1. Copyright © 2017, 2014 Pearson Education, Inc. Slide 1 Chapter 8 Hypothesis Testing for Population Proportions

2. Copyright © 2017, 2014 Pearson Education, Inc. Slide 2 Chapter 8 Topics Conduct hypothesis tests involving one and two population proportions

3. Copyright © 2017, 2014 Pearson Education, Inc. Slide 3 THE ESSENTIAL INGREDIENTS OF HYPOTHESIS TESTING Section 8.1 Write a Null and Alternative Hypothesis for a Test Involving a Population Proportion Compute and Interpret a Test Statistic and p-value Videowokart. Shutterstock

4. Copyright © 2017, 2014 Pearson Education, Inc. Slide 4 Copyright © 2017, 2014 Pearson Education, Inc. Slide 4 Hypothesis Testing A procedure that enables us to choose between two claims when we have variability in our measurements.

5. Copyright © 2017, 2014 Pearson Education, Inc. Slide 5 Copyright © 2017, 2014 Pearson Education, Inc. Slide 5 In General: Four Steps 1.Hypothesize – State a hypothesis (claim) that will be weighed against a neutral “skeptical” claim. 2.Prepare – Determine how you’ll use data to make your decision and make sure you have enough data to minimize the probability of making mistakes.

6. Copyright © 2017, 2014 Pearson Education, Inc. Slide 6 Copyright © 2017, 2014 Pearson Education, Inc. Slide 6 In General: Four Steps 3.Compute to Compare – Collect data and compare them to your expectation. 4.Interpret – State your conclusion. – Do you believe the claim or do you find that the claim doesn’t have enough evidence to back it up?

7. Copyright © 2017, 2014 Pearson Education, Inc. Slide 7 Copyright © 2017, 2014 Pearson Education, Inc. Slide 7 An Introductory Example A coin is spun on a flat surface. You hypothesize it is more likely to come up heads because that side of the coin bulges out slightly compared to the tails side. You spin the coin 20 times and it comes up heads all 20 times. If it was equally likely that the coin stopped on heads or tails after spinning, it would be very unusual to find 20 heads in 20 spins of a coin. This is so unusual that you would conclude that your hypothesis that the coin is more likely to come up heads when spinning is correct.

8. Copyright © 2017, 2014 Pearson Education, Inc. Slide 8 Copyright © 2017, 2014 Pearson Education, Inc. Slide 8 An Introductory Example 1.You stated a hypothesis that the coin in more likely to come up heads and test this against the neutral hypothesis that the coin is fair. 2.You decided that you would use the proportion of heads in 20 spins of the coin to test your hypothesis. 3.You collected data (20 spins resulted in 20 heads) and compared them to your expectations. If the coin was not more likely to land on heads, you would expect about 50% of the spins (or about 10 spins) to result in heads. However, you got a result of 20 heads which is much greater than you expected.

9. Copyright © 2017, 2014 Pearson Education, Inc. Slide 9 Copyright © 2017, 2014 Pearson Education, Inc. Slide 9 An Introductory Example 4.Because it would be so unlikely to get 20 heads in 20 spins if the coin was equally likely to land on heads or tails when spun, you conclude that the coin is more likely to land on heads when spun.

10. Copyright © 2017, 2014 Pearson Education, Inc. Slide 10 Copyright © 2017, 2014 Pearson Education, Inc. Slide 10 A Pair of Hypotheses H 0 The Null Hypothesis -The neutral, status quo, skeptical statement about a population parameter -Always contains = H a The Alternative Hypothesis -The research hypothesis; the statement about a population parameter we intend to demonstrate is true -Always contains, or ≠ NOTE: Hypotheses are always about population PARAMETERS; they are never about sample statistics.

