470 Chapter 9 Hypothesis Testing 8. In a random sample of 500 people who took their driver’s test, 445 passed. Let p be the population proportion who pass. A test will be made of H0 : p = 0.85 versus H1: p > 0.85. a. Compute the value of the test statistic. 2.50 b. Do you reject H0 at the �� = 0.05 level? Yes c. State a conclusion. 9. For testing H0 : �� = 3 versus H1: �� < 3, a P-value of 0.024 is obtained. a. If the significance level is �� = 0.05, would you conclude that �� < 3? Explain. Yes b. If the significance level is �� = 0.01, would you conclude that �� < 3? Explain. No 10. True or false: When we reject H0, we are certain that H1 is true. False 11. The result of a hypothesis test is reported as follows: ‘‘We reject H0 at the �� = 0.05 level.’’ What additional information should be included? P-value or value of test statistic 12. In a test of H0 : �� = 5 versus H1: �� > 5, the value of the test statistic is t = 2.96. There are 17 degrees of freedom. Do you reject H0 at the �� = 0.05 level? Yes 13. True or false: We can perform a test for a standard deviation only when the population is almost exactly normal. True 14. A random sample of size 20 from a normal population has sample standard deviation s = 10. Test H0 : �� = 15 versus H1: �� < 15. Use the �� = 0.05 level. Reject H0. 15. A test of H0 : �� = 50 versus H1: �� > 50 will be made at a significance level of �� = 0.05. The population standard deviation is �� = 10 and the sample size is n = 60. Find the power of the test against the alternative ��1 = 55. 0.9871 Review Exercises 1. What’s the conclusion? A hypothesis test is performed, and P = 0.02. Which of the following is the best conclusion? i i. H0 is rejected at the 0.05 level. ii. H0 is rejected at the 0.01 level. iii. H1 is rejected at the 0.05 level. iv. H1 is rejected at the 0.01 level. 2. Scoring runs: In 2012, the mean number of runs scored by both teams in a Major League Baseball game was 8.62. Following are the numbers of runs scored in a sample of 24 games in 2013. 2 10 3 9 15 10 7 4 3 7 5 9 5 9 15 15 4 5 13 6 14 11 6 12 a. Construct a boxplot of the data. Is it appropriate to perform a hypothesis test? Yes b. If appropriate, perform a hypothesis test to determine whether the mean number of runs in 2013 is less than it was in 2012. Use the �� = 0.05 level. Do not reject H0. 3. Facebook: A popular blog reports that 60% of college students log in to Facebook on a daily basis. The Dean of Students at a certain university thinks that the proportion may be different at her university. She polls a simple random sample of 200 students, and 134 of them report that they log in to Facebook daily. Can you conclude that the proportion of students who log in to Facebook daily differs from 0.60? a. State the null and alternate hypotheses. H0 : p = 0.6, H1: p ≠ 0.6 b. Compute the value of the test statistic. 2.02 c. Do you reject H0? Use the �� = 0.05 level. Yes d. State a conclusion. 4. Playing the market: The Russell 2000 is a group of 2000 small-company stocks. On June 21, 2013, a random sample of 35 of these stocks had a mean price of $26.89, with a standard deviation of $23.41. A stock market analyst predicted that the mean price of all 2000 stocks would be $25.00. Can you conclude that the mean price differs from $25.00? a. State the null and alternate hypotheses. H0 : �� = 25, H1: �� ≠ 25 b. Should we perform a z-test or a t-test? Explain. t-test c. Compute the value of the test statistic. 0.478 d. Do you reject H0? Use the �� = 0.05 level. No e. State a conclusion. 5. Power: A test of H0 : �� = 100 versus H1: �� > 100 will be made at a significance level of �� = 0.01. The population standard deviation is �� = 50 and the sample size is n = 75. Find the power of the test against the alternative ��1 = 110. 0.2776 Tech: 0.2762 6. More power: Refer to Exercise 5. If the test is made at the �� = 0.05 level with the same sample size, would the power be greater than or less than in Exercise 5? Explain. Greater 7. Household size: For the past several years, the mean number of people in a household has been declining. A social scientist believes that in a certain large city, the mean number of people per household is less than 2.5. To investigate this, she takes a simple
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