SECTION 7.2 Hypothesis Testing for the Mean (s Known) 369 When you cannot find the exact area in Table 4, use the area that is closest. For an area that is exactly midway between two areas in the table, use the z@score midway between the corresponding z@scores. Finding a Critical Value for a Left-Tailed Test Find the critical value and rejection region for a left-tailed test with a = 0.01. SOLUTION The figure shows the standard normal curve with a shaded area of 0.01 in the left tail. In Table 4, the z@score that is closest to an area of 0.01 is -2.33. So, the critical value is z0 = -2.33. The rejection region is to the left of this critical value. You can check your answer using technology, as shown below. EXCEL =NORM.S.INV(0.01) –2.326347874 A 1 TRY IT YOURSELF 7 Find the critical value and rejection region for a left-tailed test with a = 0.10. Answer: Page A41 Because normal distributions are symmetric, in a two-tailed test the critical values are opposites, as shown in the next example. Finding Critical Values for a Two-Tailed Test Find the critical values and rejection regions for a two-tailed test with a = 0.05. SOLUTION The figure shows the standard normal curve with shaded areas of 1 2a = 0.025 in each tail. The area to the left of -z0 is 1 2a = 0.025, and the area to the left of z0 is 1 - 1 2a = 0.975. In Table 4, the z@scores that correspond to the areas 0.025 and 0.975 are -1.96 and 1.96, respectively. So, the critical values are -z0 = -1.96 and z0 = 1.96. The rejection regions are to the left of -1.96 and to the right of 1.96. TRY IT YOURSELF 8 Find the critical values and rejection regions for a two-tailed test with a = 0.08. Answer: Page A41 EXAMPLE 7 z 1 2 3 −1 −2 −3 0 α= 0.01 z0 = −2.33 1% Level of Significance EXAMPLE 8 z 1 2 3 −1 −2 −3 0 α= 0.025 z0 = 1.96 −z0 = −1.96 1 2 α= 0.025 1 2 α 1 − = 0.95 5% Level of Significance Study Tip The table lists the critical values for commonly used levels of significance. Alpha Tail z 0.10 Left Right Two -1.28 1.28 {1.645 0.05 Left Right Two -1.645 1.645 {1.96 0.01 Left Right Two -2.33 2.33 {2.575
RkJQdWJsaXNoZXIy NjM5ODQ=