1. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Chapter Hypothesis Tests Regarding a Parameter 10

2. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section The Language of Hypothesis Testing 10.1

3. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-3 1.Determine the null and alternative hypotheses 2.Explain Type I and Type II errors 3.State conclusions to hypothesis tests Objectives

4. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-4 Objective 1 Determine the Null and Alternative Hypotheses

5. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-5 A hypothesis is a statement regarding a characteristic of one or more populations. Ex: A newspaper headline makes the claim that most workers get their jobs through networking. In this chapter, we look at hypotheses regarding a single population parameter.

6. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-6 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. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-7 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

8. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-8 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. 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. Examples of Claims Regarding a Characteristic of a Single Population

9. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-9 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. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-10 Hypothesis testing is a procedure, based on sample evidence and probability, used to test statements regarding a characteristic of one or more populations.

11. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-11 1.Make a statement regarding the nature of the population. 2.Collect evidence (sample data) to test the statement. 3.Analyze the data to assess the plausibility of the statement. Steps in Hypothesis Testing

12. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-12 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 and is assumed true until evidence indicates otherwise. Its states that the value of a population parameter(mean,proportion…) is equal to some claimed value

13. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-13 The alternative hypothesis, denoted H 1, is a statement that we are trying to find evidence to support.

14. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-14 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 1 : parameter ≠ some value

15. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-15 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 1 : parameter ≠ some value 2.Equal versus less than (left-tailed test) H 0 : parameter = some value H 1 : parameter < some value

16. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-16 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 1 : parameter ≠ some value 2.Equal versus less than (left-tailed test) H 0 : parameter = some value H 1 : parameter < some value 3.Equal versus greater than (right-tailed test) H 0 : parameter = some value H 1 : parameter > some value

17. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-17 “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. We seek evidence that supports the statement in the alternative hypothesis.

18. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-18 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 believes that the mean length of a call has increased since then. 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. Parallel Example 2: Forming Hypotheses

19. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-19 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 1 : p≠0.62. Solution

20. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-20 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, μ. 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 : μ = 3.25. Since the researcher believes that the mean call length has increased, the alternative hypothesis is: H 1 : μ > 3.25, a right-tailed test. Solution

21. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-21 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, σ. If the standard deviation with the new equipment has not changed, it will be 0.23 ounces so the null hypothesis is H 0 : σ = 0.23. Since the quality control manager believes that the standard deviation has decreased, the alternative hypothesis is: H 1 : σ < 0.23, a left-tailed test. Solution

22. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-22 Objective 2 Explain Type I and Type II Errors

23. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-23 1.Reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. Four Outcomes from Hypothesis Testing

24. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-24 1.Reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. 2.Do not reject the null hypothesis when the null hypothesis is true. This decision would be correct. Four Outcomes from Hypothesis Testing

25. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-25 1.Reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. 2.Do not reject the null hypothesis when the null hypothesis is true. This decision would be correct. 3.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

26. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-26 1.Reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. 2.Do not reject the null hypothesis when the null hypothesis is true. This decision would be correct. 3.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.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

27. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-27 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

28. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-28 a)In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today. H 0 : p=0.62 H 1 : p≠0.62 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

29. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-29 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 μ > 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

30. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-30 α = P(Type I Error) = P(rejecting H 0 when H 0 is true) β =P(Type II Error) = P(not rejecting H 0 when H 1 is true)

31. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-31 The probability of making a Type I error, α, is chosen by the researcher before the sample data is collected. The level of significance, α, is the probability of making a Type I error.

32. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-32 “In Other Words” As the probability of a Type I error increases, the probability of a Type II error decreases, and vice-versa.

33. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-33 Objective 3 State Conclusions to Hypothesis Tests

34. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-34 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. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-35 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. Parallel Example 4: Stating the Conclusion

36. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-36 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 (μ = 3.25) is rejected, 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

37. Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 10-37 b)Suppose the sample evidence indicates that the null hypothesis should not be rejected. State the wording of the conclusion. Since the null hypothesis (μ = 3.25) is not rejected, 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