1. Author(s): Brenda Gunderson, Ph.D., 2011 License: Unless otherwise noted, this material is made available under the terms of the Creative Commons Attribution–Non-commercial–Share Alike 3.0 License: http://creativecommons.org/licenses/by-nc-sa/3.0/ We have reviewed this material in accordance with U.S. Copyright Law and have tried to maximize your ability to use, share, and adapt it. The citation key on the following slide provides information about how you may share and adapt this material. Copyright holders of content included in this material should contact open.michigan@umich.edu with any questions, corrections, or clarification regarding the use of content. For more information about how to cite these materials visit http://open.umich.edu/education/about/terms-of-use. Any medical information in this material is intended to inform and educate and is not a tool for self-diagnosis or a replacement for medical evaluation, advice, diagnosis or treatment by a healthcare professional. Please speak to your physician if you have questions about your medical condition. Viewer discretion is advised: Some medical content is graphic and may not be suitable for all viewers.

2. Attribution Key for more information see: http://open.umich.edu/wiki/AttributionPolicy Use + Share + Adapt Make Your Own Assessment Creative Commons – Attribution License Creative Commons – Attribution Share Alike License Creative Commons – Attribution Noncommercial License Creative Commons – Attribution Noncommercial Share Alike License GNU – Free Documentation License Creative Commons – Zero Waiver Public Domain – Ineligible: Works that are ineligible for copyright protection in the U.S. (17 USC § 102(b)) *laws in your jurisdiction may differ Public Domain – Expired: Works that are no longer protected due to an expired copyright term. Public Domain – Government: Works that are produced by the U.S. Government. (17 USC § 105) Public Domain – Self Dedicated: Works that a copyright holder has dedicated to the public domain. Fair Use: Use of works that is determined to be Fair consistent with the U.S. Copyright Act. (17 USC § 107) *laws in your jurisdiction may differ Our determination DOES NOT mean that all uses of this 3rd-party content are Fair Uses and we DO NOT guarantee that your use of the content is Fair. To use this content you should do your own independent analysis to determine whether or not your use will be Fair. { Content the copyright holder, author, or law permits you to use, share and adapt. } { Content Open.Michigan believes can be used, shared, and adapted because it is ineligible for copyright. } { Content Open.Michigan has used under a Fair Use determination. }

3. Yellow Card on the Big 5 Parameters We have covered the first column, moving into column 2

4. The Independent Samples Scenario page 95 Two samples are said to be independent samples when the measurements in one sample are not related to the measurements in the other sample.

5. Independent Samples Scenario Ways that independent samples can occur: Random samples are taken separately from two populations and same response variable is recorded for each individual. One random sample is taken and a variable recorded for each individual, but then units are categorized as belonging to one population or another (e.g. M/F). Participants randomly assigned to one of two treatment conditions, same response variable recorded for each unit.

6. The Independent Samples Scenario If response variable is categorical, a researcher might compare the two independent groups by looking at the difference between the two proportions p 1 – p 2

7. Example of Two Proportion Scenario We want to … estimate value of the difference p 1 – p 2 test hypothesis that difference is 0, p 1 – p 2 = 0  two population proportions are equal

8. 9.5 SD Module 2: Sampling Distribution for the Difference Between Two Sample Proportions Want to compare men versus women: p 1 = population proportion of men who respond yes p 2 = population proportion of women who respond yes Want to learn about p 1 – p 2 … estimate with Will it be a good estimate? What are the possible values for if we took many sets of independent random samples of same sizes from two populations? What can we say about the distribution of the difference in two sample proportions? Driving Safely: Time Poll of American Adults … page 96 “Have you ever driven a car when you probably had too much alcohol to drive safely?”

