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3. What if? Researchers convinced popul variances equal. Estimate of common popul std dev: Pooled standard error (same!): df = 16 + 16 – 2 = 30, so t* = 2.04 95% Pooled CI: (13.88)  (2.04)(6.41)  (0.80, 26.96) Also excludes 0; but is a bit narrower.

4. 13.4 Testing about Difference in Two Popul Means  1 -  2 Typical Summary of Responses for a Two Independent Samples Problem:  1 = population mean for first population  2 = population mean for second population Parameter =  1 -  2 Estimate = Standard error = Pooled standard error = PopulationSample Size Sample Mean Sample Standard Deviation 1 n1n1 2 n2n2

5. 13.4 Testing about Difference in Two Popul Means  1 -  2 Both pooled and unpooled require … Independent random samples from normal populations (but if sample sizes are large, assumption of normality not so crucial). 1.H 0 :H a : 2.H 0 : H a : 3.H 0 : H a :

6. Test Statistic Test statistic = Sample statistic – Null value Standard error

7. Learning about the Difference in Two Popul Means Pooled or Unpooled? Guidelines: If sample std devs similar  POOL If sample std devs not similar   If larger std dev from group with larger sample size, pooled acceptable, conservative (produces larger p-value).  If smaller std dev from group with larger sample size, pooled produces misleading smaller p-value. Bottom-line: Pool if reasonable; but if sample std devs not similar, we have unpooled.

8. Try It! Effect of Beta-blockers on pulse rates Do beta-blockers reduce the pulse rate? 60 subjects randomly divided into 2 groups of 30 1 = beta-blocker, 2 = placebo Think about part (a) stating the hypotheses and be ready to clicker in your answer. Group Sample size Sample mean Sample std deviation 1=beta-block3065.27.8 2=placebo3070.38.4

9. a. State hypotheses to assess if beta-blockers reduce pulse rate on average. 1 = beta-blocker, 2 = placebo A) H 0 :  1 =  2 vs H a :  1 ≠  2 B) H 0 :  1 =  2 vs H a :  1 <  2 C) H 0 :  1 =  2 vs H a :  1 >  2

10. b. Which test will you perform? Pooled OR Unpooled? Group Sample size Sample mean Sample std deviation 1=beta-bl3065.27.8 2=placebo3070.38.4

11. c. Perform the t-test of H 0 :  1 =  2 vs H a :  1 <  2 Are the results significant at a 5% level? Group Sample size Sample mean Sample std deviation 1=beta-bl3065.27.8 2=placebo3070.38.4

12. c. Test of H 0 :  1 =  2 vs H a :  1 <  2 (continued) Are the results significant at a 5% level?

13. Try It! Does the Drug Speed Learning? Researcher to assess if a drug speeds learning 5 rats (Group 1 = control group) required to run a maze without use of drug Second independent group of 8 rats (Group 2 = experimental group) administered drug Running times (time to complete the maze) for rats in each group entered into SPSS. Conduct test at 5% level.

14. Try It! Does the Drug Speed Learning? H 0 : ________________________ H a : ________________________ Test statistic: ________________ p-value: ____________________ Decision: Fail to reject H 0 Reject H 0 Thus …

15. Try It! Eat that Dark Chocolate Researchers fed small 1.6-ounce bar of dark chocolate to each of 22 volunteers daily for two weeks.  Half of subjects randomly selected and received bars containing dark chocolate’s high levels of flavonoids,  Other half received placebo bars with trace amts flavonoids Ability of brachial artery to dilate significantly improved for high-flavonoid group compared to placebo group.   1 = population avg improvement in blood flow for flavonoid   2 = population avg improvement in blood flow for placebo

16. Try It! Eat that Dark Chocolate (continued) Researchers tested: high-flavonoid group have higher average improvement in blood flow. a. State the hypotheses. H 0 : _____________________ H a : _____________________

17. Try It! Eat that Dark Chocolate (continued) b.Researchers conducted pooled two sample t-test. Two assumptions about data are that the two samples are independent random samples. i. State a remaining assumption regarding populations. ii. Explain how to use data to assess that assumption. Think about parts (c) and (d) and be ready to clicker in your answers.

