2. Copyright © 2009 Pearson Education, Inc. Chapter 22 Comparing Two Proportions

3. Copyright © 2009 Pearson Education, Inc. Slide 1- 3 Objectives: The student will be able to: Find and interpret a confidence interval for the difference between two proportions, checking the necessary assumptions. Perform a two-proportion z-test, to include: writing appropriate hypotheses, checking the necessary assumptions, drawing an appropriate diagram, computing the P-value, making a decision, and interpreting the results in the context of the problem.

4. Copyright © 2009 Pearson Education, Inc. Slide 1- 4 Comparing Two Proportions Comparisons between two percentages are much more common than questions about isolated percentages. And they are more interesting. We often want to know how two groups differ, whether a treatment is better than a placebo control, or whether this year’s results are better than last year’s.

5. Copyright © 2009 Pearson Education, Inc. Slide 1- 5 The Sampling Distribution We already know that for under certain conditions for large enough samples, each of our proportions has an approximately Normal sampling distribution. The same is true of their difference.

6. Copyright © 2009 Pearson Education, Inc. Slide 1- 6 The Sampling Distribution (cont.) Provided that the sampled values are independent, the samples are independent, and the samples sizes are large enough, the sampling distribution of is modeled by a Normal model with Mean: Standard deviation:

7. Copyright © 2009 Pearson Education, Inc. Slide 1- 7 The Standard Deviation of the Difference Between Two Proportions Proportions observed in independent random samples are independent. Thus, we can add their variances. So… The standard deviation of the difference between two sample proportions is When the population proportions are unknown we can use the standard error:

8. Copyright © 2009 Pearson Education, Inc. Slide 1- 8 Assumptions and Conditions Independence Assumptions: Randomization Condition: The data in each group should be drawn independently and at random from a homogeneous population or generated by a randomized comparative experiment. The 10% Condition: If the data are sampled without replacement, the sample should not exceed 10% of the population. Independent Groups Assumption: The two groups we’re comparing must be independent of each other.

9. Copyright © 2009 Pearson Education, Inc. Slide 1- 9 Assumptions and Conditions (cont.) Sample Size Condition: Each of the groups must be big enough… Success/Failure Condition: Both groups are big enough that at least 10 successes and at least 10 failures have been observed in each.

10. Copyright © 2009 Pearson Education, Inc. Two Proportion Z-interval Since the sampling distribution of the difference between p 1 and p 2 is modeled by And since this can be approximated by We can find confidence intervals around using invNorm But what is even easier is to use the 2-proportion Z-interval function on your calculator Stat... Tests… 2-prop Zint Slide 1- 10

11. Copyright © 2009 Pearson Education, Inc. Slide 1- 11 FYI --- Two-Proportion z-Interval by hand When the conditions are met, we are ready to find the confidence interval for the difference of two proportions: The confidence interval is where The critical value z* depends on the particular confidence level, C, that you specify.

12. Copyright © 2009 Pearson Education, Inc. 2-prop Zint using calculator To find the confidence interval for the difference between the proportions of two Populations 84.9% of 12,460 males and 88.1% of 12,678 females indicated that they had high school diplomas. Find the 95% confidence interval for the difference in graduation rates between males and females. Stat... Tests...#B for 2-prop Zint enter x1: number of males with diploma;.849 times 12,460 or approximately 10,579 n1: 12460 x2: number of females with a diploma;.881 times 12,678 or approximately 11,170 n2: 12678 C-level:.95 You should get a confidence level from -4.1% to -2.4%. The percentages are negative because we let n1 represent the males and their percentage of graduates was lower than the female percentage. This means that we are 95% confident that the difference in the proportions of males and females that graduate is between 2.4 and 4.1 percentage points. NEVER report negative differences- not really appropriate. Slide 1- 12

