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1. What is the essential idea about hypothesis testing for the mean? (p. 498)

2. What does $Z_{\text{data}}$ represent? (p. 499)

3. Explain what a test statistic is. (p. 499)

4. Describe the difference between the critical region and the noncritical region. (p. 500)

5. Clearly describe what $Z_{\text{crit}}$ is. (p. 500)

6. Suppose we reject $H_0$ for the hypothesis test $H_0:\mu =5$ versus $H_a:\mu <5$. Provide the generic interpretation. (p. 502)

7. How did the right-tailed test get its name? (p. 501)

8. True or false: The value of $Z_{\text{crit}}$ does not depend at all on the sample data. (p. 500)

9. $H_0:\mu =75$ vs. $H_a:\mu >75,\hspace{0.17em}\overline{x}=79,\hspace{0.17em}\sigma =20,\hspace{0.17em}n=25$

10. $H_0:\mu =75$ vs. $H_a:\mu >75,\overline{x}=81,\sigma =20,n=25$

11. Right-tailed test with ${\mu }_0=50$ and $\sigma =10$. A sample of size 100 has a mean of 51.

12. Right-tailed test with ${\mu }_0=50$ and $\sigma =10$. A sample of size 100 has a mean of 54.

13. $H_0:\mu =98.6$ vs. $H_a:\mu <98.6,\overline{x}=98.35,\sigma =1,n=16$

14. $H_0:\mu =98.6$ vs. $H_a:\mu <98.6,\overline{x}=97.85,\sigma =1,n=16$

15. Left-tailed test with ${\mu }_0=20$ and $\sigma =5$. A sample of size 100 has a mean of 19.

16. Left-tailed test with ${\mu }_0=20$ and $\sigma =5$. A sample of size 100 has a mean of 18.5.

17. $H_0:\mu =1000$ vs. $H_a:\mu \ne 1000,\overline{x}=1005,\sigma =100,n=400$

18. $H_0:\mu =1000$ vs. $H_a:\mu \ne 1000,\overline{x}=995,\sigma =100,n=400$

19. Two-tailed test with ${\mu }_0=2.5$ and $\sigma =0.5$. A sample of size 100 has a mean of 2.45.

20. Two-tailed test with ${\mu }_0=2.5$ and $\sigma =0.5$. A sample of size 100 has a mean of 2.55.

21. Consider your results from Exercises 9–12. Describe what happens to $Z_{\text{data}}$ for a right-tailed test as $\overline{x}$ increases its distance above ${\mu }_0$, and everything else stays the same.

22. Consider your results from Exercises 13–16. Describe what happens to $Z_{\text{data}}$ for a left-tailed test as $\overline{x}$ increases its distance below ${\mu }_0$, and everything else stays the same.

1. Find the critical value $Z_{\text{crit}}$.
2. Sketch the critical region, using the figures in Table 4 as a guide.
3. State the rejection rule.

23. $H_0:\mu =75$ vs. $H_a:\mu >75$, level of significance $\alpha =0.10$

24. $H_0:\mu =75$ vs. $H_a:\mu >75$, level of significance $\alpha =0.05$

25. Right-tailed test with ${\mu }_0=50$, level of significance $\alpha =0.05$

26. Right-tailed test with ${\mu }_0=50$, level of significance $\alpha =0.01$

27. $H_0:\mu =98.6$ vs. $H_a:\mu <98.6$, level of significance $\alpha =0.05$

28. $H_0:\mu =98.6$ vs. $H_a:\mu <98.6$, level of significance $\alpha =0.01$

29. Left-tailed test with ${\mu }_0=20$, level of significance $\alpha =0.10$

30. Left-tailed test with ${\mu }_0=20$, level of significance $\alpha =0.05$

31. $H_0:\mu =1000$ vs. $H_a:\mu \ne 1000$, level of significance $\alpha =0.05$

32. $H_0:\mu =1000$ vs. $H_a:\mu \ne 1000$, level of significance $\alpha =0.01$

33. Two-tailed test with ${\mu }_0=2.5$, level of significance $\alpha =0.10$

34. Two-tailed test with ${\mu }_0=2.5$, level of significance $\alpha =0.05$

35. Consider your results from Exercises 23–26. Describe what happens to (a) $Z_{\text{crit}}$ and (b) the critical region, for a right-tailed test when the only change is the decrease in the level of significance $\alpha$.