18. c.Significance level of 0.05 used. Based on statements, what can say about the p-value? A) p-value > 0.05 B) p-value ≤ 0.05 C) Can’t tell

19. d.Researchers found concentrations of cocoa flavonoid epicatechin soared in group that received high-flavonoid chocolate, rising from baseline of 25.6 nmol/L to 204.4 nmol/L. Average improvement of 204.4–25.6= 178.8 nmol/L is a … CLICKER SELECT ALL THAT APPLY A) Parameter B) Statistic C) Sample mean D) Population mean E) Sampling distribution

20. Yellow Card on the Big 5 Parameters

21. 13.6 Choosing an Appropriate Inference Procedure Determining the Appropriate Parameter p: One population proportion  : One population mean  d : Paired difference population mean  1 -  2 : Difference between two population means p 1 - p 2 : Difference between two population proportions

22. 13.6 Choosing an Appropriate Inference Procedure Questions to Ask: Is the response measured quantitative or categorical? How many samples? What is the main purpose? From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with permission.

23. It’s time for … Name that Scenario Take out your copy of that worksheet that you were to bring to class today! Look it over and we will start in a few minutes!

24. Background: Bobby and Barney are the owners of a bakery. They are considering packaging their “mini” chocolate chip cookies in individual bags for vending machine sales. Before they embark on this endeavor, they have many plans to formulate and decisions to make. Bobby is in charge of production and Barney is in charge of marketing. They need your help. For each issue below, select the most appropriate statistical analysis technique for addressing that issue.

25. 1. Is the average number of chocolate chips in our cookies higher than the average number of chocolate chips in our competitor’s cookies? Enter your choice. A. 1-sample t-test for a popul mean B. Paired t-test C. 2-sample t-test for comparing two popul means D. 1-sample z-test for a popul proportion E. 2-sample z-test for comparing two popul proportions Name that Scenario …

26. 1.Is the average number of chocolate chips in our cookies higher than the average number of chocolate chips in our competitor’s cookies? C.2-sample t-test for comparing two popul means What would the hypotheses look like? What would the test statistic look like? What is the distribution of the test statistic under the null hypothesis? What conditions are required?

27. 2.Is the percentage of defective seals under the first option different from the percentage of defective seals under the second option? Enter your choice. A. 1-sample t-test for a popul mean B. Paired t-test C. 2-sample t-test for comparing two popul means D. 1-sample z-test for a popul proportion E. 2-sample z-test for comparing two popul proportions

28. 2.Is the percentage of defective seals under the first option different from the percentage of defective seals under the second option? E.2-sample z-test for comparing two popul proportions What would the hypotheses look like? What would the test statistic look like? What is the distribution of the test statistic under the null hypothesis? What conditions are required?

29. 3. Two scales: We weighed the same 10 bags of cookies on each scale. Do the two scales produce different weights on average? Enter your choice. A. 1-sample t-test for a popul mean B. Paired t-test C. 2-sample t-test for comparing two popul means D. 1-sample z-test for a popul proportion E. 2-sample z-test for comparing two popul proportions

30. 3. Do the two scales produce different weights on average? B.Paired t-test What would the hypotheses look like? What would the test statistic look like? What is the distribution of the test statistic under the null hypothesis? What conditions are required?

31. 4. Our bags of cookies are listed as 20 oz. A sample of bags is selected and weighed to assess if the average weight is less than 20 oz. Enter your choice. A. 1-sample t-test for a popul mean B. Paired t-test C. 2-sample t-test for comparing two popul means D. 1-sample z-test for a popul proportion E. 2-sample z-test for comparing two popul proportions

32. 4. A sample of bags is selected and weighed to assess if the average weight is less than 20 oz. A.1-sample t-test for a population mean What would the hypotheses look like? What would the test statistic look like? What is the distribution of the test statistic under the null hypothesis? What conditions are required?

33. 5. A sample of bags are weighed to assess if more than 90% are meeting the specifications (meeting specs = weigh at least 20 oz). Enter your choice. A. 1-sample t-test for a popul mean B. Paired t-test C. 2-sample t-test for comparing two popul means D. 1-sample z-test for a popul proportion E. 2-sample z-test for comparing two popul proportions

34. 5. A sample of bags are weighed to assess if more than 90% are meeting the specifications (meeting specs = weigh at least 20 oz). D.1-sample z-test for a population proportion What would the hypotheses look like? What would the test statistic look like? What is the distribution of the test statistic under the null hypothesis? What conditions are required?