1. Rejection Regions and Critical Values Rejection region (or critical region) The range of values for which the null hypothesis is not probable. If a test statistic falls in this region, the null hypothesis is rejected. A critical value z 0 separates the rejection region from the nonrejection region. © 2012 Pearson Education, Inc. All rights reserved. 1 of 101

2. Rejection Regions and Critical Values Finding Critical Values in a Normal Distribution 1.Specify the level of significance . 2.Decide whether the test is left-, right-, or two-tailed. 3.Find the critical value(s) z 0. If the hypothesis test is a.left-tailed, find the z-score that corresponds to an area of , b.right-tailed, find the z-score that corresponds to an area of 1 – , c.two-tailed, find the z-score that corresponds to ½  and 1 – ½ . 4.Sketch the standard normal distribution. Draw a vertical line at each critical value and shade the rejection region(s). © 2012 Pearson Education, Inc. All rights reserved. 2 of 101

3. Example: Finding Critical Values Find the critical value and rejection region for a two-tailed test with  = 0.05. z 0z0z0 z0z0 The rejection regions are to the left of -z 0 = and to the right of z 0 =. Solution: © 2012 Pearson Education, Inc. All rights reserved. 3 of 101

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5. Decision Rule Based on Rejection Region To use a rejection region to conduct a hypothesis test, calculate the standardized test statistic, z. If the standardized test statistic 1. is in the rejection region, then reject H 0. 2. is not in the rejection region, then fail to reject H 0. z 0 z0z0 Fail to reject H 0. Reject H 0. Left-Tailed Test z < z 0 z 0 z0z0 Reject H o. Fail to reject H o. z > z 0 Right-Tailed Test z 0 z0z0 Two-Tailed Test z0z0 z < -z 0 z > z 0 Reject H 0 Fail to reject H 0 Reject H 0 © 2012 Pearson Education, Inc. All rights reserved. 5 of 101

6. Using Rejection Regions for a z-Test for a Mean μ 1.State the claim mathematically and verbally. Identify the null and alternative hypotheses. 2.Specify the level of significance. 3.Sketch the sampling distribution. 4.Determine the critical value(s). 5.Determine the rejection region(s). State H 0 and H a. Identify . Use Table 4 in Appendix B. In WordsIn Symbols © 2012 Pearson Education, Inc. All rights reserved. 6 of 101

7. Using Rejection Regions for a z-Test for a Mean μ 6.Find the standardized test statistic. 7.Make a decision to reject or fail to reject the null hypothesis. 8.Interpret the decision in the context of the original claim. If z is in the rejection region, reject H 0. Otherwise, fail to reject H 0. In WordsIn Symbols © 2012 Pearson Education, Inc. All rights reserved. 7 of 101

8. Example: Testing with Rejection Regions Employees at a construction and mining company claim that the mean salary of the company’s mechanical engineers is less than that of the one of its competitors, which is $68,000. A random sample of 30 of the company’s mechanical engineers has a mean salary of $66,900 with a standard deviation of $5500. At α = 0.05, test the employees’ claim. © 2012 Pearson Education, Inc. All rights reserved. 8 of 101

9. Solution: Testing with Rejection Regions

10. Example: Testing with Rejection Regions The U.S. Department of Agriculture claims that the mean cost of raising a child from birth to age 2 by husband-wife families in the U.S. is $13,120. A random sample of 500 children (age 2) has a mean cost of $12,925 with a standard deviation of $1745. At α = 0.10, is there enough evidence to reject the claim? (Adapted from U.S. Department of Agriculture Center for Nutrition Policy and Promotion) © 2012 Pearson Education, Inc. All rights reserved. 10 of 101

11. Solution: Testing with Rejection Regions

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13. Section 7.2 Summary Found P-values and used them to test a mean μ Used P-values for a z-test Found critical values and rejection regions in a normal distribution Used rejection regions for a z-test © 2012 Pearson Education, Inc. All rights reserved. 13 of 101