Chapter

1. Chapter 10 Hypothesis Tests Regarding a Parameter

2. Section 10.2 Hypothesis Tests for a Population Mean-Population Standard Deviation Known

3. Objectives • Explain the logic of hypothesis testing • Test the hypotheses about a population mean with  known using the classical approach • Test hypotheses about a population mean with  known using P-values • Test hypotheses about a population mean with  known using confidence intervals • Distinguish between statistical significance and practical significance.

4. Objective 1 • Explain the Logic of Hypothesis Testing

5. To test hypotheses regarding the population mean assuming the population standard deviation is known, two requirements must be satisfied: • A simple random sample is obtained. • The population from which the sample is drawn is normally distributed or the sample size is large (n≥30). If these requirements are met, the distribution of is normal with mean  and standard deviation .

6. Recall the researcher who believes that the mean length of a cell phone call has increased from its March, 2006 mean of 3.25 minutes. Suppose we take a simple random sample of 36 cell phone calls. Assume the standard deviation of the phone call lengths is known to be 0.78 minutes. What is the sampling distribution of the sample mean? Answer: is normally distributed with mean 3.25 and standard deviation .

7. Suppose the sample of 36 calls resulted in a sample mean of 3.56 minutes. Do the results of this sample suggest that the researcher is correct? In other words, would it be unusual to obtain a sample mean of 3.56 minutes from a population whose mean is 3.25 minutes? What is convincing or statistically significant evidence?

8. When observed results are unlikely under the assumption that the null hypothesis is true, we say the result is statistically significant. When results are found to be statistically significant, we reject the null hypothesis.

9. The Logic of the Classical Approach One criterion we may use for sufficient evidence for rejecting the null hypothesis is if the sample mean is too many standard deviations from the assumed (or status quo) population mean. For example, we may choose to reject the null hypothesis if our sample mean is more than 2 standard deviations above the population mean of 3.25 minutes.

10. Recall that our simple random sample of 36 calls resulted in a sample mean of 3.56 minutes with standard deviation of 0.13. Thus, the sample mean is standard deviations above the hypothesized mean of 3.25 minutes. Therefore, using our criterion, we would reject the null hypothesis and conclude that the mean cellular call length is greater than 3.25 minutes.

11. Why does it make sense to reject the null hypothesis if the sample mean is more than 2 standard deviations above the hypothesized mean?

12. If the null hypothesis were true, then 1-0.0228=0.9772=97.72% of all sample means will be less than 3.25+2(0.13)=3.51.

13. Because sample means greater than 3.51 are unusual if the population mean is 3.25, we are inclined to believe the population mean is greater than 3.25.

14. The Logic of the P-Value Approach A second criterion we may use for sufficient evidence to support the alternative hypothesis is to compute how likely it is to obtain a sample mean at least as extreme as that observed from a population whose mean is equal to the value assumed by the null hypothesis.

15. We can compute the probability of obtaining a sample mean of 3.56 or more using the normal model.

16. Recall So, we compute The probability of obtaining a sample mean of 3.56 minutes or more from a population whose mean is 3.25 minutes is 0.0087. This means that fewer than 1 sample in 100 will give us a mean as high or higher than 3.56 if the population mean really is 3.25 minutes. Since this outcome is so unusual, we take this as evidence against the null hypothesis.

17. Premise of Testing a Hypothesis Using the P-value Approach Assuming that H0 is true, if the probability of getting a sample mean as extreme or more extreme than the one obtained is small, we reject the null hypothesis.

18. Objective 2 • Test Hypotheses about a Population Mean with  Known Using the Classical Approach

19. Testing Hypotheses Regarding the Population Mean with σ Known Using the Classical Approach To test hypotheses regarding the population mean with  known, we can use the steps that follow, provided that two requirements are satisfied: • The sample is obtained using simple random sampling. • The sample has no outliers, and the population from which the sample is drawn is normally distributed or the sample size is large (n ≥ 30).

20. Step 1:Determine the null and alternative hypotheses. Again, the hypotheses can be structured in one of three ways:

21. Step 2:Select a level of significance, , based on the seriousness of making a Type I error.

22. Step 3:Provided that the population from which the sample is drawn is normal or the sample size is large, and the population standard deviation, , is known, the distribution of the sample mean, , is normal with mean and standard deviation . Therefore, represents the number of standard deviations that the sample mean is from the assumed mean. This value is called the test statistic.

