Section 9.2 Hypothesis Tests for a Population Mean, Standard Deviation Known 419 We now summarize the steps in testing a hypothesis with the P-value method. Performing a Hypothesis Test for a Population Mean with �� Known Using the P-Value Method Check to be sure the assumptions are satisfied. If they are, then proceed with the following steps. Step 1: State the null and alternate hypotheses. The null hypothesis specifies a value for the population mean ��. We will call this value ��0. So the null hypothesis is of the form H0 : �� = ��0. The alternate hypothesis can be stated in one of three ways: Left-tailed: H1: �� < ��0 Right-tailed: H1: �� > ��0 Two-tailed: H1: �� ≠ ��0 Step 2: If making a decision, choose a significance level ��. Step 3: Compute the test statistic z = ̄x − ��0 ��∕ √ n . Step 4: Compute the P-value of the test statistic. The P-value is the probability, assuming that H0 is true, of observing a value for the test statistic that is as extreme or more extreme than the value actually observed. The P-value is an area under the standard normal curve; it depends on the type of alternate hypothesis. Note that the inequality in the alternate hypothesis points in the direction of the tail that contains the area for the P-value. The P-value is the area to the left of z. z The P-value is the area to the right of z. The P-value is the sum of the areas in the two tails. z −|z| |z| Left-tailed: H1: �� < ��0 Right-tailed: H1: �� > ��0 Two-tailed: H1: �� ≠ ��0 Step 5: Interpret the P-value. If making a decision, reject H0 if the P-value is less than or equal to the significance level ��. Step 6: State a conclusion. EXAMPLE 9.13 Perform a hypothesis test The National Health and Nutrition Examination Surveys (NHANES) are designed to assess the health and nutritional status of adults and children in the United States. According to a recent NHANES survey, the mean height of adult men in the United States is 69.7 inches, with a standard deviation of 3 inches. A sociologist believes that taller men may be more likely to be promoted to positions of leadership, so the mean height �� of male business executives may be greater than the mean height of the entire male population. A simple random sample of 100 male business executives has a mean height of 69.9 in. Assume that the standard deviation of male executive heights is �� = 3 inches. Can we conclude that male business executives are taller, on the average, than the general male population at the �� = 0.05 level? Solution We first check the assumptions. We have a simple random sample, the sample size is large (n > 30), and the population standard deviation is known. The assumptions are satisfied. Step 1: State H0 and H1. The null hypothesis, H0, says that there is no difference between the mean heights of executives and others. Therefore, we have H0 : �� = 69.7 We are interested in determining whether the mean height of executives is greater than 69.7. Therefore, we have H1: �� > 69.7 At this point, we assume that H0 is true.
navidi_monk_elementary_statistics_2e_ch7-9
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