11. Copyright © 2017, 2014 Pearson Education, Inc. Slide 11 Copyright © 2017, 2014 Pearson Education, Inc. Slide 11 The Null Hypothesis The null hypothesis always gets the benefit of the doubt throughout the hypothesis-testing procedure. We only reject the null hypothesis if the observed outcome is extremely unusual if the null hypothesis were true. It is analogous to assume that a defendant in a jury trial is innocent unless proven guilty “beyond a reasonable doubt.”

12. Copyright © 2017, 2014 Pearson Education, Inc. Slide 12 Copyright © 2017, 2014 Pearson Education, Inc. Slide 12 Example: Global Warming A 2014 Pew Poll found that 61% of Americans believed in global warming. A researcher believes this rate has declined. State the null and alternative hypotheses in words and in symbols.

13. Copyright © 2017, 2014 Pearson Education, Inc. Slide 13 Copyright © 2017, 2014 Pearson Education, Inc. Slide 13 Example: Global Warming H 0 : p = 0.61 The same proportion of Americans believe in global warming as in the past. H a : p < 0.61 The proportion of Americans who believe in global warming has decreased.

14. Copyright © 2017, 2014 Pearson Education, Inc. Slide 14 Copyright © 2017, 2014 Pearson Education, Inc. Slide 14 One-Sided vs. Two-Sided Hypotheses The sign in the alternative hypothesis determines whether a hypothesis is one-sided or two-sided.

15. Copyright © 2017, 2014 Pearson Education, Inc. Slide 15 Copyright © 2017, 2014 Pearson Education, Inc. Slide 15 Example: Internet Sales During the previous fiscal year, 30% of a retailer’s sales were due to online sales. In an effort to increase this percentage the retailer has purchased ads on social media sites. The retailer gathers data on types of sales on a sample of 20 days during the current fiscal year. Has there been a change in online sales? 1.Write the H 0 and H a for the retailer. 2.Is this a one-sided or two-sided hypothesis? 3.What other hypotheses could the retailer have posed?

16. Copyright © 2017, 2014 Pearson Education, Inc. Slide 16 Copyright © 2017, 2014 Pearson Education, Inc. Slide 16 Example: Internet Sales H 0 : p = 0.30 H a : p ≠ 0.30 This is a two-sided hypothesis. The retailer could also have asked, “Have sales increased?” Then H 0 : p = 0.30 H a : p > 0.30 This would be a one-sided test (right) If the retailer asked, “Have sales decreased?” Then H 0 : p = 0.30, H a : p < 0.30, and this would be a one- side (left) test.

17. Copyright © 2017, 2014 Pearson Education, Inc. Slide 17 Copyright © 2017, 2014 Pearson Education, Inc. Slide 17 Significance Level The significance level is the probability of making the mistake of rejecting the null hypothesis when, in fact, the null hypothesis is true. The symbol for the significance level is α. For most applications a significance level of 0.05 is used, but 0.01 and 0.10 are also sometimes used.

18. Copyright © 2017, 2014 Pearson Education, Inc. Slide 18 Copyright © 2017, 2014 Pearson Education, Inc. Slide 18 Test Statistic The test statistic compares our observed outcome with the outcome we would get if the null hypothesis is true. When the test statistic is far away from the value we would expect that if the null hypothesis is true, we reject the null hypothesis and conclude the evidence supports the alternative hypothesis.

19. Copyright © 2017, 2014 Pearson Education, Inc. Slide 19 Copyright © 2017, 2014 Pearson Education, Inc. Slide 19 Test Statistic: Hypothesis Test of One Population Proportion The test statistic has the structure: So we have a test statistic of: where is the sample proportion (from your data). is the proportion found in your null hypothesis (H 0 ).

20. Copyright © 2017, 2014 Pearson Education, Inc. Slide 20 Copyright © 2017, 2014 Pearson Education, Inc. Slide 20 Example: Global Warming In 2001, the Gallup Poll reported that 12% of Americans were “cool skeptics” who reported worrying “a little or not at all” about global warming. In 2014, 25% of those responding to a Gallup poll classified themselves as “cool skeptics.” Assume that the sample size for the 2014 poll was 500. Can we conclude that the percentage of Americans who identify themselves as cool skeptics has increased since 2001? Calculate the test statistic and explain the value in context.