13. Copyright © 2009 Pearson Education, Inc. Lets think about this… In the prior example we are 95% confident that the difference in the proportions of males and females that graduate is between 2.4 and 4.1 percentage points. Is it likely that the proportions of males and females that graduate are actually the same? What would have to be true of the difference of the proportions for this to be the case? What would our 95% confidence interval look like? It turns out we have an easier way of testing whether or not two sample proportions are really from the same population (i.e. the same) Slide 1- 13

14. Copyright © 2009 Pearson Education, Inc. 2 Proportion Z-test The 2 proportion z-test computes uses the z-score of the the difference between p^ 1 and p^ 2 (our proportions of 2 samples) to determine the probability that the true proportions are the same. Because our null hypothesis is that the two proportions are the same, our target difference is 0. The formula is below (don’t memorize this… we’ll use our calculator instead). Slide 1- 14

15. Copyright © 2009 Pearson Education, Inc. Slide 1- 15 FYI -- Two-Proportion z-Test (by hand) We use the pooled value to estimate the standard error: Now we find the test statistic: When the conditions are met and the null hypothesis is true, this statistic follows the standard Normal model, so we can use that model to obtain a P-value.

16. Copyright © 2009 Pearson Education, Inc. 2-propZtest … using calculator 120 subjects, 60 men and 60 women, were given the “water-level task” to test their perceptual and conceptual skills. Subjects were shown a drawing of a glass tilted at a 45 degree angle. They were then asked to draw a line representing the surface of the water. The researchers recorded whether or not the line drawn was within 5 degrees of the true line. These are the results of people who were within the 5 degrees. Are the proportions the same (no difference) using an alpha of 5%. x n women 24 60 men 39 60 Stat…Tests…2-propZtest You are testing to see if there is a difference, so choose p1 NOT equal to p2. You get a p value of.006 so you would reject the null indicating that there is a difference. Slide 1- 16

17. Copyright © 2009 Pearson Education, Inc. Slide 1- 17 FYI -- Everyone into the Pool – explanation of pooled successes The typical hypothesis test for the difference in two proportions is the one of no difference. In symbols, H 0 : p 1 – p 2 = 0. Since we are hypothesizing that there is no difference between the two proportions, that means that the standard deviations for each proportion are the same. Since this is the case, we combine (pool) the counts to get one overall proportion.

18. Copyright © 2009 Pearson Education, Inc. Slide 1- 18 FYI -- Everyone into the Pool (cont.) The pooled proportion is where and If the numbers of successes are not whole numbers, round them first. (This is the only time you should round values in the middle of a calculation.)

19. Copyright © 2009 Pearson Education, Inc. Slide 1- 19 FYI -- Everyone into the Pool (cont.) We then put this pooled value into the formula, substituting it for both sample proportions in the standard error formula:

20. Copyright © 2009 Pearson Education, Inc. Slide 1- 20 FYI -- Compared to What? We’ll reject our null hypothesis if we see a large enough difference in the two proportions. How can we decide whether the difference we see is large? Just compare it with its standard deviation. Unlike previous hypothesis testing situations, the null hypothesis doesn’t provide a standard deviation, so we’ll use a standard error (here, pooled).

21. Copyright © 2009 Pearson Education, Inc. Slide 1- 21 FYI -- Two-Proportion z-Test The conditions for the two-proportion z-test are the same as for the two-proportion z-interval. We are testing the hypothesis H 0 : p 1 = p 2. Because we hypothesize that the proportions are equal, we pool them to find

22. Copyright © 2009 Pearson Education, Inc. 18. A survey of 430 randomly selected adults found that 21% of the 222 men and 18% of the 208 women had purchased books online. Is there evidence that men are more likely to make online purchases of books? Use an alpha level of 0.05. H0: p1=p2 HA: p1>p2 Note that in this problem p1 refers to the men. Test statistic = z = 0.88 P-value = 0.19 Conclusion: Since the P-value is larger than alpha, we cannot reject the hypothesis that the true percentage of men and women who purchase books online are equal. The statistical evidence suggests that men don’t appear to be more likely to make online book purchases. Slide 1- 22

