1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 10.4 Hypothesis Testing for Population Proportions

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Perform hypothesis tests for means, proportions, and variances.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Hypothesis Testing for Population Proportions Test Statistic for a Hypothesis Test for a Population Proportion When the sample taken is a simple random sample, the conditions for a binomial distribution are met, and the sample size is large enough to ensure that np ≥ 5 and n(1 − p) ≥ 5, the test statistic for a hypothesis test for a population proportion is given by

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Hypothesis Testing for Population Proportions Test Statistic for a Hypothesis Test for a Population Proportion (cont.) Where is the sample proportion, p is the presumed value of the population proportion from the null hypothesis, and n is the sample size.

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.23: Determining Which Distribution to Use for the Test Statistic in a Hypothesis Test For each scenario, identify which distribution should be used to test the claim. a.At least 15% of listeners of nonprofit radio stations generally favor commercials for other nonprofit organizations on the station. A local nonprofit radio station believes that less than 15% of its listeners favor such commercials. The station plans to survey a simple random sample of 50 of its listeners to test its claim.

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.23: Determining Which Distribution to Use for the Test Statistic in a Hypothesis Test (cont.) b.The college paper reports that at least 10% of students turn off their cell phones during their classes. Aggravated with the number of cell phones that ring during his classes, a professor believes that fewer than 10% of his students turn off their cell phones during the class period. He plans on testing his claim by asking a simple random sample of 20 of his students to show him their cell phones one day during class.

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.23: Determining Which Distribution to Use for the Test Statistic in a Hypothesis Test (cont.) c.The mean amount of rainfall in the southern part of Texas during the month of September is commonly believed to be no more than 3.02 inches. A meteorologist claims that the amount of rainfall this September has been higher than normal. He tests his claim by measuring the amounts of rainfall in a simple random sample of 12 locations across the region. Assume that the population standard deviation is unknown and the population distribution is approximately normal.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.23: Determining Which Distribution to Use for the Test Statistic in a Hypothesis Test (cont.) Solution a.The claim refers to a population proportion, thus we must test the conditions necessary to use the normal distribution. First note that the sample will be a simple random sample. The station plans to survey 50 listeners, so n = 50, and the claim is referencing 15%, so p = 0.15.

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.23: Determining Which Distribution to Use for the Test Statistic in a Hypothesis Test (cont.) Therefore, we can calculate the following. All of the conditions are met, so the station should use the normal distribution to test its claim.

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.23: Determining Which Distribution to Use for the Test Statistic in a Hypothesis Test (cont.) b.The claim refers to a population proportion, thus we must test the conditions necessary to use the normal distribution. The professor plans to survey 20 students, so n = 20, and the claim is referencing 10%, so p = 0.10. Therefore, we can calculate the following.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.23: Determining Which Distribution to Use for the Test Statistic in a Hypothesis Test (cont.) The condition that the sample size must be large enough such that np ≥ 5 is not met; therefore, the normal distribution cannot be used with the sample size given. If it is possible to increase the sample size to at least 50, then the normal distribution could be used. Otherwise, other methods are necessary.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.23: Determining Which Distribution to Use for the Test Statistic in a Hypothesis Test (cont.) c.The claim refers to a population mean, therefore we must use one of the distributions discussed in the previous sections. Note that the population standard deviation is unknown and the population is assumed to be normally distributed. Therefore, a Student’s t-distribution should be used.

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Hypothesis Testing for Population Proportions Conclusions Using p-Values If p-value ≤ , then reject the null hypothesis. If p-value > , then fail to reject the null hypothesis.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.24: Performing a Hypothesis Test for a Population Proportion (Left-Tailed) The local school board has been advertising that at least 65% of voters favor a tax increase to pay for a new school. A local politician believes that less than 65% of his constituents favor this tax increase. To test his belief, his staff asked a simple random sample of 50 of his constituents whether they favor the tax increase and 27 said that they would vote in favor of the tax increase. If the politician wishes to be 95% confident in his conclusion, does this information support his belief?

