Statistical Analysis: Inference and Testing Procedures

1. Stat 31, Section 1, Last Time • T distribution • For unknown , replace with • Compute with TDIST & TINV (different!) • Paired Samples • Similar to above, work with differences • Inference for Proportions • Counts & Proportions • CIs: Best Guess & Conservative

2. Inference for proportions Case 2: Choice of Sample Size: Idea: Given the margin of error , find sample size to make: i.e. Dist’n i.e. Dist’n 0.95 0.975

3. Sample Size for Proportions i.e. find so that i.e. Problem: in both cases, can’t “get at” Solution: Standardize, i.e. put on N(0,1) scale

4. Inference for proportions I.e. Find so that N(0,1) dist’n 0.975

5. Sample Size for Proportions i.e. find so that: Now solve to get: Problem: don’t know

6. Sample Size for Proportions Solution 1: Best Guess Use from: • Earlier Study • Previous Experience • Prior Idea

7. Sample Size for Proportions Solution 2: Conservative Recall So “safe” to use:

8. Sample Size for Proportions E.g. Old textbook problem 8.14 (now 8.16) An opinion poll found that 44% of adults agree that parents should be given vouchers for education at a school of their choice. The result was based on a small sample. How large an SRS is required to obtain a margin of error of +- 0.03, in a 95% CI?

9. Sample Size for Proportions E.g. Old textbook problem 8.14 (now 8.16) See Class Example 26, Part 2: https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg26.xls

10. Sample Size for Proportions Note: conservative version not much bigger, since 0.44 ~ 0.5 so gap is small 0.44 0.5

11. Sample Size for Proportions HW: 8.23, 8.25, give both “best guess” and “conservative” answers

12. Hypo. Tests for Proportions Case 3: Hypothesis Testing General Setup: Given Value

13. Hypo. Tests for Proportions Assess strength of evidence by: P-value = P{what saw or m.c. | B’dry} = = P{observed or m.c. | p = } Problem: sd of

14. Hypo. Tests for Proportions Problem: sd of Solution: (different from above “best guess” and “conservative”) calculation is done base on:

15. Hypo. Tests for Proportions e.g. Old Text Problem 8.16 (now 8.18) Of 500 respondents in a Christmas tree marketing survey, 44% had no children at home and 56% had at least one child at home. The corresponding figures from the most recent census are 48% with no children, and 52% with at least one. Test the null hypothesis that the telephone survey has a probability of selecting a household with no children that is equal to the value of the last census. Give a Z-statistic and P-value.

16. Hypo. Tests for Proportions e.g. Old Text Problem 8.16 (now 8.18) Let p = % with no child (worth writing down)

17. Hypo. Tests for Proportions Observed , from P-value =

18. Hypo. Tests for Proportions P-value = 2 * NORMDIST(0.44,0.48,sqrt(0.48*(1-0.48)/500),true) See Class Example 26, Part 3 https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg26.xls = 0.0734 Yes-No: no strong evidence Gray-level: somewhat strong evidence

19. Hypo. Tests for Proportions Z-score version: P-value = So Z-score is: = 1.79

20. Hypo. Tests for Proportions Note also 1-sided version: Yes-no: is strong evidence Gray Level: stronger evidence HW: 8.19, 8.21, interpret from both yes-no and gray-level viewpoints

21. And now for somethingcompletely different…. Another fun movie Thanks to Trent Williamson

22. Chapter 9: Two-Way Tables Main idea: Divide up populations in two ways • E.g. 1: Age & Sex • E.g. 2: Education & Income • Typical Major Question: How do divisions relate? • Are the divisions independent? • Similar idea to indepe’nce in prob. Theory • Statistical Inference?

23. Two-Way Tables Class Example 40, Textbook Problem 9.20 Market Researchers know that background music can influence mood and purchasing behavior. A supermarket compared three treatments: No music, French accordion music and Italian string music. Under each condition, the researchers recorded the numbers of bottles of French, Italian and other wine purshased.

24. Two-Way Tables Class Example 40, Textbook Problem 9.20 Here is the two way table that summarizes the data: Are the type of wine purchased, and the background music related?

25. Two-Way Tables Class Example 40: Visualization Shows how counts are broken down by: music type wine type

26. Two-Way Tables Big Question: Is there a relationship? Note: tallest bars French Wine  French Music Italian Wine  Italian Music Other Wine  No Music Suggests there is a relationship

27. Two-Way Tables General Directions: • Can we make this precise? • Could it happen just by chance? • Really: how likely to be a chance effect? • Or is it statistically significant? • I.e. music and wine purchase are related?

28. Two-Way Tables Class Example 40, a look under the hood… Excel Analysis, Part 1: https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg40.xls Notes: • Read data from file • Only appeared as column • Had to re-arrange • Better way to do this??? • Made graphic with chart wizard

29. Two-Way Tables HW: Make 2-way bar graphs, and discuss relationships between the divisions, for the data in: 9.1 (younger people tend to be better educated) 9.13 (you try these…) 9.15

30. Two-Way Tables An alternate view: Replace counts by proportions (or %-ages) Class Example 40 (Wine & Music), Part 2 https://www.unc.edu/~marron/UNCstat31-2005/Stat31Eg40.xls Advantage: May be more interpretable Drawback: No real difference (just rescaled)