Section 9.3 Hypothesis Tests for a Population Mean, Standard Deviation Unknown 433 9. a. P = 0.4532. If H0 is true, then the probability of observing a test statistic as extreme as or more extreme than the value we actually observed is 0.4532. This result is not unusual, so the evidence against H0 is not strong. b. P = 0.0278. If H0 is true, then the probability of observing a test statistic as extreme as or more extreme than the value we actually observed is 0.0278. This result is fairly unusual, so the evidence against H0 is fairly strong. c. z = −2.20 10. ii 11. a. No b. No c. Yes d. Yes 12. a. No b. Yes c. Yes d. No 13. a. Yes b. No c. Yes d. No 14. a. z = 3.27 b. P = 0.0005 c. If H0 is true, then the probability of observing a test statistic greater than or equal to the value we actually observed is 0.0005. This result is very unusual, so the evidence against H0 is very strong. d. Yes e. Yes 15. a. H0: �� = 15, H1: �� ≠ 15 b. z = 2.072750901 c. P = 0.0381953363 d. Yes 16. a. H0: �� = 53.5, H1: �� ≠ 53.5 b. z = −1.29 c. P = 0.196 d. No 17. a. No b. Yes c. Yes d. Yes 18. a. Yes b. No 19. 0.05 20. a. Charlie b. Felice 21. a. Contains enough information b. The value of the test statistic z needs to be added. c. Contains enough information d. The P-value needs to be added. 22. a. z = 3.16, P = 0.0008, so H0 is rejected at the �� = 0.01 level. b. No. The difference between ̄x = 101 and 100 is not large enough to be of practical significance. SECTION 9.3 Hypothesis Tests for a Population Mean, Standard Deviation Unknown Objectives 1. Test a hypothesis about a mean using the P-value method 2. Test a hypothesis about a mean using the critical value method Objective 1 Test a hypothesis about a mean using the P-value method Do low-fat diets work? The following study was reported in the Journal of the American Medical Association (297:969–977). A total of 76 subjects were placed on a low-fat diet. After 12 months, their sample mean weight loss was ̄x = 2.2 kilograms, with a sample standard deviation of s = 6.1 kilograms. How strong is the evidence that people who adhere to this diet will lose weight, on the average? To answer this question, we need to perform a hypothesis test on a population mean. Assume that the subjects in the study constitute a simple random sample from a population of interest. We are interested in their population mean weight loss ��. We know the sample mean ̄x = 2.2. We do not know the population standard deviation ��, but we know that the sample standard deviation is s = 6.1. Because we do not know ��, we cannot use the z-score z = ̄x − �� ��∕ √ n as our test statistic. Instead, we replace �� with the sample standard deviation s and use the t statistic t = ̄x − �� s∕ √ n Recall: When x̄ is the mean of a sample from a normal population, the quantity x̄ − �� s∕ √ n has a Student’s t distribution with n − 1 degrees of freedom. When the null hypothesis is true, the t statistic has a Student’s t distribution with n − 1 degrees of freedom. We described the Student’s t distribution in Section 8.2. When we perform a test using the t statistic, we call the test a t-test. We can perform a t-test for a population mean whenever the following assumptions are satisfied. Assumptions for a Test of a Population Mean �� When �� Is Unknown 1. We have a simple random sample. 2. The sample size is large (n > 30), or the population is approximately normal.
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