1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley  Be able to identify the parameter of interest and write both Hypotheses for your test  Be able to check the appropriate conditions for a one-proportion z-test AP Statistics Objectives Ch21

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley  Be able to calculate the test statistic and P-value for your test  Know when to reject or fail to reject your null hypothesis  Be able to state the conclusion of your test in context AP Statistics Objectives Ch21

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley  Be able to interpret your P- value  Understand how to use PHANTOMS to preform a one-proportion z-test AP Statistics Objectives Ch21

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley  Null Hypothesis  Alternative Hypothesis  Two-sided alternative  One-sided alternative  P-value  One-proportion z-test Vocabulary

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Classroom Notes Chapter 21 Assignments Chapter 21 Answers False Positive – Medicine Tesing Diagram 2 Diagram 1

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 20- 6 Chapter 20 Assignments Part I Pages 491-492 #6, 8,10,12 Part II Pages 492-493 #14,16, 20, 24

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 7 Zero In on the Null 1. How do we choose the null hypothesis? There is a temptation to state your claim as the null hypothesis. DON’T! You cannot prove a null hypothesis true. So, it makes more sense to use what you want to show as the alternative. This way, when you reject the null, you are left with what you want to show.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 8 2. How to Think About P-Values A P-value is a conditional probability—the probability of the observed statistic given that the null hypothesis is true. The P-value is NOT the probability that the null hypothesis is true. It’s not even the conditional probability that null hypothesis is true given the data. Be careful to interpret the P-value correctly.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 9 Alpha Levels 3. When the P-value is small, it tells us that our data are rare given the null hypothesis. How rare is “rare”? We can define “rare event” by setting a arbitrary threshold for our P-value. If our P-value falls below that point, we’ll reject H 0. We call such results statistically significant. The threshold is called an alpha level, denoted by .

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 10 Alpha Levels (cont.) 4. Common alpha levels are 0.10, 0.05, and 0.01. The alpha level is also called the significance level. When we reject the null hypothesis, we say that the test is “significant at that level.”

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 11 Alpha Levels (cont.) 5. What can you say if the P-value does not fall below  ? You should say that “The data have failed to provide sufficient evidence to reject the null hypothesis.” Don’t say that you “accept the null hypothesis.” Recall that, in a jury trial, if we do not find the defendant guilty, we say the defendant is “not guilty”—we don’t say that the defendant is “innocent.”

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 12 Alpha Levels (cont.) 6. The P-value gives the reader far more information than just stating that you reject or fail to reject the null. In fact, by providing a P-value to the reader, you allow that person to make his or her own decisions about the test. What you consider to be statistically significant might not be the same as what someone else considers statistically significant. There is more than one alpha level that can be used, but each test will give only one P-value.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 13 What Not to Say About Significance 7. What do we mean when we say that a test is statistically significant? All we mean is that the test statistic had a P-value lower than our alpha level. For large samples, even small, unimportant (“insignificant”) deviations from the null hypothesis can be statistically significant. If the sample is not large enough, even large, financially or scientifically “significant” differences may not be statistically significant.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 14 What Not to Say About Significance (cont.)

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 15 Critical Values Again 9. When making a confidence interval, we’ve found a critical value to correspond to our selected confidence level. 10. Prior to the use of technology, P-values were difficult to find, and it was easier to select a few common alpha values and learn the corresponding critical values for the Normal model.

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 16 Critical Values Again (cont.) 11. Rather than looking up your z-score in the table, you could just check it directly against these critical values. Any z-score larger in magnitude than a particular critical value leads us to reject H 0. Any z-score smaller in magnitude than a particular critical value leads us to fail to reject H 0.

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 17 Critical Values Again (cont.) 12. Here are the traditional critical values from the Normal model:  1-sided2-sided 0.051.6451.96 0.012.282.575 0.0013.093.29

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 18 Critical Values Again (cont.) 13. When the alternative is one-sided, the critical value puts all of  on one side: 14. When the alternative is two-sided, the critical value splits  equally into two tails:

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 19 Confidence Intervals and Hypothesis Tests 15. Confidence intervals and hypothesis tests are built from the same calculations. They have the same assumptions and conditions. You can approximate a hypothesis test by examining a confidence interval. Just ask whether the null hypothesis value is consistent with a confidence interval for the parameter at the corresponding confidence level.

