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

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

3 Focus Points Review the general procedure for testing using P-values. Test  when  is known using the normal distribution.

4 Testing the Mean  In this section, we continue our study of testing the mean . The method we are using is called the P-value method. It was used extensively by the famous statistician R. A. Fisher and is the most popular method of testing in use today.

5 In recent years, the use of this method has been declining. It is important to realize that for a fixed, preset level of significance , both methods are logically equivalent. Let’s quickly review the basic process of hypothesis testing using P-values. 1. We first state a proposed value for a population parameter in the null hypothesis H 0. The alternate hypothesis H 1 states alternative values of the parameter, either, or ≠ the value proposed in H 0. We also set the level of significance . This is the risk we are willing to take of committing a type I error. That is,  is the probability of rejecting H 0 when it is, in fact, true. Testing the Mean 

6 2. We use a corresponding sample statistic from a simple random sample to challenge the statement made in H 0. We convert the sample statistic to a test statistic, which is the corresponding value of the appropriate sampling distribution. 3. We use the sampling distribution of the test statistic and the type of test to compute the P-value of this statistic. Under the assumption that the null hypothesis is true, the P-value is the probability of getting a sample statistic as extreme as or more extreme than the observed statistic from our random sample.

7 Testing the Mean  4. Next, we conclude the test. If the P-value is very small, we have evidence to reject H 0 and adopt H 1. What do we mean by “very small”? We compare the P-value to the preset level of significance . If the P-value  , then we say that we have evidence to reject H 0 and adopt H 1. Otherwise, we say that the sample evidence is insufficient to reject H Finally, we interpret the results in the context of the application.

8 Testing the Mean  Knowing the sampling distribution of the sample test statistic is an essential part of the hypothesis testing process. For tests of , we use one of two sampling distributions for : the standard normal distribution or a Student’s t distribution. The appropriate distribution depends upon our knowledge of the population standard deviation , the nature of the x distribution, and the sample size.

9 The P-value of a Statistical Test

10 The P-value of a Statistical Test The P-value, sometimes called the probability of chance, can be thought of as the probability that the results of a statistical experiment are due only to chance. The lower the P-value, the greater the likelihood of obtaining the same (or very similar) results in a repetition of the statistical experiment.

11 The P-value of a Statistical Test Thus, a low P-value is a good indication that your results are not due to random chance alone. The P-value associated with the observed test statistic takes on different values depending on the alternate hypothesis and the type of test. Let’s look at P-values and types of tests when the test involves the mean and standard normal distribution. Notice that in Example 2, part (c), we computed a P-value for a left-tailed test.

12 The P-value of a Statistical Test

13 The P-value of a Statistical Test cont’d

14 Part I: Testing  when  Is Known

15 Part I: Testing  when  Is Known In most real-world situations,  is simply not known. However, in some cases a preliminary study or other information can be used to get a realistic and accurate value for .

16 Part I: Testing  when  Is Known Procedure:

17 Part I: Testing  when  Is Known cont’d

18 Part I: Testing  when  Is Known We have examined P-value tests for normal distributions with relatively small sample sizes (n < 30). The next example does not assume a normal distribution, but has a large sample size (n  30).

19 Example 3 – Testing ,  known Sunspots have been observed for many centuries. Records of sunspots from ancient Persian and Chinese astronomers go back thousands of years. Some archaeologists think sunspot activity may somehow be related to prolonged periods of drought in the southwestern United States.

20 Example 3 – Testing ,  known Let x be a random variable representing the average number of sunspots observed in a four-week period. A random sample of 40 such periods from Spanish colonial times gave the following data (Reference: M. Waldmeir, Sun Spot Activity, International Astronomical Union Bulletin). cont’d

21 Example 3 – Testing ,  known The sample mean is  47.0 Previous studies of sunspot activity during this period indicate that  = 35. It is thought that for thousands of years, the mean number of sunspots per four-week period was about  = 41. Sunspot activity above this level may (or may not) be linked to gradual climate change. Do the data indicate that the mean sunspot activity during the Spanish colonial period was higher than 41? Use  = cont’d

22 Example 3 – Solution (a) Establish the null and alternate hypotheses. Since we want to know whether the average sunspot activity during the Spanish colonial period was higher than the long-term average of  = 41, H 0 :  = 41 and H 1 :  > 41

23 Example 3 – Solution (b) Check Requirements What distribution do we use for the sample test statistic? Compute the test statistic from the sample data. Since n  30 and we know , we use the standard normal distribution. Using = 47, from the sample,  = 35,  = 41 from H 0, and n = 40, cont’d

24 Example 3 – Solution (c) Find the P-value of the test statistic. Figure 9-3 shows the P-value. Since we have a right-tailed test, the P-value is the area to the right of z = 1.08 shown in Figure 9-3. Using Table 3 of Appendix, we find that P-value = P(z > 1.08)  P-value Area Figure 9-3 cont’d

25 Example 3 – Solution (d) Conclude the test. Since the P-value of > 0.05 for  we do not reject H 0. (e) Interpretation Interpret the results in the context of the problem. At the 5% level of significance, the evidence is not sufficient to reject H 0. Based on the sample data, we do not think the average sunspot activity during the Spanish colonial period was higher than the long-term mean. cont’d

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