Chapter

1. Chapter 10 Hypothesis Tests Regarding a Parameter

2. Section 10.1 The Language of Hypothesis Testing

3. Objectives • Determine the null and alternative hypotheses • Explain Type I and Type II errors • State conclusions to hypothesis tests

4. Objective 1 • Determine the Null and Alternative Hypotheses

5. A hypothesis is a statement regarding a characteristic of one or more populations. In this chapter, we look at hypotheses regarding a single population parameter.

6. Examples of Claims Regarding a Characteristic of a Single Population • In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. Source: ReadersDigest.com poll created on 2008/05/02

7. Examples of Claims Regarding a Characteristic of a Single Population • In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. • According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

8. Examples of Claims Regarding a Characteristic of a Single Population • In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. • According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher wonders if the mean length of a call has increased since then. • Using an old manufacturing process, the standard deviation of the amount of wine put in a bottle was 0.23 ounces. With new equipment, the quality control manager believes the standard deviation has decreased.

9. CAUTION! We test these types of statements using sample data because it is usually impossible or impractical to gain access to the entire population. If population data are available, there is no need for inferential statistics.

10. Hypothesistesting is a procedure, based on sample evidence and probability, used to test statements regarding a characteristic of one or more populations.

11. Steps in Hypothesis Testing • A statement is made regarding the nature of the population. • Evidence (sample data) is collected in order to test the statement. • The data is analyzed to assess the plausibility of the statement.

12. The null hypothesis, denoted H0, is a statement to be tested. The null hypothesis is a statement of no change, no effect or no difference. The null hypothesis is assumed true until evidence indicates otherwise. In this chapter, it will be a statement regarding the value of a population parameter.

13. The alternative hypothesis, denoted H1, is a statement that we are trying to find evidence to support. In this chapter, it will be a statement regarding the value of a population parameter.

14. In this chapter, there are three ways to set up the null and alternative hypotheses: • Equal versus not equal hypothesis (two-tailed test) H0: parameter = some value H1: parameter ≠ some value

15. In this chapter, there are three ways to set up the null and alternative hypotheses: • Equal versus not equal hypothesis (two-tailed test) H0: parameter = some value H1: parameter ≠ some value • Equal versus less than (left-tailed test) H0: parameter = some value H1: parameter < some value

16. In this chapter, there are three ways to set up the null and alternative hypotheses: • Equal versus not equal hypothesis (two-tailed test) H0: parameter = some value H1: parameter ≠ some value • Equal versus less than (left-tailed test) H0: parameter = some value H1: parameter < some value • Equal versus greater than (right-tailed test) H0: parameter = some value H1: parameter > some value

17. “In Other Words” The null hypothesis is a statement of “status quo” or “no difference” and always contains a statement of equality. The null hypothesis is assumed to be true until we have evidence to the contrary. The claim that we are trying to gather evidence for determines the alternative hypothesis.

18. Parallel Example 2: Forming Hypotheses For each of the following claims, determine the null and alternative hypotheses. State whether the test is two-tailed, left-tailed or right-tailed. • In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. • According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher wonders if the mean length of a call has increased since then. • Using an old manufacturing process, the standard deviation of the amount of wine put in a bottle was 0.23 ounces. With new equipment, the quality control manager believes the standard deviation has decreased.

19. Solution • In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. The hypothesis deals with a population proportion, p. If the percentage participating in charity work is no different than in 2008, it will be 0.62 so the null hypothesis isH0: p=0.62. Since the researcher believes that the percentage is different today, the alternative hypothesis is a two-tailed hypothesis: H1: p≠0.62.

20. Solution b) According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then. The hypothesis deals with a population mean, . If the mean call length on a cellular phone is no different than in 2006, it will be 3.25 minutes so the null hypothesis isH0: =3.25. Since the researcher believes that the mean call length has increased, the alternative hypothesis is: H1:  > 3.25, a right-tailed test.