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 16. Because confidence intervals are two-sided, they correspond to two-sided tests. In general, a confidence interval with a confidence level of C% corresponds to a two-sided hypothesis test with an  -level of 100 – C%. Slide 21- 20 Confidence Intervals and Hypothesis Tests 95% _____% CI and a Two-sided Test 95% 0.025

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 21 Confidence Intervals and Hypothesis Tests The relationship between confidence intervals an one-sided hypothesis tests is a little more complicated. 17. A confidence interval with a confidence level of C% corresponds to a one-sided hypothesis test with an  -level of ½(100 – C)%. 90% _____% CI and a One-sided Test 90% 0.05

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 22 Making Errors 18. Here’s some shocking news for you: nobody’s perfect. Even with lots of evidence we can still make the wrong decision. 19. When we perform a hypothesis test, we can make mistakes in two ways: I. The null hypothesis is true, but we mistakenly reject it. (Type I error) II. The null hypothesis is false, but we fail to reject it. (Type II error)

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Making Errors 20. Which type of error is more serious depends on the situation at hand. In other words, the gravity of the error is context dependent.

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 24 Making Errors 21. Here’s an illustration of the four situations in a hypothesis test: Type I Error Type II Error

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 22. Medical Testing Slide 21- 25 What is a False Positive? What is a False Negative? Test shows they are not healthy, but they are healthy. TYPE I Error Test shows they are healthy, but they are not healthy. TYPE II Error

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1. Public health officials believe that 90% of children have been vaccinated against measles. A random survey of medical records at many schools across the country found that among more than 13,000 children only 89.4% had been vaccinated. A statistician would reject the 90% null hypothesis with a P-value of P = 0.011. Slide 21- 26 Measles

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1. Public health officials believe that 90% of children have been vaccinated against measles. A random survey of medical records at many schools across the country found that among more than 13,000 children only 89.4% had been vaccinated. A statistician would reject the 90% null hypothesis with a P-value of P = 0.011. a) Explain what the P-value means in this context. Slide 21- 27 There is only a 1.1% chance of seeing a sample proportion of 89.4% or lower vaccinated from sample error, if 90% have really been vaccinated.

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1. Public health officials believe that 90% of children have been vaccinated against measles. A random survey of medical records at many schools across the country found that among more than 13,000 children only 89.4% had been vaccinated. A statistician would reject the 90% null hypothesis with a P-value of P = 0.011. a) Explain what the P-value means in this context. b) The result is statistically significant, but is it important? Comment. 89.4% indicates 11,622 were vaccinated. While 90% would be 11,700 that were vaccinated. This is f 78 Slide 21- 28 The 90% in the null hypothesis would be 11,700 children and the 89.4% from the sample is 11,622 children. The difference of 0.6% between the two is only 78 children. This may not be important. The large sample size has probably exaggerated the significance.

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 29 How often will a Type I error occur? 2. Since a Type I error is rejecting a true null hypothesis, The probability of a Type I error is our  level.

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 30 Making Errors When H 0 is false and we reject it, we have done the right thing. 3. A test’s ability to detect a false hypothesis is called the power of the test. (more to come)

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 31 Making Errors 4. When H 0 is false and we fail to reject it, we have made a Type II error.  is the probability of a Type II error. It’s harder to assess the value of  because we don’t know what the value of the parameter really is. 5. If we decrease the chance of a Type I error, , we increase the chance of a Type II error, . 6. If we increase the chance of a Type I error, , we decrease the chance of a Type II error, .

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 32 Making Errors 7. The only way to reduce both types of errors is to collect more data. 8. Otherwise, we just wind up trading off one kind of error against the other.

33. 9. A sample distribution can be estimated using a Normal model when the Independence and Large Enough Assumptions are satisfied. p

34. What happens to this sample distribution when the sample size is increased? p

35. 10. When the sample size increases, the standard deviation decreases. p

36. What happens to this sample distribution when the sample size is decreased? p

37. p 11. When the sample size decreases, the standard deviation increases.