21. Copyright © 2017, 2014 Pearson Education, Inc. Slide 21 Copyright © 2017, 2014 Pearson Education, Inc. Slide 21 Example: Global Warming H 0 : p = 0.12 H a : p > 0.12 Test statistic: The observed value of the test statistic is 8.95. This means that while the null hypothesis claims a proportion of 0.12, the observed proportion was much higher – 8.95 standard errors higher – than what the null hypothesis claims.

22. Copyright © 2017, 2014 Pearson Education, Inc. Slide 22 Copyright © 2017, 2014 Pearson Education, Inc. Slide 22 Measuring Surprise: The p-value The null hypothesis tells us what to expect when we look at our data. If we see something unexpected – that is, when we are surprised – then we should doubt the null hypothesis. If we are really surprised, we should reject it altogether. The p-value gives us a way to numerically measure our surprise. It reports the probability that, if the null hypothesis is true, our test statistic will have a value as extreme as or more extreme than the value we actually observe. Small p-values (close to 0) mean we are really surprised. Large p-values (close to 1) mean we are not surprised at all.

23. Copyright © 2017, 2014 Pearson Education, Inc. Slide 23 Copyright © 2017, 2014 Pearson Education, Inc. Slide 23 p-values and the Significance Level If we get a p-value that is less than our significance level, our p-value is considered small (we are really surprised!) and we reject the null hypothesis. If we get a p-value greater than or equal to our significance level, our p-value is not considered small (we are not surprised) and we do not reject the null hypothesis.

24. Copyright © 2017, 2014 Pearson Education, Inc. Slide 24 Copyright © 2017, 2014 Pearson Education, Inc. Slide 24 Global Warming: The p-value In our example on Global Warming, we had a test statistic z = 8.95. Using StatCrunch we can calculate the probability of getting a test statistic at least as extreme at 8.95. The p-value approximately 0. This p-value gives us strong evidence to reject the null hypothesis. We conclude that the percentage of Americans who identify themselves as “cool skeptics” has increased since 2001.

25. Copyright © 2017, 2014 Pearson Education, Inc. Slide 25 HYPOTHESIS TESTING IN FOUR STEPS Section 8.2 Conduct a Hypothesis Test Involving a Population Proportion connel. Shutterstock

26. Copyright © 2017, 2014 Pearson Education, Inc. Slide 26 Copyright © 2017, 2014 Pearson Education, Inc. Slide 26 Four Steps in Hypothesis Testing 1.Hypothesize State your hypotheses about the population parameter. 2.Prepare State a significance level, choose an appropriate test statistic, state and check conditions required for future computations, state any assumptions that must be made. 3.Compute to Compare Compute the observed value of the test statistic and compare it to what the null hypothesis said you would get. Find the p-value in order to measure your level of surprise. 4.Interpret Do you reject or not reject your null hypothesis? What does this mean in the context of the data?

27. Copyright © 2017, 2014 Pearson Education, Inc. Slide 27 Copyright © 2017, 2014 Pearson Education, Inc. Slide 27 Hypothesis Test about a Population Proportion: Conditions to Check 1.Random sample 2.Large sample size np 0 ≥ 10 and n(1 – p 0 ) ≥ 10 3.Large population Population is at least 10 times bigger than the sample size if the sample is collected without replacement. 4.Independence Each observation has no influence on any others.

28. Copyright © 2017, 2014 Pearson Education, Inc. Slide 28 Copyright © 2017, 2014 Pearson Education, Inc. Slide 28 Example: Death Penalty In 2009, it was estimated that 63% of Americans supported the death penalty. In 2014, a Rasmussen Poll found that, in a random sample of 943 Americans, 538 supported the death penalty. Do we have evidence that support of the death penalty has declined among Americans since 2009?