9. Now we need a little from Section 8.8 Sums, Differences, Combin of Random Variables pg 63 Sum = X + YDifference = X – Y Rules for Means: Mean(X + Y) = Mean(X) + Mean(Y) Mean(X – Y) = Mean(X) – Mean(Y) Rules for Variances (if X and Y are independent): Variance(X + Y) = Variance(X) + Variance(Y) Variance(X – Y) = Variance(X) + Variance(Y)

10. So from Rules for Means and Variances… Section 8.8: Mean of difference = difference in means Variance of difference = sum of variances (if indep) Remember standard deviation of a sample proportion? = Apply this to our difference of …

11. Sampling Distribution of the Difference in Two (Independent) Sample Proportions If two sample proportions are based on independent random samples from the two populations and if all quantities,, and are at least 10, then is (approximately)…

12. Standard Error for the Difference in Two Sample Proportions … estimates, roughly, the average distance of the possible values from p 1 – p 2. We will use this standard error to form a CI for and test hypotheses about p 1 – p 2.

13. 10.4 Confidence Interval for Difference between Two Popul Proportions (page 99) p 1 = population proportion for first population p 2 = population proportion for second population Parameter = p 1 – p 2 Estimate = Standard error =

14. CI for the Difference in Population Proportions Two Independent-Samples z Confidence Interval for p 1 - p 2 where and z* is an appropriate value from N(0,1) distribution. This interval requires the sample proportions are based on independent random samples from the two populations. Also,,,, and be preferably at least 10. Sample Estimate  Multiplier x Std error

15. Try It! DWI Rates Question: How much of a difference between men and women with regard to proportion who have driven a car when they had too much alcohol to drive safely? Results of Time poll: Of 300 men who answered, 189 (63%) said yes and 108 (36%) said no. Remaining 3 weren’t sure. Of 300 women, 87 (29%) said yes and 210 (70%) said no; remaining 3 weren’t sure. Compute approx 95% CI for difference in proportions of men and women who would say yes.

16. Try It! DWI Rates 95% CI for difference in population proportions Of 300 men who answered, 189 (63%) said yes. Of 300 women, 87 (29%) said yes.

17. Try It! DWI Rates 95% CI for p M - p W : (0.265, 0.415) Complete the sentence to interpret the interval … With 95% confidence, we estimate the difference in the proportion of men versus women who have driven after having too much to drink to be somewhere between __________ and _________.

18. Try It! DWI Rates 95% CI for p M - p W : (0.265, 0.415) What value do you notice is not in this interval? _________ Q: Does there appear to be a significant difference between popul rates for men versus women? Click in: A = YES or B = NO

19. 12.4 Testing about Difference in Two Population Proportions page 103 p 1 = population proportion for first population p 2 = population proportion for second population Parameter = p 1 – p 2 Estimate = Standard error = Standard error used in constructing CI is not the same as that used for the standardized z test statistic.

20. 12.4 Testing about Difference in Two Popul Proportions Possible Hypotheses … 1.H 0 :H a : 2.H 0 : H a : 3.H 0 : H a : Test statistic = Sample statistic – Null value Standard error

21. Developing the Test Statistic If H 0 is true, p 1 - p 2 = 0 or p 1 = p 2 = p. Reasonable estimate of common population proportion p? General Standard error: but if H 0 true, then is best estimate  use in standard error. So null standard error:

22. Two Sample Z Test Statistic Standardized test statistic is: If H 0 is true, z will have a _____________ distribution. This distribution used to find p-value. Conditions: Test requires sample proportions based on indep random samples from 2 populations. Also need sample sizes to be large enough (note: using ).

23. Try It! Returning Lost Money page 105 Sample of college students asked if would return $$ if found a wallet on the street. Of 93 women, 83 said they would. Of 75 men, 53 said they would. Assume these represent all college students. Test hypothesis that equal proportions of college men and women would return money versus alternative hypothesis that a higher proportion of women would do so. Use a 5% significance level. Let 1 = women and 2 = men. Try stating the hypotheses and clicker in answer.

24. Which hypotheses for assessing if higher proportion of women would return $$? 1 = women, 2 = men A) H 0 : p 1 = p 2 vs H a : p 1 ≠ p 2 B) H 0 : p 1 = p 2 vs H a : p 1 < p 2 C) H 0 : p 1 = p 2 vs H a : p 1 > p 2

25. Perform the z-test of H 0 : p 1 = p 2 vs H a : p 1 > p 2 at a 5% level.  Of 93 women, 83 said they would.  Of 75 men, 53 said they would.

26. Test of H 0 : p 1 = p 2 vs H a : p 1 > p 2 (continued) Are the results significant at a 5% level?