38. 12. How does sample size affect Type I Errors? p As the sample size increases the probability of a Type I Error decreases.

39. 13. How does sample size affect the Type II Errors p As the sample size increases the probability of a Type II Error decreases.

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 40 Power 14. POWER – The probability that a hypothesis test will correctly reject a false null hypothesis. Power = 1 -  15. Effect Size – The difference between the null hypothesis, p 0, and the true value of a model parameter, p. 16. The larger the EFFECT SIZE, the easier it is to detect a false null hypothesis…the more POWERFUL the test.

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 41 A Picture Worth a Thousand Words 17. This diagram shows the relationship between these concepts:

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 42 Reducing Both Type I and Type II Error

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 43 Reducing Both Type I and Type II Error Original comparison of errors: Comparison of errors with a larger sample size:

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 44 What Can Go Wrong? Don’t interpret the P-value as the probability that H 0 is true. 20. The P-value is the probability of the data given that H 0 is true, not the other way around. 21. Don’t believe too strongly in arbitrary alpha levels. 22. It’s better to report your P-value and a confidence interval so that the reader can make her/his own decision. 23. Report magnitude of difference too.

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 45 What Can Go Wrong? Don’t confuse practical and statistical significance. 24. Just because a test is statistically significant doesn’t mean that it is significant in practice. And, sample size can impact your decision about a null hypothesis, making you miss an important difference or find an “insignificant” difference. 25. Don’t forget that in spite of all your care, you might make a wrong decision.

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 21- 46 What Can Go Wrong? Don’t confuse practical and statistical significance. 24. Just because a test is statistically significant doesn’t mean that it is significant in practice. And, sample size can impact your decision about a null hypothesis, making you miss an important difference or find an “insignificant” difference. 25. Don’t forget that in spite of all your care, you might make a wrong decision.

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Assignment Answers Slide 21- 47 Part IPart II Page 491-492 #6, 8Pages 491 - 492 #10,12 Part IIIPart IV Page 492 #14,16Page 493 #20, 24 681012 14 1620 24

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #6 Slide 21- 48 6. A new reading program may reduce the number of elementary students who read below grade level. The company that developed this program supplied materials and teacher training for a large-scale test involving nearly 8500 children in several different school districts. Statistical analysis of the results showed that the percentage of students who did not attain the grade level standard was reduced from 15.9% to 15.1%. The hypothesis that the new reading program produced no improvement was rejected with a P-value of 0.023.

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #6 Slide 21- 49 6. A new reading program may reduce the number of elementary students who read below grade level. The company that developed this program supplied materials and teacher training for a large-scale test involving nearly 8500 children in several different school districts. Statistical analysis of the results showed that the percentage of students who did not attain the grade level standard was reduced from 15.9% to 15.1%. The hypothesis that the new reading program produced no improvement was rejected with a P-value of 0.023. a) Explain what the P-value means in this context.

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #6 Slide 21- 50 6. A new reading program may reduce the number of elementary students who read below grade level. The company that developed this program supplied materials and teacher training for a large-scale test involving nearly 8500 children in several different school districts. Statistical analysis of the results showed that the percentage of students who did not attain the grade level standard was reduced from 15.9% to 15.1%. The hypothesis that the new reading program produced no improvement was rejected with a P-value of 0.023. b) Even though this reading method has been shown to be significantly better, why might you not recommend that your local school adopt it?

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #6 Slide 21- 51 6. A new reading program may reduce the number of elementary students who read below grade level. The company that developed this program supplied materials and teacher training for a large-scale test involving nearly 8500 children in several different school districts. Statistical analysis of the results showed that the percentage of students who did not attain the grade level standard was reduced from 15.9% to 15.1%. The hypothesis that the new reading program produced no improvement was rejected with a P-value of 0.023. b) Even though this reading method has been shown to be significantly better, why might you not recommend that your local school adopt it?

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #8 Slide 21- 52 8. Soon after the Euro was introduced as currency in Europe, it was widely reported that someone had spun a Euro coin 250 times and gotten heads 140 times. We wish to test a hypothesis about the fairness of spinning the coin.