29. Copyright © 2017, 2014 Pearson Education, Inc. Slide 29 Copyright © 2017, 2014 Pearson Education, Inc. Slide 29 Example: Death Penalty 1.Hypothesize H 0 : p = 0.63 (Support has not changed since 2009) H a : p < 0.63 (Support has declined) 2.Prepare The sample is random and independent. Large sample size: np 0 = 943(0.63) ≥ 10 n(1 – p 0 ) = 943(1 – 0.63) ≥ 10 Large population: The population of Americans is greater than 10(943).

30. Copyright © 2017, 2014 Pearson Education, Inc. Slide 30 Copyright © 2017, 2014 Pearson Education, Inc. Slide 30 Example: Death Penalty 3.Compute to Compare Standard error = Test statistic = We know it is very unusual to be 3.82 standard errors below the mean. In practice, we usually use technology to compute the test statistic and p-value.

31. Copyright © 2017, 2014 Pearson Education, Inc. Slide 31 p-value Using Technology 4. Interpret Our p-value is very small, so we would reject our null hypothesis. Support for the death penalty has declined among Americans since 2009.

32. Copyright © 2017, 2014 Pearson Education, Inc. Slide 32 Copyright © 2017, 2014 Pearson Education, Inc. Slide 32 Example: Death Penalty We usually use technology to generate both the test statistic and p-value: StatCrunch Stat > Proportion Stat > One Sample > with summary.

33. Copyright © 2017, 2014 Pearson Education, Inc. Slide 33 Copyright © 2017, 2014 Pearson Education, Inc. Slide 33 Using the TI-84 Calculator To conduct a hypothesis test for a population proportion using the TI-84 calculator: 1.Push STAT > TESTS then select the 1-PropZTest option. 2.Enter p 0, the proportion in the null hypothesis. 3.Enter x and n (make sure to round x to a whole number, if necessary). 4.Select the sign that corresponds with the sign in your alternative hypothesis. 5.Select Calculate or Draw. The test statistic (z) and p-value (p) will be displayed.

34. Copyright © 2017, 2014 Pearson Education, Inc. Slide 34 Copyright © 2017, 2014 Pearson Education, Inc. Slide 34 More about the p-value Calculation of the p-value depends on the sign in the alternative hypothesis. If H a contains ≠, we say the alternative hypothesis is two-sided and the p-value would be the total of the shaded regions (a “two-tailed” p-value).

35. Copyright © 2017, 2014 Pearson Education, Inc. Slide 35 Copyright © 2017, 2014 Pearson Education, Inc. Slide 35 More about the p-value If H a contains a, we have a one-tailed p-value. If H a contains a <, we have a left-tailed test and the p-value is the area to the left of the test statistic.

36. Copyright © 2017, 2014 Pearson Education, Inc. Slide 36 Copyright © 2017, 2014 Pearson Education, Inc. Slide 36 More about the p-value If H a contains a >, we have a right-tailed test and the p-value is the area to the right of the test statistic.

37. Copyright © 2017, 2014 Pearson Education, Inc. Slide 37 Copyright © 2017, 2014 Pearson Education, Inc. Slide 37 Example: Increased Library Funding Suppose a poll of 250 randomly selected voters in a city found that 140 of them favored increased funding for public libraries. Based on this poll, could we conclude that a majority of voters in this city (more than 50%) favor increased funding for public libraries? Use a significance level of 0.01.

38. Copyright © 2017, 2014 Pearson Education, Inc. Slide 38 Copyright © 2017, 2014 Pearson Education, Inc. Slide 38 Example: Increased Library Funding 1.Hypothesize H 0 : p = 0.50 H a : p > 0.50 2.Prepare The sample is random and independent. Large sample? Large population? We’ll assume there are more than 10(250) = 2500 voters in the city. Since 125 ≥ 10 the sample size is large enough.

