1. © 2010 Pearson Prentice Hall. All rights reserved STA220: 6.1-6.2 Formulating Hypotheses and Setting Up the Rejection Region

2. © 2010 Pearson Prentice Hall. All rights reserved 10-103 1.Determine the null and alternative hypotheses 2.Explain Type I and Type II errors 3.State conclusions to hypothesis tests Objectives

3. © 2010 Pearson Prentice Hall. All rights reserved 10-104 Objective 1 Determine the Null and Alternative Hypotheses

4. © 2010 Pearson Prentice Hall. All rights reserved 10-105 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.

5. © 2010 Pearson Prentice Hall. All rights reserved 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

6. © 2010 Pearson Prentice Hall. All rights reserved 10-107 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. Examples of Claims Regarding a Characteristic of a Single Population

7. © 2010 Pearson Prentice Hall. All rights reserved 10-108 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. Examples of Claims Regarding a Characteristic of a Single Population

8. © 2010 Pearson Prentice Hall. All rights reserved 10-109 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.

9. © 2010 Pearson Prentice Hall. All rights reserved 10-110 Hypothesis testing is a procedure, based on sample evidence and probability, used to test statements regarding a characteristic of one or more populations.

10. © 2010 Pearson Prentice Hall. All rights reserved 10-111 1.A statement is made regarding the nature of the population. 2.Evidence (sample data) is collected in order to test the statement. 3.The data is analyzed to assess the plausibility of the statement. Steps in Hypothesis Testing

11. © 2010 Pearson Prentice Hall. All rights reserved 10-112 The null hypothesis, denoted H 0, 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.

12. © 2010 Pearson Prentice Hall. All rights reserved 10-113 The alternative hypothesis, denoted H a, 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.

13. © 2010 Pearson Prentice Hall. All rights reserved 10-114 In this chapter, there are three ways to set up the null and alternative hypotheses: 1.Equal versus not equal hypothesis (two-tailed test) H 0 : parameter = some value H a : parameter ≠ some value

14. © 2010 Pearson Prentice Hall. All rights reserved 10-115 In this chapter, there are three ways to set up the null and alternative hypotheses: 1.Equal versus not equal hypothesis (two-tailed test) H 0 : parameter = some value H a : parameter ≠ some value 2.Equal versus less than (left-tailed test) H 0 : parameter = some value H a : parameter < some value

15. © 2010 Pearson Prentice Hall. All rights reserved 10-116 In this chapter, there are three ways to set up the null and alternative hypotheses: 1.Equal versus not equal hypothesis (two-tailed test) H 0 : parameter = some value H a : parameter ≠ some value 2.Equal versus less than (left-tailed test) H 0 : parameter = some value H a : parameter < some value 3.Equal versus greater than (right-tailed test) H 0 : parameter = some value H a : parameter > some value

16. © 2010 Pearson Prentice Hall. All rights reserved 10-117 “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.

17. © 2010 Pearson Prentice Hall. All rights reserved 10-118 For each of the following claims, determine the null and alternative hypotheses. State whether the test is two-tailed, left-tailed or right-tailed. a)In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. 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 wonders if the mean length of a call has increased since then. Forming Hypotheses

18. © 2010 Pearson Prentice Hall. All rights reserved 10-119 a)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 is H 0 : p=0.62. Since the researcher believes that the percentage is different today, the alternative hypothesis is a two-tailed hypothesis: H a : p≠0.62. Solution

19. © 2010 Pearson Prentice Hall. All rights reserved 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 is H 0 :  =3.25. Since the researcher believes that the mean call length has increased, the alternative hypothesis is: H a :  > 3.25, a right-tailed test. Solution

20. © 2010 Pearson Prentice Hall. All rights reserved The test statistic is a sample statistic, computed from information provided in the sample, that the researcher uses to decide between the null and alternative hypotheses.