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #8 Slide 21- 53 8. Soon after the Euro was introduced as currency in Europe, it was widely reported that someone had spun a Euro coin 250 times and gotten heads 140 times. We wish to test a hypothesis about the fairness of spinning the coin. a) Estimate the true proportion of heads. Use a 95% confidence interval. Don’t forget to check the conditions first. The parameter of interest is true proportion of heads when a Euro coin is spun. Independence is reasonable, because coin spins are independent of each other. One spin doesn’t influence the other spins. There were 140 heads and 110 tails. Since both are greater than 10, the sample size is reasonably large enough.

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #8 Slide 21- 54 8. Soon after the Euro was introduced as currency in Europe, it was widely reported that someone had spun a Euro coin 250 times and gotten heads 140 times. We wish to test a hypothesis about the fairness of spinning the coin. a) Estimate the true proportion of heads. Use a 95% confidence interval. Don’t forget to check the conditions first. Since the conditions are met, we can preform a one-proportion z-interval. (0.499, 0.621)

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #8 Slide 21- 55 8. Soon after the Euro was introduced as currency in Europe, it was widely reported that someone had spun a Euro coin 250 times and gotten heads 140 times. We wish to test a hypothesis about the fairness of spinning the coin. a) Estimate the true proportion of heads. Use a 95% confidence interval. Don’t forget to check the conditions first. We are 95% confident that the true proportion of heads when a Euro is spun is between 0.499 and 0.621.

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #8 Slide 21- 56 8. Soon after the Euro was introduced as currency in Europe, it was widely reported that someone had spun a Euro coin 250 times and gotten heads 140 times. We wish to test a hypothesis about the fairness of spinning the coin. b) Does your confidence interval provide evidence that the coin is unfair when spun? Explain.

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #8 Slide 21- 57 8. Soon after the Euro was introduced as currency in Europe, it was widely reported that someone had spun a Euro coin 250 times and gotten heads 140 times. We wish to test a hypothesis about the fairness of spinning the coin. c) What is the significance level of this test? Explain.

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #8 Slide 21- 58 8. Soon after the Euro was introduced as currency in Europe, it was widely reported that someone had spun a Euro coin 250 times and gotten heads 140 times. We wish to test a hypothesis about the fairness of spinning the coin. c) What is the significance level of this test? Explain.

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #10 Slide 21- 59 10. The Spike network commissioned a telephone poll of randomly sampled U.S. men. Of the 712 respondents who had children, 22% said “yes” to the question “Are you a stay-at-home dad?” [Time, August 23, 2004]

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #10 Slide 21- 60 10. The Spike network commissioned a telephone poll of randomly sampled U.S. men. Of the 712 respondents who had children, 22% said “yes” to the question “Are you a stay-at-home dad?” [Time, August 23, 2004] a) To help them market commercial time, Spike wants an accurate estimate of the true percentage of stay-at-home dads. Construct a 95% confidence interval. The parameter of interest is true proportion of dads stay-at-home dads.

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #10 Slide 21- 61 10. The Spike network commissioned a telephone poll of randomly sampled U.S. men. Of the 712 respondents who had children, 22% said “yes” to the question “Are you a stay-at-home dad?” [Time, August 23, 2004] a) To help them market commercial time, Spike wants an accurate estimate of the true percentage of stay-at-home dads. Construct a 95% confidence interval. Independence is reasonable for the respondents, because The dads were sampled randomly and 712 men is less than 10% of all dads. They found 157 dads that stay-at-home and 555 that did not. Since both are at least 10, the sample is reasonably large enough.

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #10 Slide 21- 62 10. The Spike network commissioned a telephone poll of randomly sampled U.S. men. Of the 712 respondents who had children, 22% said “yes” to the question “Are you a stay-at-home dad?” [Time, August 23, 2004] a) To help them market commercial time, Spike wants an accurate estimate of the true percentage of stay-at-home dads. Construct a 95% confidence interval. Since the conditions are met, we can preform a one-proportion z-interval.

