n=153.

  17. 8.33

    1. The margin of error does not take into account those who did not answer the questions.
    2. The P-value represents that there is a 5% chance of obtaining results as extreme as those observed.
    3. A confidence level cannot be larger than 100%.

  18. 8.35

    1. The population consists of all male customers in a similar environment. X1=40, n1=69, X2=130, n2=349. p^1=0.5797, p^2=0.3725, difference=0.2072.
    2. The population is all runners X1=11, n1=20, X2=14, n2=20. p^1=0.65, p^2=0.70, difference=−0.15.

  19. 8.37

    1. The 95% CI is (0.0802, 0.3343). With 95% confidence, the percent of male customers tipping a red-shirted server is between 8.02% and 33.43% higher than the percent of male customers tipping a server with a shirt of a different color.
    2. The 95% CI is (–0.4464, 0.1464). With 95% confidence, the percent of runners satisfied with the first stretching routine is between 44.64% lower and 14.64% higher than the percent of runners satisfied with the second stretching routine.

  20. 8.39

    1. H0: p1=p2 versus Ha: p1≠p2 ·p^1=0.5797, p^2=0.3725, p^=0.4067. z=3.20. P-value=0.0014. The data provide evidence that, for male customers, the percent who tip a red-shirted server is different than the percent who tip servers wearing shirts of different colors.
    2. H0: p1=p2 versus Ha: p1≠p2. p^1=0.55, p^2=0.70, p^=0.625. z=−0.98. P-value=0.3270. The data do not provide evidence that there is a difference in satisfaction between percent of runners satisfied with each of the two routines.

  21. 8.41

    1. Type of college is explanatory; response is whether physical education is required.
    2. The populations are private and public colleges and universities.
    3. X1=101, n1=129, p^1=0.7829. X2=60, n2=225, p^2=0.2667.
    4. (0.4245, 0.6079).
    5. H0: p1=p2. Ha: p1≠p2; p^=60+101225+129=0.4548, z=9.39, P-value≈0.
    6. All these counts are greater than five. We do not know if the samples were SRSs.

  22. 8.43 (0.0363, 0.1457). Among youth who stress about their health, between 3.6% and 14.6% more are exergamers.

  23. 8.45

    1. X1=947, n1=1064, p^1=0.89. X2=574, n2=1063, p^2=0.54.
    2. 0.35.
    3. Yes.
    4. (0.3146, 0.3854).
    5. 35%; 31.5% to 38.5%.
    6. A possible concern is that adults were surveyed only before Christmas.

  24. 8.47

    1. X1=809, n1=1064, p^1=0.76. X2=776, n2=1063, p^2=0.73.
    2. 0.03.
    3. Yes.
    4. (−0.0070, 0.0670).
    5. 3%; −0.7% to 6.7%.
    6. A possible concern is that adults were surveyed only before Christmas.

  25. 8.49 n=48 for both groups. When p1*=0.65, n=62. When p2*=0.35, n=34.

  26. 8.51

      1. n=197.
      2. n=291.
      3. n=354.
      4. n=385.
      5. n=385.
      6. n=354.
      7. n=291.
      8. 197.
    2. The max sample size was found when p1=0.4 and p2=0.5 or p1=0.5 and p2=0.6.

  27. 8.53 Focusing on those who thought there would be no major increases in gamification by 2020, the 95% CI is (0.3897, 0.4503). With 95% confidence, the proportion of people who think that there will be no major increases in gamification by 2020 is between 38.97% and 45.03%. Focusing on those who believed there would be significant advances in the adoption of gamification by 2020, the 95% CI is (0.4994, 0.5606). With 95% confidence, the proportion of people who believe there will be significant advances in the adoption of gamification by 2020 is between 49.94% and 56.06%. A higher proportion of people think there will be significant adoption of gamification by 2020 than think there will be no major increases in gamification.

  28. 8.55 The populations are children 5 to 10 years old (population 1) and children 11 to 13 years old (population 2). Let Xi=the number who met the requirement in each population. X1=861, n1=1055. X2=417, n2=974.

  29. 8.57 H0: p1=p2 versus Ha: p1≠p2. p^=0.6299. z=18.08. P-value≈0. We have significant evidence that children 5 to 10 years old are more likely than children 11 to 13 years old to get the calcium required in their diets. The test is valid because we have large samples, assuming that both are SRSs.

  30. 8.59

    1. p^=0.4749, SE=0.01599; the 95% CI is (0.4435, 0.5062).
    2. The confidence interval is 44.35% to 50.62%. (c) About 16,632 to 18,983 students.

  31. 8.61

    1. 288.
    2. (0.1431, 0.1769).
    3. About 14.3% to 17.7% of all Internet users use Twitter, at 95% confidence.

  32. 8.63 Racing (0.7141, 0.7659); Puzzle (0.6935, 0.7465); Sports (0.6525, 0.7075); Action (0.6422, 0.6978); Adventure (0.6320, 0.6880); Rhythm (0.5812, 0.6388).

  33. 8.65 (0.0747, 0.1453).

  34. 8.67 New interval for 800 is (0.0687, 0.1513), z=5.11, P-value<0.0001. New interval for 2500 is (0.0785, 0.1415), z=6.91, P-value<0.0001.

  35. 8.69 Answers will vary.

  36. 8.71 The margin of error decreases as sample size increases, but the rate of decrease is noticeably less for large n.

  37. 8.73 Assume SRS. There may be an issue with the assumption of independence in the event that there are multiple births to the same set of parents. Despite these issues, a 95% CI is (0.00175, 0.04923). With the plus four method, the 95% CI is (−0.0020, 0.0489). Significance test has H0: p1=p2 versus Ha: p1>p2. p^=0.02960. z=1.82. P-value=0.0344. Both suggest that the proportions are different.

  38. 8.75 H0: p1=p2 versus Ha: p1≠p2. p^=0.545. z=3.41. P-value<0.0007.