39. Copyright © 2017, 2014 Pearson Education, Inc. Slide 39 3. Compute to Compare Test statistic: z = 1.90 p-value: 0.0289

40. Copyright © 2017, 2014 Pearson Education, Inc. Slide 40 Copyright © 2017, 2014 Pearson Education, Inc. Slide 40 Example: Increased Library Funding 4.Interpret In this problem we were given a significance level of 0.01. Our p-value of 0.0289 is greater than our significance level so we will NOT reject the null hypothesis. We cannot conclude that the majority of voters support increased public library funding.

41. Copyright © 2017, 2014 Pearson Education, Inc. Slide 41 HYPOTHESIS TESTS IN DETAIL Section 8.3 Identify Two Possible Sources of Error in Hypothesis Testing Distinguish between Practical and Statistical Significance Identify Similarities and Differences between a Hypothesis Test and Confidence Interval Approach Cynthia Farmer. Shutterstock

42. Copyright © 2017, 2014 Pearson Education, Inc. Slide 42 Copyright © 2017, 2014 Pearson Education, Inc. Slide 42 Revisiting Small p-values A small p-value means our test statistic is extreme. An extreme test statistic means something unusual, and therefore unexpected, has happened. Small p-values lead us to reject the null hypothesis.

43. Copyright © 2017, 2014 Pearson Education, Inc. Slide 43 Copyright © 2017, 2014 Pearson Education, Inc. Slide 43 If Conditions Fail… If the conditions concerning the sampling distribution of the z-statistic fail to be met, then we cannot find a p-value using the Normal curve. Conditions can fail for the following reasons: 1.The sample size is too small. 2.Samples are not randomly selected – in this case conclusions may not generalize to the population.

44. Copyright © 2017, 2014 Pearson Education, Inc. Slide 44 Copyright © 2017, 2014 Pearson Education, Inc. Slide 44 Potential Mistakes in Hypothesis Testing There are two types of potential mistakes that can occur in hypothesis testing:

45. Copyright © 2017, 2014 Pearson Education, Inc. Slide 45 Copyright © 2017, 2014 Pearson Education, Inc. Slide 45 Example: Describing Mistakes Suppose a political analyst is interested in predicting whether a school bond measure on the ballot will pass. Her hypotheses are: H 0 : p = 0.50 H a : p > 0.50 Describe two types of errors she might make in conducting this test. Explain what a 5% significance level means in the context of this problem.

46. Copyright © 2017, 2014 Pearson Education, Inc. Slide 46 Copyright © 2017, 2014 Pearson Education, Inc. Slide 46 Example: Making Mistakes 1.Rejecting the null hypothesis when it is true. This means concluding that more than 50% favor the bond when the actual proportion who favor the bond is 50% or less. 2.Failing to reject the null hypothesis when it is false. This means concluding that 50% or less favor the bond when the actual proportion who favor the bond is more than 50%.

47. Copyright © 2017, 2014 Pearson Education, Inc. Slide 47 Copyright © 2017, 2014 Pearson Education, Inc. Slide 47 Example: Making Mistakes 3.The 5% significance level means there is only a 5% chance that the analyst mistakenly concludes that more than 50% favor the bond when, in fact, they don’t.

48. Copyright © 2017, 2014 Pearson Education, Inc. Slide 48 Copyright © 2017, 2014 Pearson Education, Inc. Slide 48 Reducing Mistakes The only way to reduce the probability of both types of mistakes is to increase the sample size. Increasing the sample size improves the precision of the test so we make mistakes less often. We cannot make the significance level arbitrarily small because this increases the probability the we mistakenly fail to reject the null hypothesis.

49. Copyright © 2017, 2014 Pearson Education, Inc. Slide 49 Copyright © 2017, 2014 Pearson Education, Inc. Slide 49 Statistical Significance vs. Practical Significance A result is “statistically significant” when the null hypothesis is rejected – the difference between the data-estimated value for a parameter and the null hypothesis value for a parameter is so large it cannot be convincingly explained by chance. However statistically significant findings do not necessarily mean the results are useful.