23. Step 4:The level of significance is used to determine the critical value. The critical region represents the maximum number of standard deviations that the sample mean can be from 0 before the null hypothesis is rejected. The critical region or rejection region is the set of all values such that the null hypothesis is rejected.

24. Two-Tailed (critical value)

25. Left-Tailed (critical value)

26. Right-Tailed (critical value)

27. Step 5:Compare the critical value with the test statistic:

28. Step 6:State the conclusion.

29. The procedure is robust, which means that minor departures from normality will not adversely affect the results of the test. However, for small samples, if the data have outliers, the procedure should not be used.

30. Parallel Example 2: The Classical Approach to Hypothesis Testing A can of 7-Up states that the contents of the can are 355 ml. A quality control engineer is worried that the filling machine is miscalibrated. In other words, she wants to make sure the machine is not under- or over-filling the cans. She randomly selects 9 cans of 7-Up and measures the contents. She obtains the following data. 351 360 358 356 359 358 355 361 352 Is there evidence at the =0.05 level of significance to support the quality control engineer’s claim? Prior experience indicates that =3.2ml. Source: Michael McCraith, Joliet Junior College

31. Solution The quality control engineer wants to know if the mean content is different from 355 ml. Since the sample size is small, we must verify that the data come from a population that is approximately normal with no outliers.

32. Normal Probability Plot for Contents (ml) Assumption of normality appears reasonable.

33. No outliers.

34. Solution Step 1:H0: =355 versus H1: ≠355 Step 2: The level of significance is =0.05. Step 3: The sample mean is calculated to be 356.667. The test statistic is then The sample mean of 356.667 is 1.56 standard deviations above the assumed mean of 355 ml.

35. Solution Step 4: Since this is a two-tailed test, we determine the critical values at the =0.05 level of significance to be -z0.025= -1.96 and z0.025=1.96 Step 5: Since the test statistic, z0=1.56, is less than the critical value 1.96, we fail to reject the null hypothesis. Step 6: There is insufficient evidence at the =0.05 level of significance to conclude that the mean content differs from 355 ml.

36. Objective 3 • Test Hypotheses about a Population Mean with  Known Using P-values.

37. A P-valueis the probability of observing a sample statistic as extreme or more extreme than the one observed under the assumption that the null hypothesis is true.

38. Testing Hypotheses Regarding the Population Mean with σ Known Using P-values To test hypotheses regarding the population mean with  known, we can use the steps that follow to compute the P-value, provided that two requirements are satisfied: • The sample is obtained using simple random sampling. • The sample has no outliers, and the population from which the sample is drawn is normally distributed or the sample size is large (n ≥ 30).

39. Step 1:A claim is made regarding the population mean. The claim is used to determine the null and alternative hypotheses. Again, the hypothesis can be structured in one of three ways:

40. Step 2: Select a level of significance, , based on the seriousness of making a Type I error.

41. Step 3:Compute the test statistic,

42. Step 4: Determine the P-value

46. Step 5: Reject the null hypothesis if the P- value is less than the level of significance, . The comparison of the P-value and the level of significance is called the decision rule. Step 6: State the conclusion.

47. Parallel Example 3: The P-Value Approach to Hypothesis Testing: Left-Tailed, Large Sample The volume of a stock is the number of shares traded in the stock in a day. The mean volume of Apple stock in 2007 was 35.14 million shares with a standard deviation of 15.07 million shares. A stock analyst believes that the volume of Apple stock has increased since then. He randomly selects 40 trading days in 2008 and determines the sample mean volume to be 41.06 million shares. Test the analyst’s claim at the =0.10 level of significance using P-values.

48. Solution Step 1: The analyst wants to know if the stock volume has increased. This is a right-tailed test with H0: =35.14 versus H1: >35.14. We want to know the probability of obtaining a sample mean of 41.06 or more from a population where the mean is assumed to be 35.14.

49. Solution Step 2: The level of significance is =0.10. Step 3: The test statistic is

50. Solution Step 4: P(Z > z0)=P(Z > 2.48)=0.0066. The probability of obtaining a sample mean of 41.06 or more from a population whose mean is 35.14 is 0.0066.