434 Chapter 9 Hypothesis Testing When these assumptions are met, a hypothesis test can be performed. Either the critical value method or the P-value method may be used. Following are the steps for the P-value method. Performing a Hypothesis Test on a Population Mean with �� Unknown Using the P-Value Method Check to determine whether 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 t = ̄x − ��0 s∕ √ 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 disagrees as strongly as or more strongly with H0 than the value actually observed. The P-value is an area under the Student’s t curve with n − 1 degrees of freedom. The area is in the left tail, the right tail, or in both tails, depending on the type of alternate hypothesis. Note that the inequality 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 t. t The P-value is the area to the right of t. The P-value is the sum of the areas in the two tails. t −|t| |t| 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 9.16 Perform a hypothesis test In a recent medical study, 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. Can we conclude that the mean weight loss is greater than 0? Use the �� = 0.05 level of significance. Source: Journal of the American Medical Association 297:969–977 Solution We first check the assumptions. We have a simple random sample. The sample size is 76, so n > 30. The assumptions are satisfied. Step 1: State H0 and H1. The issue is whether the mean weight loss �� is greater than 0. So the null and alternate hypotheses are H0 : �� = 0 H1: �� > 0 Note that we have a right-tailed test, because we are particularly interested in whether the diet results in a weight loss.
navidi_monk_elementary_statistics_2e_ch7-9
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