50. Copyright © 2017, 2014 Pearson Education, Inc. Slide 50 Copyright © 2017, 2014 Pearson Education, Inc. Slide 50 Example: Statistical vs. Practical Significance Suppose the proportion of people who get a certain type of cancer is 1 in 10 million and that a statistical analysis shows that talking on a cell phone daily doubles the risk of getting cancer. This means that the risk of getting cancer is now 2 in 10 million. Would this, practically speaking, cause a person to stop talking on a cell phone? A practically significant result is both statistically significant and meaningful.

51. Copyright © 2017, 2014 Pearson Education, Inc. Slide 51 Copyright © 2017, 2014 Pearson Education, Inc. Slide 51 Cautions about Writing Conclusion Because we can never be 100% certain that our conclusion in hypothesis testing is true, when your p-value is greater than your significance level, AVOID using any of the following phrases: “We accept H 0.” “We have proved H 0 is true.” Say instead: “We have failed to reject H 0.” “We cannot reject H 0.”

52. Copyright © 2017, 2014 Pearson Education, Inc. Slide 52 Copyright © 2017, 2014 Pearson Education, Inc. Slide 52 Confidence Intervals and Hypothesis Tests Confidence intervals and hypothesis tests are closely related but ask slightly different questions. Confidence intervals – “What is the value of this parameter?” Hypothesis test – “Are the data consistent with the parameter being one particular value or might the parameter be something else?

53. Copyright © 2017, 2014 Pearson Education, Inc. Slide 53 Copyright © 2017, 2014 Pearson Education, Inc. Slide 53 Confidence Intervals and Hypothesis Tests Even though they are designed to answer different questions, they are similar enough to lead us to reach the same types of conclusions. A confidence interval can lead us to the same type of conclusion as a two-sided hypothesis test.

54. Copyright © 2017, 2014 Pearson Education, Inc. Slide 54 Copyright © 2017, 2014 Pearson Education, Inc. Slide 54 Example: Death Penalty Revisited In 2009, it was estimated that 63% of Americans supported the death penalty. In 2014, a Rasmussen Poll found that, in a random sample of 943 Americans, 538 supported the death penalty. Do we have evidence that support of the death penalty has changed among Americans since 2009? (Note: Previously we asked if support had declined.) 1)Conduct a hypothesis test. 2)Construct a 95% confidence interval for the proportion of Americans who support the death penalty. 3)Explain how your confidence interval supports the conclusion of your hypothesis test.

55. Copyright © 2017, 2014 Pearson Education, Inc. Slide 55 Copyright © 2017, 2014 Pearson Education, Inc. Slide 55 Example: Death Penalty Hypothesis Test 1.Hypothesize H 0 : p = 0.63 H a : p ≠0.63 2.Prepare Previously we verified the sample size was random and large enough to conduct the hypothesis test. 3.Compute to Compare z = -3.78, p =.0002 (use technology) 4.Interpret Reject H 0. Support for the death penalty has changed among Americans since 2009.

56. Copyright © 2017, 2014 Pearson Education, Inc. Slide 56 Copyright © 2017, 2014 Pearson Education, Inc. Slide 56 Example: Death Penalty Confidence Interval The 95% confidence interval for the proportion of Americans who support the death penalty is (0.54, 0.60). (Use technology) This supports the hypothesis test conclusion because 0.63 is not contained in the interval; in other words, the population proportion has changed.

