Hypothesis Testing for the Mean (s Unknown) 7.3 SECTION 7.3 Hypothesis Testing for the Mean (s Unknown) 377 What You Should Learn How to find critical values in a t-distribution How to use the t-test to test a mean m when s is not known How to use technology to find P-values and use them with a t-test to test a mean m when s is not known Critical Values in a t@Distribution The t@Test for a Mean m Using P-Values with t@Tests Critical Values in a t@Distribution In Section 7.2, you learned how to perform a hypothesis test for a population mean when the population standard deviation is known. In many real-life situations, the population standard deviation in not known. When either the population has a normal distribution or the sample size is at least 30, you can still test the population mean m. To do so, you can use the t@distribution with n - 1 degrees of freedom. Finding Critical Values in a t@Distribution 1. Specify the level of significance a. 2. Identify the degrees of freedom, d.f. = n - 1. 3. Find the critical value(s) using Table 5 in Appendix B in the row with n - 1 degrees of freedom. When the hypothesis test is a. left-tailed, use the “One Tail, a” column with a negative sign. b. right-tailed, use the “One Tail, a” column with a positive sign. c. two-tailed, use the “Two Tails, a” column with a negative and a positive sign. See the figures below. GUIDELINES Finding a Critical Value for a Left-Tailed Test Find the critical value t0 for a left-tailed test with a = 0.05 and n = 21. SOLUTION The degrees of freedom are d.f. = n - 1 = 21 - 1 = 20. To find the critical value, use Table 5 in Appendix B with d.f. = 20 and a = 0.05 in the “One Tail, a” column. Because the test is left-tailed, the critical value is negative. So, t0 = -1.725, as shown in the figure at the left. TRY IT YOURSELF 1 Find the critical value t0 for a left-tailed test with a = 0.01 and n = 14. Answer: Page A41 5% Level of Significance t −3 −2 −1 0 1 2 3 α= 0.05 = −1.725 t0 t 0 α t 0 t 0 α t 0 t 0 α −t 0 t 0 1 2 α 1 2 Left-Tailed Test Right-Tailed Test Two-Tailed Test EXAMPLE 1
RkJQdWJsaXNoZXIy NjM5ODQ=