440 Chapter 9 Hypothesis Testing Objective 2 Test a hypothesis about a mean using the critical value method Testing a Hypothesis About a Population Mean Using the Critical Value Method The critical value method when �� is unknown is the same as that when �� is known, except that we use the Student’s t distribution rather than the normal distribution. The critical value can be found in Table A.3 or with technology. The procedure depends on whether the alternate hypothesis is left-tailed, right-tailed, or two-tailed. Critical Values for the t Statistic Let �� denote the chosen significance level and let n denote the sample size. The critical value depends on whether the alternate hypothesis is left-tailed, right-tailed, or two-tailed. We use the Student’s t distribution with n−1 degrees of freedom. Critical region: Area = α −tα For left-tailed H1: The critical value is −t��, which has area �� to its left. Reject H0 if t ≤ −t��. Critical region: tα Area = α For right-tailed H1: The critical value is t��, which has area �� to its right. Reject H0 if t ≥ t��. Critical region: Area = α/2 Critical region: Area = α/2 −tα/2 tα/2 For two-tailed H1: The critical values are t��∕2, which has area ��∕2 to its right, and −t��∕2, which has area ��∕2 to its left. Reject H0 if t ≥ t��∕2 or t ≤ −t��∕2. Check Your Understanding 6. Find the critical value or values for the following values of the significance level ��, sample size n, and alternate hypothesis H1. a. �� = 0.05, n = 3, H1: �� > ��0 2.920 b. �� = 0.01, n = 26, H1: �� ≠ ��0 −2.787, 2.787 c. �� = 0.10, n = 81, H1: �� < ��0 −1.292 d. �� = 0.05, n = 14, H1: �� ≠ ��0 −2.160, 2.160 Answers are on page 447. The assumptions for the critical value method are the same as those for the P-value method. We repeat them here. 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. When these assumptions are satisfied, a hypothesis test can be performed using the following steps. Performing a Hypothesis Test on a Population Mean with �� Unknown Using the Critical Value Method Check to be sure that 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
navidi_monk_elementary_statistics_2e_ch7-9
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