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #10 Slide 21- 63 10. The Spike network commissioned a telephone poll of randomly sampled U.S. men. Of the 712 respondents who had children, 22% said “yes” to the question “Are you a stay-at-home dad?” [Time, August 23, 2004] a) To help them market commercial time, Spike wants an accurate estimate of the true percentage of stay-at-home dads. Construct a 95% confidence interval. (0.189, 0.251)

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #10 Slide 21- 64 10. The Spike network commissioned a telephone poll of randomly sampled U.S. men. Of the 712 respondents who had children, 22% said “yes” to the question “Are you a stay-at-home dad?” [Time, August 23, 2004] a) To help them market commercial time, Spike wants an accurate estimate of the true percentage of stay-at-home dads. Construct a 95% confidence interval. We are 95% confident that the proportion of dads that stay-at-home is between 18.9% and 25.1%. (0.189, 0.251)

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #10 Slide 21- 65 10. The Spike network commissioned a telephone poll of randomly sampled U.S. men. Of the 712 respondents who had children, 22% said “yes” to the question “Are you a stay-at-home dad?” [Time, August 23, 2004] b) An advertiser of baby-carrying slings for dads will buy commercial time if at least 25% of men are stay-at-home dads. Use your confidence interval to test an appropriate hypothesis and make a recommendation to the advertiser.

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #10 Slide 21- 66 10. The Spike network commissioned a telephone poll of randomly sampled U.S. men. Of the 712 respondents who had children, 22% said “yes” to the question “Are you a stay-at-home dad?” [Time, August 23, 2004] c) Could Spike claim to the advertiser that it is possible that 25% of men with young children are stay-at-home dads? What is wrong with the reasoning?

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #10 Slide 21- 67 10. The Spike network commissioned a telephone poll of randomly sampled U.S. men. Of the 712 respondents who had children, 22% said “yes” to the question “Are you a stay-at-home dad?” [Time, August 23, 2004] c) Could Spike claim to the advertiser that it is possible that 25% of men with young children are stay-at-home dads? What is wrong with the reasoning?

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #12 Slide 21- 68 12. Testing for Alzheimer’s disease can be a long and expensive process, consisting of lengthy tests and medical diagnosis. Recently, a group of researchers (Solomon et al., 1998) devised a 7- minute test to serve as a quick screen for the disease for use in the general population of senior citizens. A patient who tested positive would then go through the more expensive battery of tests and medical diagnosis. The authors reported a false positive rate of 4% and a false negative rate of 8%.

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #12 Slide 21- 69 12. Testing for Alzheimer’s disease can be a long and expensive process, consisting of lengthy tests and medical diagnosis. Recently, a group of researchers (Solomon et al., 1998) devised a 7-minute test to serve as a quick screen for the disease for use in the general population of senior citizens. A patient who tested positive would then go through the more expensive battery of tests and medical diagnosis. The authors reported a false positive rate of 4% and a false negative rate of 8%. a) Put this in the context of a hypothesis test. What are the null and alternative hypotheses?

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #12 Slide 21- 70 12. Testing for Alzheimer’s disease can be a long and expensive process, consisting of lengthy tests and medical diagnosis. Recently, a group of researchers (Solomon et al., 1998) devised a 7-minute test to serve as a quick screen for the disease for use in the general population of senior citizens. A patient who tested positive would then go through the more expensive battery of tests and medical diagnosis. The authors reported a false positive rate of 4% and a false negative rate of 8%. b) What would a Type I error mean?

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #12 Slide 21- 71 12. Testing for Alzheimer’s disease can be a long and expensive process, consisting of lengthy tests and medical diagnosis. Recently, a group of researchers (Solomon et al., 1998) devised a 7-minute test to serve as a quick screen for the disease for use in the general population of senior citizens. A patient who tested positive would then go through the more expensive battery of tests and medical diagnosis. The authors reported a false positive rate of 4% and a false negative rate of 8%. c) What would a Type II error mean?

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #12 Slide 21- 72 12. Testing for Alzheimer’s disease can be a long and expensive process, consisting of lengthy tests and medical diagnosis. Recently, a group of researchers (Solomon et al., 1998) devised a 7-minute test to serve as a quick screen for the disease for use in the general population of senior citizens. A patient who tested positive would then go through the more expensive battery of tests and medical diagnosis. The authors reported a false positive rate of 4% and a false negative rate of 8%. d) Which is worse here, a Type I or Type II error? Explain.