57. Copyright © 2017, 2014 Pearson Education, Inc. Slide 57 Copyright © 2017, 2014 Pearson Education, Inc. Slide 57 Death Penalty Hypothesis Test: StatCrunch

58. Copyright © 2017, 2014 Pearson Education, Inc. Slide 58 Copyright © 2017, 2014 Pearson Education, Inc. Slide 58 Death Penalty Confidence Interval: StatCrunch

59. Copyright © 2017, 2014 Pearson Education, Inc. Slide 59 Copyright © 2017, 2014 Pearson Education, Inc. Slide 59 Relationship between Confidence Levels and Significance Levels Significance levels and corresponding confidence levels Confidence Level (1 – α) Alternative Hypothesis Significance Level (α) 90%Two-Sided ≠10% 95%Two-Sided ≠5% 99%Two-Sided ≠1%

60. Copyright © 2017, 2014 Pearson Education, Inc. Slide 60 COMPARING PROPORTIONS FROM TWO POPULATIONS Section 8.4 Conduct a Hypothesis Test Comparing Two Population Proportions Monkey Business Images. Shutterstock

61. Copyright © 2017, 2014 Pearson Education, Inc. Slide 61 Copyright © 2017, 2014 Pearson Education, Inc. Slide 61 Example: Economic Confidence In January 2014, the Gallup organization reported that 45% of Americans reported feeling “pretty good” about the amount of money they had to spend. In January 2015, Gallup reported that 49% of Americans felt this way. Both samples had a sample size of 3500. Can we conclude that economic confidence has improved since 2014 or could this difference be due to chance variation during the sampling procedure?

62. Copyright © 2017, 2014 Pearson Education, Inc. Slide 62 Copyright © 2017, 2014 Pearson Education, Inc. Slide 62 Changes to the Hypotheses p 1 represents the proportion of Americans who felt “pretty good” about the amount of money they had to spend in 2014. p 2 represents the proportion who felt this way in 2015. We are interested in the relationship between these two parameters. In comparing two population proportions, the null hypothesis is H 0 : p 1 = p 2 the alternative hypothesis is one of these 3 possibilities: a) H a : p 1 ≠ p 2 b) H a : p 1 > p 2 c) H a : p 1 < p 2

63. Copyright © 2017, 2014 Pearson Education, Inc. Slide 63 Copyright © 2017, 2014 Pearson Education, Inc. Slide 63 Hypotheses: One-Sided and Two-Sided Tests

64. Copyright © 2017, 2014 Pearson Education, Inc. Slide 64 Copyright © 2017, 2014 Pearson Education, Inc. Slide 64 Changes to The Test Statistic The test statistic has the same general form: Estimator: because we are estimating p 1 – p 2 Null value: 0 because our null hypothesis is that p 1 = p 2, so p 1 – p 2 = 0.

65. Copyright © 2017, 2014 Pearson Education, Inc. Slide 65 Copyright © 2017, 2014 Pearson Education, Inc. Slide 65 Two Proportions: The Test Statistic In practice, the test statistic and corresponding p-value are computed using an appropriate technology.

66. Copyright © 2017, 2014 Pearson Education, Inc. Slide 66 Copyright © 2017, 2014 Pearson Education, Inc. Slide 66 Changes to Checking Conditions 1.Large samples Both sample sizes must be large enough. We use, the pooled sample proportion, where Check that and 2. Random Samples If we are not told explicitly that the sample was randomly drawn we may have to assume this condition is satisfied.

67. Copyright © 2017, 2014 Pearson Education, Inc. Slide 67 Copyright © 2017, 2014 Pearson Education, Inc. Slide 67 Changes to Checking Conditions 3.Independent Samples The samples are independent of each other. 4.Independent within Samples The observations within each sample must be independent of one another.

68. Copyright © 2017, 2014 Pearson Education, Inc. Slide 68 Copyright © 2017, 2014 Pearson Education, Inc. Slide 68 Example: Gallup Poll on Consumer Confidence Researchers from the Gallup Poll interviewed two random samples. Both samples, one taken in January 2014 and one taken in January 2015, had 3500 people. In January 2014, 1575 respondents said they felt “pretty good” about the amount of money they had to spend. In January 2015, 1715 respondents felt this way. Can we conclude that economic confidence has improved since 2014? Use a significance level of 0.05.