36. Consider your results from Exercises 27–30. Explain what happens to (a) $Z_{\text{crit}}$ and (b) the critical region, for a left-tailed test as the level of significance a decreases but everything else stays the same.

1. State the hypotheses.
2. Find $Z_{\text{crit}}$ and state the rejection rule.
3. State the value of $Z_{\text{data}}$ from the indicated exercise.
4. State the conclusion and the interpretation.

37. Use $Z_{\text{data}}$ from Exercise 9 and $Z_{\text{crit}}$ from Exercise 23.

38. Use $Z_{\text{data}}$ from Exercise 10 and $Z_{\text{crit}}$ from Exercise 24.

39. Use $Z_{\text{data}}$ from Exercise 11 and $Z_{\text{crit}}$ from Exercise 25.

40. Use $Z_{\text{data}}$ from Exercise 12 and $Z_{\text{crit}}$ from Exercise 26.

41. Use $Z_{\text{data}}$ from Exercise 13 and $Z_{\text{crit}}$ from Exercise 27.

42. Use $Z_{\text{data}}$ from Exercise 14 and $Z_{\text{crit}}$ from Exercise 28.

43. Use $Z_{\text{data}}$ from Exercise 15 and $Z_{\text{crit}}$ from Exercise 29.

44. Use $Z_{\text{data}}$ from Exercise 16 and $Z_{\text{crit}}$ from Exercise 30.

45. Use $Z_{\text{data}}$ from Exercise 17 and $Z_{\text{crit}}$ from Exercise 31.

46. Use $Z_{\text{data}}$ from Exercise 18 and $Z_{\text{crit}}$ from Exercise 32.

47. Use $Z_{\text{data}}$ from Exercise 19 and $Z_{\text{crit}}$ from Exercise 33

48. Use $Z_{\text{data}}$ from Exercise 20 and $Z_{\text{crit}}$ from Exercise 34.

1. State the hypotheses.
2. Find $Z_{\text{crit}}$ and the critical region.
3. Find $Z_{\text{data}}$. Also, draw a standard normal $Z$ curve showing $Z_{\text{crit}}$, the critical region, and $Z_{\text{data}}$.
4. State the conclusion and the interpretation.

49. Facebook Connections. According to Facebook.com, the mean number of community pages, groups, and events that users are connected to is 80. A random sample of 64 Facebook users showed a mean of 86 connections to community pages, groups, and events. Assume $\sigma =48$. Test, using level of significance $\alpha =0.05$, whether the population mean number of connections to community pages, groups, and events is greater than 80.

50. Nurses' Study Time. The National Survey of Student Engagement reported that the mean number of hours spent studying per week for nursing majors is 18. Suppose a medical researcher obtained a random sample of 100 nursing majors, which yielded a sample mean study time of 19 hours. Assume $\sigma =10$ hours. Test, using level of significance $\alpha =0.05$, whether the population mean study time for nursing majors is greater than 18 hours per week.

51. Text Messages. The Pew Internet and American Life Project reports that young people ages 12–17 send a mean of 60 text messages per day. A random sample of 100 young people showed a mean of 69 text messages per day. Assume $\sigma =30$. Test, using level of significance $\alpha =0.01$, whether the population mean number of text messages per day differs from 60.

52. Video Gamers. Can't pry the PlayStation away from your dad? In 2015, the Entertainment Software Association reported that the mean age of video gamers was 35 years old. A recent random sample of 36 video gamers had a mean age of 34. Assume $\sigma =6$. Test, using level of significance $\alpha =0.05$, whether the population mean age of video gamers is less than 35.