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.24: Performing a Hypothesis Test for a Population Proportion (Left-Tailed) (cont.) Solution Step 1: State the null and alternative hypotheses. The politician’s belief is that less than 65% of the constituents favor a tax increase. Written mathematically, this claim is p < 0.65. The logical opposite of this claim is p ≥ 0.65. Thus, the null and alternative hypotheses are stated as follows.

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.24: Performing a Hypothesis Test for a Population Proportion (Left-Tailed) (cont.) Step 2: Determine which distribution to use for the test statistic, and state the level of significance. We are testing a population proportion, so we must check the necessary conditions to use the normal distribution and the z-test statistic. A simple random sample of 50 constituents were surveyed, giving us n = 50, and we know from Step 1 that p = 0.65.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.24: Performing a Hypothesis Test for a Population Proportion (Left-Tailed) (cont.) Therefore, we check the conditions for the sample size as follows. Since all of the conditions are satisfied, we can use the z-test statistic for the sample proportion.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.24: Performing a Hypothesis Test for a Population Proportion (Left-Tailed) (cont.) For this hypothesis test, the level of confidence is 95%, so c = 0.95 and the level of significance is calculated as follows.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.24: Performing a Hypothesis Test for a Population Proportion (Left-Tailed) (cont.) Step 3: Gather data and calculate the necessary sample statistics. The sample data show that 27 out of 50 constituents favor the tax increase. Thus, the sample proportion is computed as follows.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.24: Performing a Hypothesis Test for a Population Proportion (Left-Tailed) (cont.) Substituting the necessary values into the formula for the test statistic for the sample proportion, we obtain the following z-score.

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.24: Performing a Hypothesis Test for a Population Proportion (Left-Tailed) (cont.) Step 4: Draw a conclusion and interpret the decision. The alternative hypothesis tells us that we are conducting a left-tailed test. Therefore, the p- value for this test statistic is the probability of obtaining a test statistic less than or equal to z =  1.63, written p-value To find the p-value, we need to find the area under the standard normal curve to the left of z =  1.63.

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.24: Performing a Hypothesis Test for a Population Proportion (Left-Tailed) (cont.) The p-value is 0.0516. Comparing this p-value to the level of significance, we see that 0.0516 > 0.05, so p- value > .

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.24: Performing a Hypothesis Test for a Population Proportion (Left-Tailed) (cont.) Thus, we fail to reject the null hypothesis. This means that, at the 0.05 level of significance, this evidence does not sufficiently support the politician’s belief that less than 65% of his constituents favor a tax increase to pay for a new school.

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.25: Performing a Hypothesis Test for a Population Proportion (Two-Tailed) According to a report published by the Mathematical Association of America, 1% of bachelor’s degrees in the United States are awarded in the fields of mathematics and statistics. One recent mathematics graduate believes that this number is incorrect. She has no indication if the actual percentage is more or less than 1%. To perform a hypothesis test, she uses a simple random sample of 12,317 recent college graduates for her sample.

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.25: Performing a Hypothesis Test for a Population Proportion (Two-Tailed) (cont.) According to the graduation programs from the colleges, she notes that bachelor’s degrees in mathematics or statistics were awarded to 148 graduates in her sample. Does this evidence support this graduate’s belief that the percentage of bachelor’s degrees awarded in the fields of mathematics and statistics is not 1%? Use a 0.10 level of significance. Source: Bressoud, David. “Status of the Math-Intensive Majors.” Launchings. Feb. 2011. http://www.maa.org/columns/launchings/launchings_02_11.html (1 Dec. 2011).

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.25: Performing a Hypothesis Test for a Population Proportion (Two-Tailed) (cont.) Solution Step 1: State the null and alternative hypotheses. The graduate is trying to show that the percentage of bachelor’s degrees awarded in the fields of mathematics and statistics is not 1%. Written mathematically, this is p ≠ 0.01. The logical opposite of this is p = 0.01. Thus, the null and alternative hypotheses are stated as follows.