69. Copyright © 2017, 2014 Pearson Education, Inc. Slide 69 Copyright © 2017, 2014 Pearson Education, Inc. Slide 69 Example: Consumer Confidence The data is summarized in the table below. Let p 1 = proportion in 2014 who felt “pretty good.” Let p 2 = proportion in 2015 who felt “pretty good.” 1.Hypothesize H 0 : p 1 = p 2 H a : p 1 < p 2 20142015Total Felt “pretty good157517153290 Did not feel “pretty good”192517853710 Total3500 7000

70. Copyright © 2017, 2014 Pearson Education, Inc. Slide 70 Copyright © 2017, 2014 Pearson Education, Inc. Slide 70 Example: Consumer Confidence 2.Prepare Sample sizes large? Since both sample sizes are the same (3500) it is enough to show the check that n 1 is large enough. Samples are random, independent, and independent of each other.

71. Copyright © 2017, 2014 Pearson Education, Inc. Slide 71 Copyright © 2017, 2014 Pearson Education, Inc. Slide 71 Example: Consumer Confidence 3.Compute to Compare We use technology to compute the test statistic and p-value. StatCrunch: Stat > Proportion Stats > Two Sample > with summary.

72. Copyright © 2017, 2014 Pearson Education, Inc. Slide 72 Test Statistic: z = –3.35 p-value: 0.0004

73. Copyright © 2017, 2014 Pearson Education, Inc. Slide 73 Copyright © 2017, 2014 Pearson Education, Inc. Slide 73 Example: Consumer Confidence 4. Interpret Our p-value is small (less than our significance level) so we reject the null hypothesis. We can conclude that consumer confidence has increased.

74. Copyright © 2017, 2014 Pearson Education, Inc. Slide 74 Copyright © 2017, 2014 Pearson Education, Inc. Slide 74 Using the TI-84 Calculator To conduct a hypothesis test for comparing proportions from two populations using the TI-84 calculator: Push STAT > TESTS then select the 2-PropZTest option. 1.Enter p 0, the proportion in the null hypothesis. 2.Enter x1, n1, x2, and n2 (make sure to round x to a whole number, if necessary). 3.Select the sign that corresponds with the sign in your alternative hypothesis. 4.Select Calculate or Draw. The test statistic (z) and p-value (p) will be displayed.

75. Copyright © 2017, 2014 Pearson Education, Inc. Slide 75 Copyright © 2017, 2014 Pearson Education, Inc. Slide 75 Example: Texting While Driving A student is interested in determining if there are gender differences in texting while driving. She collects data from a random sample of 80 men and 60 women. Of the 80 men, 38 reported they had texted while driving. Of the 60 women, 27 reported they had texted while driving. Assume the samples were random and independent. Can the student conclude that the proportion of men and women who text while driving are different? Use a significance level of 0.05.

76. Copyright © 2017, 2014 Pearson Education, Inc. Slide 76 Copyright © 2017, 2014 Pearson Education, Inc. Slide 76 Example: Texting While Driving The data are summarized in the table below. Let p 1 = proportion of men who text while driving. Let p 2 = proportion of women who text while driving. 1. Hypothesize H 0 : p 1 = p 2 H a : p 1 ≠ p 2 MenWomenTotal Text – Yes382765 Text – No423375 Total8060140

77. Copyright © 2017, 2014 Pearson Education, Inc. Slide 77 Copyright © 2017, 2014 Pearson Education, Inc. Slide 77 Example: Texting While Driving 2.Prepare Sample sizes large? Samples are random, independent, and independent of each other. All these values are ≥ 10, so the sample sizes are large enough.

78. Copyright © 2017, 2014 Pearson Education, Inc. Slide 78 3.Compute to Compare Test statistic: z = 0.29 p-value = 0.77

79. Copyright © 2017, 2014 Pearson Education, Inc. Slide 79 Copyright © 2017, 2014 Pearson Education, Inc. Slide 79 Example: Texting While Driving 4.Interpret Since our p-value is large (greater than our significance level) we will not reject the H 0. We cannot conclude that there is a difference in the proportions of men and women who text while driving.