Chapter

1. Chapter 10 Hypothesis Tests Regarding a Parameter

2. Section 10.1 The Language of Hypothesis Testing

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

4. 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. 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. 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. 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

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

10. Make a statement regarding the nature of the population. Collect evidence (sample data) to test the statement. Analyze the data to assess the plausibility of the statement. Steps in Hypothesis Testing

11. 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 and is assumed true until evidence indicates otherwise.

12. The alternative hypothesis, denoted H1, is a statement that we are trying to find evidence to support.

13. 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

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 Equal versus less than (left-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 Equal versus greater than (right-tailed test) H0: parameter = some value H1: parameter > some value

16. “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.

17. 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 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. Parallel Example 2: Forming Hypotheses

18. 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. Solution

19. 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 isH0: μ =3.25. Since the researcher believes that the mean call length has increased, the alternative hypothesis is:H1: μ> 3.25, a right-tailed test. Solution

20. 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 isH0: σ= 0.23. Since the quality control manager believes that the standard deviation has decreased, the alternative hypothesis is: H1: σ < 0.23, a left-tailed test. Solution

21. Objective 2 Explain Type I and Type II Errors

22. Reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. Four Outcomes from Hypothesis Testing

23. Reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. Do not reject the null hypothesis when the null hypothesis is true. This decision would be correct. Four Outcomes from Hypothesis Testing

24. Reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. Do not reject the null hypothesis when the null hypothesis is true. This decision would be correct. 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

25. Reject the null hypothesis when the alternative hypothesis is true. This decision would be correct. Do not reject the null hypothesis when the null hypothesis is true. This decision would be correct. 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. 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

26. 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. Parallel Example 3: Type I and Type II Errors

27. 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 thatp ≠ 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

28. 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

29. α = P(Type I Error) = P(rejecting H0 when H0 is true) β = P(Type II Error) = P(not rejecting H0 when H1 is true)

30. 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.

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

32. Objective 3 State Conclusions to Hypothesis Tests

33. 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”.

34. 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. Parallel Example 4: Stating the Conclusion

35. 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

36. 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

37. Section 10.2 Hypothesis Tests for a Population Proportion

38. Objectives Explain the logic of hypothesis testing Test the hypotheses about a population proportion Test hypotheses about a population proportion using the binomial probability distribution.

39. Objective 1 Explain the Logic of Hypothesis Testing

40. A researcher obtains a random sample of 1000 people and finds that 534 are in favor of the banning cell phone use while driving, so = 534/1000. Does this suggest that more than 50% of people favor the policy? Or is it possible that the true proportion of registered voters who favor the policy is some proportion less than 0.5 and we just happened to survey a majority in favor of the policy? In other words, would it be unusual to obtain a sample proportion of 0.534 or higher from a population whose proportion is 0.5? What is convincing, or statistically significant, evidence?

41. 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.

42. To determine if a sample proportion of 0.534 is statistically significant, we build a probability model. Since np(1 – p) = 100(0.5)(1 – 0.5) = 250 ≥ 10 and the sample size (n = 1000) is sufficiently smaller than the population size, we can use the normal model to describe the variability in . The mean of the distribution of is and the standard deviation is

43. Sampling distribution of the sample proportion

44. We may consider the sample evidence to be statistically significant (or sufficient) if the sample proportion is too many standard deviations, say 2, above the assumed population proportion of 0.5. The Logic of the Classical Approach

45. Recall that our simple random sample yielded a sample proportion of 0.534, so standard deviations above the hypothesized proportion of 0.5. Therefore, using our criterion, we would reject the null hypothesis.

46. Why does it make sense to reject the null hypothesis if the sample proportion is more than 2 standard deviations away from the hypothesized proportion? The area under the standard normal curve to the right ofz= 2 is 0.0228.

47. If the null hypothesis were true (population proportion is 0.5), then 1 – 0.0228 = 0.9772 = 97.72% of all sample proportions will be less than .5 + 2(0.016) = 0.532 and only 2.28% of the sample proportions will be more than 0.532.

48. If the sample proportion is too many standard deviations from the proportion stated in the null hypothesis, we reject the null hypothesis. Hypothesis Testing Usingthe Classical Approach

49. A second criterion we may use for testing hypotheses is to determine how likely it is to obtain a sample proportion of 0.534 or higher from a population whose proportion is 0.5. If a sample proportion of 0.534 or higher is unlikely (or unusual), we have evidence against the statement in the null hypothesis. Otherwise, we do not have sufficient evidence against the statement in the null hypothesis. The Logic of the P-Value Approach

50. We can compute the probability of obtaining a sample proportion of 0.534 or higher from a population whose proportion is 0.5 using the normal model.