Section 9.2 Hypothesis Tests for a Population Mean, Standard Deviation Known 423 �������� ���� ���� = ���� �� ���� ������ = ���� �� ������ �������������� ���������������� ������������������ = �� �������� �� *�� ���� �� �� ������ �� �������� ������ ���� ������ ���� �������� �� �������� ����~ ���� .���� ������, ���� ������/ a. What are the null and alternate hypotheses? H0 : �� = 53.5, H1: �� ≠ 53.5 b. What is the value of the test statistic? −1.29 c. What is the P-value? 0.196 d. Do you reject H0 at the �� = 0.05 level? No Answers are on page 433. Objective 3 Describe the relationship between hypothesis tests and confidence intervals The Relationship Between Hypothesis Tests and Confidence Intervals In Example 9.14, we tested the hypotheses H0 : �� = 74 versus H1: �� ≠ 74 and obtained a P-value of 0.025. Because P < 0.05, H0 is rejected at the 0.05 level. Informally, this says that the value 74 is not plausible for ��. Another way to express information about �� is through a confidence interval. A 95% confidence interval for �� is 74.247 < �� < 77.753. (This confidence interval is displayed in the MINITAB output following Example 9.14.) Note that the 95% confidence interval does not contain the null hypothesis value of 74. In this way, the 95% confidence interval agrees with the results of the hypothesis test. Informally, a confidence interval for �� contains all the values that are plausible for ��. Because 74 is not in the confidence interval, 74 is not a plausible value for ��. This relationship holds for any confidence interval for a population mean, and any two-tailed hypothesis test. If we test H0 : �� = ��0 versus H1: �� ≠ ��0, then ∙ If the 95% confidence interval contains ��0, then H0 will not be rejected at the 0.05 level. ∙ If the 95% confidence interval does not contain ��0, then H0 will be rejected at the 0.05 level. This relationship between hypothesis tests and confidence intervals holds exactly for population means, but only approximately for other parameters such as population proportions. The reason is that the standard error that is used in a hypothesis test for a proportion differs somewhat from the standard error that is used in a confidence interval for a proportion. Hypothesis tests and confidence intervals address different questions Although hypothesis tests are closely related to confidence intervals, the two address different questions. A confidence interval provides all of the values that are plausible at a specified level. A hypothesis test tells us about only one value, but it tells us much more precisely how plausible that one value is. For example, consider the 95% confidence interval 74.247 < �� < 77.753 previously mentioned for the mean satisfaction level in Example 9.14. The value �� = 74 is not in the confidence interval, so we can conclude that the hypothesis H0 : �� = 74 will be rejected at the �� = 0.05 level with a two-tailed test. However, this tells us only that P < 0.05. It does not tell us exactly how much less than 0.05 the P-value is. By performing the hypothesis test, we find that P = 0.025. This tells us much more precisely just how plausible or implausible the value of 74 is for ��.
navidi_monk_elementary_statistics_2e_ch7-9
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