1. Copyright © Cengage Learning. All rights reserved. Hypothesis Testing 9

2. Copyright © Cengage Learning. All rights reserved. Section 9.2 Testing the Mean 

3. 3 Focus Points Test  when  is known using the normal distribution. Understand the “traditional” method of testing that uses critical regions and critical values instead of P-values.

4. 4 Testing  Using Critical Regions (Traditional Method)

5. 5 Part III: Testing  Using Critical Regions (Traditional Method) The most popular method of statistical testing is the P-value method. For that reason, the P-value method is emphasized in this book. Another method of testing is called the critical region method or traditional method. For a fixed, preset value of the level of significance , both methods are logically equivalent. Because of this, we treat the traditional method as an “optional” topic and consider only the case of testing  when  is known.

6. 6 Part III: Testing  Using Critical Regions (Traditional Method) Consider the null hypothesis H 0 :  = k. We use information from a random sample, together with the sampling distribution for and the level of significance , to determine whether or not we should reject the null hypothesis. The essential question is, “How much can vary from  = k before we suspect that H 0 :  = k is false and reject it?”

7. 7 Part III: Testing  Using Critical Regions (Traditional Method) The answer to the question regarding the relative sizes of and , as stated in the null hypothesis, depends on the sampling distribution of, the alternate hypothesis H 1, and the level of significance . If the sample test statistic is sufficiently different from the claim about  made in the null hypothesis, we reject the null hypothesis. The values of for which we reject H 0 are called the critical region of the distribution.

8. 8 Part III: Testing  Using Critical Regions (Traditional Method) Depending on the alternate hypothesis, the critical region is located on the left side, the right side, or both sides of the, distribution. Figure 9-7 shows the relationship of the critical region to the alternate hypothesis and the level of significance . Critical Regions for H 0 :  = k Figure 9-7

9. 9 Part III: Testing  Using Critical Regions (Traditional Method) Notice that the total area in the critical region is preset to be the level of significance . Recall that the level of significance  should (in theory) be a fixed, preset number assigned before drawing any samples.

10. 10 Part III: Testing  Using Critical Regions (Traditional Method) The most commonly used levels of significance are  = 0.05 and  = 0.01. Critical regions of a standard normal distribution are shown for these levels of significance in Figure 9-8. Critical values are the boundaries of the critical region. Critical values designated as z 0 for the standard normal distribution are shown in Figure 9-8. Level of significance Critical Values z 0 for Tests Involving a Mean (Large Samples) Figure 9-8

11. 11 Part III: Testing  Using Critical Regions (Traditional Method) Critical Values z 0 for Tests Involving a Mean (Large Samples) Level of significance Figure 9-8

12. 12 Part III: Testing  Using Critical Regions (Traditional Method) For easy reference, they are also included in Table 3 of the Appendix, “Hypothesis Testing Critical Values z 0.” The procedure for hypothesis testing using critical regions follows the same first two steps as the procedure using P-values. However, instead of finding a P-value for the sample test statistic, we check if the sample test statistic falls in the critical region. If it does, we reject H 0. Otherwise, we do not reject H 0.

13. 13 Part III: Testing  Using Critical Regions (Traditional Method) Procedure:

14. 14 Part III: Testing  Using Critical Regions (Traditional Method) cont’d

15. 15 Example 5 – Critical region method of testing  Let x be a random variable representing the number of sunspots observed in a four-week period. A random sample of 40 such periods from Spanish colonial times gave the number of sunspots per period. The sample mean is  47.0. Previous studies indicate that for this period,  = 35. It is thought that for thousands of years, the mean number of sunspots per four-week period was about  = 41. Do the data indicate that the mean sunspot activity during the Spanish colonial period was higher than 41? Use  = 0.05.

16. 16

17. 17 Example 5 – Critical region method of testing  Solution: (a) Set the null and alternate hypotheses. As in Example 3, we use H 0 :  = 41and H 1 :  > 41

18. 18 Example 5 – Solution (b) Compute the sample test statistic. We use the standard normal distribution, with x = 47  = 35,  = 41 from H 0, and n = 40 cont’d

19. 19 Example 5 – Solution (c) Determine the critical region and critical value based on H 1 and  = 0.05. Since we have a right-tailed test, the critical region is the rightmost 5% of the standard normal distribution. According to Figure 9-8, the critical value is z 0 = 1.645. (d) Conclude the test. We conclude the test by showing the critical region, critical value, and sample test statistic z = 1.08 on the standard normal curve. cont’d

20. 20 Example 5 – Solution For a right-tailed test with  = 0.05 the critical value is z 0 = 1.645 Figure 9-9 shows the critical region. As we can see, the sample test statistic does not fall in the critical region. Therefore, we fail to reject H 0. Critical Region,  = 0.05 Figure 9-9 cont’d

21. 21 Example 5 – Solution (e) Interpretation Interpret the results in the context of the application. At the 5% level of significance, the sample evidence is insufficient to justify rejecting H 0. It seems that the average sunspot activity during the Spanish colonial period was the same as the historical average. cont’d

22. 22 Part III: Testing  Using Critical Regions (Traditional Method) Procedure:

23. 23 Part III: Testing  Using Critical Regions (Traditional Method) Procedure: cont’d

24. 24 Concluding a Statistical Test “Fail to reject the null hypothesis” simply means that the evidence in favor of rejection was not strong enough (see Table 9-4). Table 9-4 Meaning of the Terms Fail to Reject H 0 and Reject H 0

25. 25