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.25: Performing a Hypothesis Test for a Population Proportion (Two-Tailed) (cont.) Step 2: Determine which distribution to use for the test statistic, and state the level of significance. We are testing a population proportion, so we must check the necessary conditions to use the normal distribution and the z-test statistic. Recall from the information given that the sample was a simple random sample and that n = 12,317 and p = 0.01.

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.25: Performing a Hypothesis Test for a Population Proportion (Two-Tailed) (cont.) Therefore, we check the conditions for the sample size as follows. Due to the large sample size, it should have been obvious that both of these conditions would be easily satisfied.

29. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.25: Performing a Hypothesis Test for a Population Proportion (Two-Tailed) (cont.) Since all of the conditions are satisfied, we can use the z-test statistic for the sample proportion. For this hypothesis test, we were told to use a level of significance of  = 0.10.

30. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.25: Performing a Hypothesis Test for a Population Proportion (Two-Tailed) (cont.) Step 3: Gather data and calculate the necessary sample statistics. The sample data show that 148 out of 12,317 graduates obtained degrees in mathematics or statistics. Thus, the sample proportion is calculated as follows.

31. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.25: Performing a Hypothesis Test for a Population Proportion (Two-Tailed) (cont.) Note that we rounded the sample proportion to six decimal places, rather than three decimal places, to avoid additional rounding error in the following calculation of the z-score. Substituting the necessary values into the formula for the test statistic for the sample proportion, we obtain the following z-score.

32. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.25: Performing a Hypothesis Test for a Population Proportion (Two-Tailed) (cont.) Step 4: Draw a conclusion and interpret the decision. The alternative hypothesis tells us that we are conducting a two-tailed test. Therefore, the p-value for this test statistic is the probability of obtaining a test statistic that is either less than or equal to z 1 = −2.25 or greater than or equal to z 2 = 2.25, which is written mathematically as To find the p-value, we need to find the area under the standard normal curve to the left of z 1 = −2.25.

33. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.25: Performing a Hypothesis Test for a Population Proportion (Two-Tailed) (cont.) This area is 0.0122, which is just the area in the left tail. The total area is then twice this amount. Thus, the p- value is calculated as follows.

34. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.25: Performing a Hypothesis Test for a Population Proportion (Two-Tailed) (cont.) The p-value is 0.0244. Comparing this p-value to the level of significance, we see that 0.0244 ≤ 0.10, so p-value ≤ . Thus, we reject the null hypothesis. This means that, at the 0.10 level of significance, this evidence supports the belief that the percentage of bachelor’s degrees awarded in the fields of mathematics and statistics is not 1%. Notice that despite the sample proportion being larger than the assumed population proportion, our conclusion references only the original null and alternative hypotheses.

35. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.26: Performing a Hypothesis Test for a Population Proportion Using a TI-83/84 Plus Calculator (Right-Tailed) A crime watch group believes that the percentage of all inmates released from prison who were released because their sentences expired was higher in 2008– 2009 for the whole United States than it was for the state of Florida. Between July 1, 2008 and June 30, 2009, 63.9% of all prison inmates released in Florida were released because their sentences expired. Suppose that, in a simple random sample of 36,463 prison inmates in the United States who were released during the same time period, 64.6% were released because their sentences expired.

36. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.26: Performing a Hypothesis Test for a Population Proportion Using a TI-83/84 Plus Calculator (Right-Tailed) (cont.) Use a TI-83/84 Plus calculator to run a hypothesis test to determine if the watch group’s claim is correct at the 99% confidence level. Source: Florida Department of Corrections. “Annual Statistics for Fiscal Year 2008–2009: Inmate Releases and Time Served.” http://www.dc.state.fl.us/pub/ annual/0809/stats/im_release.html (1 Dec. 2011).

37. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.26: Performing a Hypothesis Test for a Population Proportion Using a TI-83/84 Plus Calculator (Right-Tailed) (cont.) Solution Step 1: State the null and alternative hypotheses. The watch group’s belief is that the proportion of inmates released because of an expired sentence in 2008–2009 was higher for the whole United States than it was for the state of Florida. Since the proportion for Florida was 63.9% in 2008–2009, we write this belief mathematically as p > 0.639. The logical opposite is p ≤ 0.639.

38. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.26: Performing a Hypothesis Test for a Population Proportion Using a TI-83/84 Plus Calculator (Right-Tailed) (cont.) Thus the null and alternative hypotheses are stated as follows. Step 2: Determine which distribution to use for the test statistic, and state the level of significance. Using the normal distribution, and thus the z-test statistic, in a hypothesis test for a population proportion requires first testing the necessary conditions.

39. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.26: Performing a Hypothesis Test for a Population Proportion Using a TI-83/84 Plus Calculator (Right-Tailed) (cont.) The sample is a simple random sample, the sample size is n = 36,463, and the presumed value of the population proportion from the null hypothesis is p = 0.639. Therefore, we check the conditions for the sample size as follows. Once again, the large sample size is an indication that both of these conditions are easily met.

40. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.26: Performing a Hypothesis Test for a Population Proportion Using a TI-83/84 Plus Calculator (Right-Tailed) (cont.) Therefore, we can use the z-test statistic for the sample proportion. For this hypothesis test, the confidence level is 99%, so c = 0.99 and the level of significance is calculated as follows.

41. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.26: Performing a Hypothesis Test for a Population Proportion Using a TI-83/84 Plus Calculator (Right-Tailed) (cont.) Step 3: Gather data and calculate the necessary sample statistics. Once again, this is where we begin to use a TI- 83/84 Plus calculator. We need three pieces of information to use the calculator to run a hypothesis test for the population proportion: n, x, and p 0. From above we know that n = 36,463 and p 0 = 0.639. The calculator wants a whole number for x and will return an error if you enter either a decimal or the percentage of the sample.

42. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.26: Performing a Hypothesis Test for a Population Proportion Using a TI-83/84 Plus Calculator (Right-Tailed) (cont.) So, we need to find the number of inmates in the sample who were released from prison because their sentences expired. Here’s where we actually have to work backwards a little bit. Since we were told that 64.6% of the inmates in the sample were released because of expired sentences, we can find x by multiplying 0.646 by the sample size, as follows.

43. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.26: Performing a Hypothesis Test for a Population Proportion Using a TI-83/84 Plus Calculator (Right-Tailed) (cont.) Because we cannot enter a decimal, we round to the nearest whole number. Approximately 23,555 inmates in the sample fit our characteristic, so x = 23,555. The last bit of information we must input into the calculator is the type of test we are running: left-tailed, right-tailed, or two-tailed. Because the alternative hypothesis states that the percentage for the United States was higher than the percentage for Florida, this is a right-tailed test, represented by >p€ on the calculator.

44. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.26: Performing a Hypothesis Test for a Population Proportion Using a TI-83/84 Plus Calculator (Right-Tailed) (cont.) To input these values into the calculator, press, scroll to TESTS, and choose option 5:1-PropZTest. After entering the values, select Calculate and press.

45. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10.26: Performing a Hypothesis Test for a Population Proportion Using a TI-83/84 Plus Calculator (Right-Tailed) (cont.) Step 4: Draw a conclusion and interpret the decision. The p-value given is approximately 0.0027, and our level of significance is  = 0.01. Since 0.0027 < 0.01, we have p-value ≤ . Thus, we reject the null hypothesis. This means that the evidence supports the watch group’s claim that the percentage of inmates who were released from prison in 2008–2009 because their sentences expired was higher for the whole United States than it was for the state of Florida.