21. Solution c) Using an old manufacturing process, the standard deviation of the amount of wine put in a bottle was 0.23 ounces. With new equipment, the quality control manager believes the standard deviation has decreased. The hypothesis deals with a population standard deviation, . If the standard deviation with the new equipment has not changed, it will be 0.23 ounces so the null hypothesis isH0:  = 0.23. Since the quality control manager believes that the standard deviation has decreased, the alternative hypothesis is: H1:  < 0.23, a left-tailed test.

22. Objective 2 • Explain Type I and Type II Errors

23. Four Outcomes from Hypothesis Testing • We reject the null hypothesis when the alternative hypothesis is true. This decision would be correct.

24. Four Outcomes from Hypothesis Testing • We reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. • We do not reject the null hypothesis when the null hypothesis is true. This decision would be correct.

25. Four Outcomes from Hypothesis Testing • We reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. • We do not reject the null hypothesis when the null hypothesis is true. This decision would be correct. • We reject the null hypothesis when the null hypothesis is true. This decision would be incorrect. This type of error is called a Type I error.

26. Four Outcomes from Hypothesis Testing • We reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. • We do not reject the null hypothesis when the null hypothesis is true. This decision would be correct. • We reject the null hypothesis when the null hypothesis is true. This decision would be incorrect. This type of error is called a Type I error. • We do not reject the null hypothesis when the alternative hypothesis is true. This decision would be incorrect. This type of error is called a Type II error.

27. Parallel Example 3: Type I and Type II Errors For each of the following claims, explain what it would mean to make a Type I error. What would it mean to make a Type II error? • In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. • According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

28. Solution • In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. A Type I error is made if the researcher concludes that p≠0.62 when the true proportion of Americans 18 years or older who participated in some form of charity work is currently 62%. A Type II error is made if the sample evidence leads the researcher to believe that the current percentage of Americans 18 years or older who participated in some form of charity work is still 62% when, in fact, this percentage differs from 62%.

29. Solution b) According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then. A Type I error occurs if the sample evidence leads the researcher to conclude that >3.25 when, in fact, the actual mean call length on a cellular phone is still 3.25 minutes. A Type II error occurs if the researcher fails to reject the hypothesis that the mean length of a phone call on a cellular phone is 3.25 minutes when, in fact, it is longer than 3.25 minutes.

30.  = P(Type I Error) = P(rejecting H0 when H0 is true)  = P(Type II Error) = P(not rejecting H0 when H1 is true)

31. The probability of making a Type I error, , is chosen by the researcher before the sample data is collected. The level of significance, , is the probability of making a Type I error.

32. “In Other Words” As the probability of a Type I error increases, the probability of a Type II error decreases, and vice-versa.

33. Objective 3 • State Conclusions to Hypothesis Tests

34. CAUTION! We never “accept” the null hypothesis because without having access to the entire population, we don’t know the exact value of the parameter stated in the null. Rather, we say that we do not reject the null hypothesis. This is just like the court system. We never declare a defendant “innocent”, but rather say the defendant is “not guilty”.

35. Parallel Example 4: Stating the Conclusion According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then. • Suppose the sample evidence indicates that the null hypothesis should be rejected. State the wording of the conclusion. • Suppose the sample evidence indicates that the null hypothesis should not be rejected. State the wording of the conclusion.

36. Solution • Suppose the sample evidence indicates that the null hypothesis should be rejected. State the wording of the conclusion. The statement in the alternative hypothesis is that the mean call length is greater than 3.25 minutes. Since the null hypothesis (=3.25) is rejected, we conclude that there is sufficient evidence to conclude that the mean length of a phone call on a cell phone is greater than 3.25 minutes.

37. Solution b) Suppose the sample evidence indicates that the null hypothesis should not be rejected. State the wording of the conclusion. Since the null hypothesis (=3.25) is not rejected, we conclude that there is insufficient evidence to conclude that the mean length of a phone call on a cell phone is greater than 3.25 minutes. In other words, the sample evidence is consistent with the mean call length equaling 3.25 minutes.

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

39. 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.

40. Objective 1 • Explain the Logic of Hypothesis Testing

41. 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 .

42. 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 .

43. 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?

44. 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.

45. 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.

46. 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.

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

48. 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.

49. 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.

50. 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.