21. © 2010 Pearson Prentice Hall. All rights reserved The Elements of a Test of Hypotheses If the test statistic has a high probability when H 0 is true, then H 0 is not rejected. If the test statistic has a (very) low probability when H 0 is true, then H 0 is rejected. 122

22. © 2010 Pearson Prentice Hall. All rights reserved 8.1: The Elements of a Test of Hypotheses 123

23. © 2010 Pearson Prentice Hall. All rights reserved 10-124 Objective 2 Explain Type I and Type II Errors

24. © 2010 Pearson Prentice Hall. All rights reserved 10-125 1.We reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. Four Outcomes from Hypothesis Testing

25. © 2010 Pearson Prentice Hall. All rights reserved 10-126 1.We reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. 2.We do not reject the null hypothesis when the null hypothesis is true. This decision would be correct. Four Outcomes from Hypothesis Testing

26. © 2010 Pearson Prentice Hall. All rights reserved 10-127 1.We reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. 2.We do not reject the null hypothesis when the null hypothesis is true. This decision would be correct. 3.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. Four Outcomes from Hypothesis Testing

27. © 2010 Pearson Prentice Hall. All rights reserved 10-128 1.We reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. 2.We do not reject the null hypothesis when the null hypothesis is true. This decision would be correct. 3.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. 4.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. Four Outcomes from Hypothesis Testing

28. © 2010 Pearson Prentice Hall. All rights reserved 10-129 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? a)In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. 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. Parallel Example 3: Type I and Type II Errors

29. © 2010 Pearson Prentice Hall. All rights reserved 10-130 a)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%. Solution

30. © 2010 Pearson Prentice Hall. All rights reserved 10-131 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. Solution

31. © 2010 Pearson Prentice Hall. All rights reserved 10-132  = P(Type I Error) = P(rejecting H 0 when H 0 is true)  =P(Type II Error) = P(not rejecting H 0 when H 1 is true)

32. © 2010 Pearson Prentice Hall. All rights reserved 10-133 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.

33. © 2010 Pearson Prentice Hall. All rights reserved 10-134 “In Other Words” As the probability of a Type I error increases, the probability of a Type II error decreases, and vice-versa.

34. © 2010 Pearson Prentice Hall. All rights reserved 10-135 Objective 3 State Conclusions to Hypothesis Tests

35. © 2010 Pearson Prentice Hall. All rights reserved 10-136 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”.

36. © 2010 Pearson Prentice Hall. All rights reserved 10-137 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)Suppose the sample evidence indicates that the null hypothesis should be rejected. State the wording of the conclusion. b)Suppose the sample evidence indicates that the null hypothesis should not be rejected. State the wording of the conclusion. Stating the Conclusion

37. © 2010 Pearson Prentice Hall. All rights reserved 10-138 a)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. Solution

38. © 2010 Pearson Prentice Hall. All rights reserved 10-139 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. Solution

39. © 2010 Pearson Prentice Hall. All rights reserved The Elements of a Test of Hypotheses Null hypothesis (H 0 ): A theory about the values of one or more parameters Ex.: H 0 : µ = µ 0 (a specified value for µ) Alternative hypothesis (H a ): Contradicts the null hypothesis Ex.: H 0 : µ ≠ µ 0 Test Statistic: The sample statistic to be used to test the hypothesis Rejection region: The values for the test statistic which lead to rejection of the null hypothesis Assumptions: Clear statements about any assumptions concerning the target population Experiment and calculation of test statistic: The appropriate calculation for the test based on the sample data Conclusion: Reject the null hypothesis (with possible Type I error) or do not reject it (with possible Type II error) 140

40. © 2010 Pearson Prentice Hall. All rights reserved The Elements of a Test of Hypotheses Suppose a new interpretation of the rules by soccer referees is expected to increase the number of yellow cards per game. The average number of yellow cards per game had been 4. A sample of 121 matches produced an average of 4.7 yellow cards per game, with a standard deviation of.5 cards. At the 5% significance level, has there been a change in infractions called? 141

41. © 2010 Pearson Prentice Hall. All rights reserved The Elements of a Test of Hypotheses 142

42. © 2010 Pearson Prentice Hall. All rights reserved 8.2: Large-Sample Test of a Hypothesis about a Population Mean 143 H0: µ = µ0 Ha: µ < µ0 Ha: µ ≠ µ0Ha: µ > µ0 The null hypothesis is usually stated as an equality … … even though the alternative hypothesis can be either an equality or an inequality.

43. © 2010 Pearson Prentice Hall. All rights reserved 10-144

44. © 2010 Pearson Prentice Hall. All rights reserved 8.2: Large-Sample Test of a Hypothesis about a Population Mean 145

45. © 2010 Pearson Prentice Hall. All rights reserved 8.2: Large-Sample Test of a Hypothesis about a Population Mean Lower TailedUpper TailedTwo tailed z < - 1.28z > 1.28| z | > 1.645 z < - 1.645z > 1.645| z | > 1.96 z < - 2.33z > 2.33| z | > 2.575 146 Rejection Regions for Common Values of 