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #12 Slide 21- 73 12. Testing for Alzheimer’s disease can be a long and expensive process, consisting of lengthy tests and medical diagnosis. Recently, a group of researchers (Solomon et al., 1998) devised a 7-minute test to serve as a quick screen for the disease for use in the general population of senior citizens. A patient who tested positive would then go through the more expensive battery of tests and medical diagnosis. The authors reported a false positive rate of 4% and a false negative rate of 8%. e) What is the power of this test?

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part I #12 Slide 21- 74 12. Testing for Alzheimer’s disease can be a long and expensive process, consisting of lengthy tests and medical diagnosis. Recently, a group of researchers (Solomon et al., 1998) devised a 7-minute test to serve as a quick screen for the disease for use in the general population of senior citizens. A patient who tested positive would then go through the more expensive battery of tests and medical diagnosis. The authors reported a false positive rate of 4% and a false negative rate of 8%. e) What is the power of this test?

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #14 Slide 21- 75 14. Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired.

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #14 Slide 21- 76 14. Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. a) In this context, what is a Type I error?

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #14 Slide 21- 77 14. Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. b) In this context what is a Type II error?

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #14 Slide 21- 78 14. Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. c) Which type of error would the factory owner consider more serious? Why?

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #14 Slide 21- 79 14. Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. d) Which type of error might customers consider more serious? Why?

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #14 Slide 21- 80 14. Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. d) Which type of error might customers consider more serious? Why?

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #16 Slide 21- 81 16. Consider again the task of the quality inspector in question 14. 14. Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. a) In this context, what is meant by the power of the test the inspectors conduct?

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #16 Slide 21- 82 16. Consider again the task of the quality inspector in question 14. 14. Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. b) They are currently testing 5 items each hour. Someone has proposed they test 10 each hour instead. What are the advantages and disadvantages of such a change.

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #16 Slide 21- 83 16. Consider again the task of the quality inspector in question 14. 14. Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. c) Their test currently uses a 5% level of significance. What are the advantages and disadvantages of changing to an alpha level of 1%?

84. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #16 Slide 21- 84 16. Consider again the task of the quality inspector in question 14. 14. Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. d) Suppose that as a day passes one of the machines on the assembly line produces more and more items that are defective. How will this affect the power of the test?

85. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #16 Slide 21- 85 16. Consider again the task of the quality inspector in question 14. 14. Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. d) Suppose that as a day passes one of the machines on the assembly line produces more and more items that are defective. How will this affect the power of the test?

86. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #20 Slide 21- 86 20. A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random phone survey of 400 people.

87. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #20 Slide 21- 87 20. A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random phone survey of 400 people. a) What are the hypotheses?

88. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #20 Slide 21- 88 20. A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random phone survey of 400 people. b) The station plans to conduct this test using a 10% level of significance, but the company wants the significance level lowered to 5%. Why?

89. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #20 Slide 21- 89 20. A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random phone survey of 400 people. c) What is meant by the power of this test?

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #20 Slide 21- 90 20. A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random phone survey of 400 people. d) For which level of significance will the power of this test be higher? Why?

91. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #20 Slide 21- 91 20. A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random phone survey of 400 people.

92. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #20 Slide 21- 92 20. A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random phone survey of 400 people.

93. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #24 Slide 21- 93 24. An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About 40% break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay form another supplier. She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks.

94. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #24 Slide 21- 94 24. An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About 40% break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay form another supplier. She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks. a) Suppose the new expensive clay really is no better than her usual clay. What’s the probability that this test convinces her to use it anyway? (Hint: Use a Binomial model.)

95. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #24 Slide 21- 95 24. An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About 40% break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay form another supplier. She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks. b) If she decides to switch to the new clay and it is no better, what kind of error did she commit?

96. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #24 Slide 21- 96 24. An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About 40% break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay form another supplier. She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks. c) If the new clay really could reduce breakage to only 20%, what’s the probability that her test will not detect the improvement? if 2 or more pieces break.

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #24 Slide 21- 97 24. An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About 40% break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay form another supplier. She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks. d) How can she improve the power of her test? Offer at least two suggestions.

98. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 21 Part II #24 Slide 21- 98 24. An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About 40% break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay form another supplier. She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks. d) How can she improve the power of her test? Offer at least two suggestions.