53. Gas Prices. The American Automobile Association reported in June 2011 that the mean price for a gallon of regular gasoline was $3.70. A recent random sample of 25 gas stations had a mean price of $3.90. Assume normality and $\sigma =\$0.50$. Test, using level of significance $\alpha =0.05$, whether the population mean price for a gallon of regular gasoline has risen since June 2011.

54. Online Shopping. The Nielsen organization reports that smartphone owners spent a mean of 93 minutes using shopping apps on their smartphones in the fourth quarter (last three months) of 2013. A random sample of 225 smartphone owners showed a mean of 99 minutes using shopping apps on their smartphones in the fourth quarter of last year. Assume $\sigma =45\text{ minutes}$. Conduct a hypothesis test, using level of significance $\alpha =0.10$, to determine whether the population mean amount of time has changed.

55. Americans' Height. A random sample of 400 American adults yields a mean height of 176 centimeters. Assume $\sigma =2.5$. Conduct a hypothesis test to investigate whether the population mean height of American adults has changed from 175 centimeters, using level of significance $\alpha =0.10$.

56. Price of Milk. The U.S. Bureau of Labor Statistics reported that the mean price for a gallon of milk in 2011 was $3.34. A random sample of 100 retail establishments this year provides a mean price of $3.39. Assume $\sigma =\$0.25$. Perform a hypothesis test, using level of significance $\alpha =0.05$, to investigate whether the population mean price of milk this year has increased from the 2011 value.

57. Automobile Operation Cost. The Bureau of Transportation Statistics reports that the mean cost of operating an automobile in the United States, including gas and oil, maintenance and tires, is 5.9 cents per mile. Suppose that a sample taken this year of 100 automobiles shows a mean operating cost of 6.2 cents per mile, and assume that the population standard deviation is 1.5 cents per mile. Test whether the population mean cost is greater than 5.9 cents per mile, using level of significance $\alpha =0.05$.

58. Automobile Operation Cost. Refer to Exercise 57.

59. Accountants' Salaries. According to accountingWEB.com, the mean starting salary for a new accountant right out of college was $52,900 in 2014. A random sample of 16 new accountants has a mean salary of $54,000. We assume that the population standard deviation equals $4000. The histogram of the salary (in $1000s) is shown here. If it is appropriate to apply the $Z$ test, then do so, using the critical-value method and level of significance $\alpha =0.05$. If not, then explain clearly why not.

60. Are the conditions met for performing the $Z$ test for $\mu$?

Use technology to obtain a normal probability plot of the data.

61. Construct the appropriate hypotheses. State the meaning of $\mu$.

62. Find the $Z$ critical value and state the rejection rule. Use level of significance $\alpha =0.10$.

63. Calculate the value of $Z_{\text{data}}$.

64. State and interpret your conclusion.

65. Try to answer the following questions by thinking about the relationship between the statistics instead of by redoing all the calculations. Note the mpg of the worst-performing Prius: 48.1. What if the 48.1 mpg is a typo? We are not sure what the actual mpg is, but it is greater than 48.1 mpg. How would this change affect the following?

66. Now, instead of 0.10, use level of significance $\alpha =0.05$. Find the new $Z$ critical value and state the rejection rule.

There is no need to recalculate the value of $Z_{\text{data}}$. State and interpret your conclusion with this new level of significance $\alpha =0.05$.

67. Note the contradiction in your conclusions from Exercises 64 and 66. Can you suggest any way to resolve this?

68. Sodium. Work with the Nutrition data set.

69. Obtain a random sample of size 100 from the data set.

70. Using your sample, test whether the population mean number of coupons used per customer differs from 0.75, using level of significance $\alpha =0.05$. Assume $\sigma =1.74$ coupons.

71. Using the 5000 customers in the entire data set as a population, find the actual value of the population mean number of coupons used per customer. Did your hypothesis test in Exercise 71 make the right decision? Explain.