23. Copyright © 2009 Pearson Education, Inc. 19. Would being part of a support group that meets regularly help people who are wearing the nicotine patch actually quit smoking? A county health department tries an experiment using several hundred volunteers who are planning to use the patch. The subjects were randomly divided into two groups. People in Group 1 were given the patch and attended a weekly discussions meeting with counselors and others trying to quit. People in Group 2 also used the patch but did not participate in the counseling groups. After six months 46 of the 143 smokers in Group 1 and 30 of the 151 smokers in Group 2 had successfully stopped smoking. Do these results suggest that such support groups could be an effective way to help people stop smoking? Use an alpha level of 0.05. H0: p1=p2 HA: p1>p2 Note that in this problem p1 refers to Group 1 (i.e. the group who get counseling). Test statistic = z = 2.41 P-value = 0.008 Conclusion: Since the P-value is smaller than alpha, we reject the hypothesis that the true percentage of people who quit smoking is equal. The statistical evidence suggests that the counseling is beneficial to quitting smoking. Slide 1- 23

24. Copyright © 2009 Pearson Education, Inc. 20. When games were sampled from throughout a season, it was found that the home team won 127 of 198 professional basketball games, and the home team won 57 of 99 professional football games. Based on these results, does there appear to be a significant difference between the proportions of home wins for the two sports? What can we conclude about home field advantage for these two sports? Do hypothesis test with alpha=0.05 (Triola 2008). H0: p1=p2 HA: p1≠p2 In this problem p1 will refer to basketball. Test statistic = z = 1.10 P-value = 0.27 Conclusion: Since the P-value is larger than alpha, we cannot reject the hypothesis that the true percentage of home games won in basketball vs. football is the same. The statistical evidence suggests that there doesn’t appear to be MORE of a home-field advantage for one sport over the other. NOTE THAT WE DID NOT TEST WHETHER EITHER OF THE SPORTS HAS A HOME-FIELD ADVANTAGE. We just tested whether one sport has MORE of a home-field advantage. Slide 1- 24

25. Copyright © 2009 Pearson Education, Inc. 21. A gender selection methodology called “XSORT” yielded the following results for parents who WANTED a girl: 295 out of 325 babies born using the method were girls. For those parents who wanted a boy (the “YSORT” method was used for these parents), 39 out of 51 babies were boys. Perform a hypothesis test with alpha=0.05 for the difference between the proportions of boys and girls being born using these gender selection methodologies (Triola 2008). H0: p1=p2 HA: p1≠p2 In this problem p1 will refer to girls. Test statistic = z = 3.01 P-value = 0.003 Conclusion: Since the P-value is smaller than alpha, we reject the hypothesis that the two population proportions are the same. The statistical evidence suggests that the XSORT (girl) methodology is superior to the YSORT (boy) methodology. NOTE THAT WE DIDN’T TEST WHETHER EITHER OF THEM IS EFFECTIVE... WE JUST COMPARED THE TWO METHODS. Slide 1- 25

26. Copyright © 2009 Pearson Education, Inc. Slide 1- 26 What Can Go Wrong? Don’t use two-sample proportion methods when the samples aren’t independent. These methods give wrong answers when the independence assumption is violated. Don’t apply inference methods when there was no randomization. Our data must come from representative random samples or from a properly randomized experiment. Don’t interpret a significant difference in proportions causally. Be careful not to jump to conclusions about causality.

27. Copyright © 2009 Pearson Education, Inc. Slide 1- 27 What have we learned? We’ve now looked at inference for the difference in two proportions. Perhaps the most important thing to remember is that the concepts and interpretations are essentially the same—only the mechanics have changed slightly.

28. Copyright © 2009 Pearson Education, Inc. Slide 1- 28 What have we learned? Hypothesis tests and confidence intervals for the difference in two proportions are based on Normal models. Both require us to find the standard error of the difference in two proportions. We do that by adding the variances of the two sample proportions, assuming our two groups are independent. When we test a hypothesis that the two proportions are equal, we pool the sample data; for confidence intervals we don’t pool.