1. Stat 31, Section 1, Last Time Paired Diff’s vs. Unmatched Samples
Compare with example
Showed graphic about Paired often better
Review of Gray Level Hypo Testing
Inference for Proportions
Confidence Intervals
Sample Size Calculation

2. Reading In Textbook Approximate Reading for Today’s Material:
Pages , ,
Approximate Reading for Next Class:
Pages ,

3. Midterm II Coming on Tuesday, April 10 Think about: Sheet of Formulas
Again single 8 ½ x 11 sheet
New, since now more formulas
Redoing HW…
Asking about those not understood
Midterm not cumulative
Covered Material: HW

4. Midterm II Extra Office Hours: Monday, 4/9, 10:00 – 12:00 12:30 – 3:00
Tuesday, 4/10, 8:30 – 10:00
11:00 – 12:00

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

6. 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

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

8. Hypo. Tests for Proportions
e.g. Old Text Problem 8.16
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.

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

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

11. Hypo. Tests for Proportions
P-value
= 2 * NORMDIST(0.44,0.48,sqrt(0.48*(1-0.48)/500),true)
See Class Example 30, Part 3 =
Yes-No: no strong evidence
Gray-level: somewhat strong evidence

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

13. Hypo. Tests for Proportions
Note also 1-sided version:
Yes-no: is strong evidence
Gray Level: stronger evidence
HW: a (0.0057), , interpret from both yes-no and gray-level viewpoints

14. 2 Sample Proportions In text Section 8.2 Skip this
Ideas are only slight variation of above
Basically mix & Match of 2 sample ideas, and proportion methods
If you need it (later), pull out text
Covered on exams to extent it is in HW

15. 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?

16. Two-Way Tables Class Example 31, Textbook Example 9.18
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.

17. Two-Way Tables Class Example 31, Textbook Example 9.18
Here is the two way table that summarizes the data:
Are the type of wine purchased, and the background music related?
Music
Wine:
None
French
Italian 30 39 11 1 19
Other 43 35

18. Two-Way Tables Class Example 31: Visualization
Shows how counts are broken down by:
music type wine type

19. 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

20. 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?

21. Two-Way Tables Class Example 31, a look under the hood…
Excel Analysis, Part 1:
Notes:
Read data from file
Only appeared as column
Had to re-arrange
Better way to do this???
Made graphic with chart wizard

22. Two-Way Tables
HW: Make 2-way bar graphs, and discuss relationships between the divisions, for the data in:
(younger people tend to be better educated)
(you try these…)
9.11

23. Class Example 31 (Wine & Music), Part 2
Two-Way Tables
An alternate view:
Replace counts by proportions (or %-ages)
Class Example 31 (Wine & Music), Part 2
Advantage:
May be more interpretable
Drawback:
No real difference (just rescaled)

24. Two-Way Tables Testing for independence: What is it?
From probability theory:
P{A | B} = P{A}
i.e. Chances of A, when B is known, are same as when B is unknown
Table version of this idea?

25. Independence in 2-Way Tables
Recall:
P{A | B} = P{A}
Counts - proportions analog of these?
Analog of P{A}?
Proportions of factor A, “not knowing B”
Called “marginal proportions”
Analog of P{A|B}???

26. Independence in 2-Way Tables
Marginal proportions (or counts):
Sums along rows
Sums along columns
Useful to write at margins of table
Hence name marginal
Number of independent interest
Also nice to put total at bottom

27. Independence in 2-Way Tables
Marginal Counts:
Class Example 31 (Wine & Music), Part 3
Marginals are of independent interest:
Other wines sold best (French second)
Italian music sold most wine…
But don’t tell whole story
E.g.Can’t see same music & wine is best…
Full table tells more than marginals

28. Independence in 2-Way Tables
Recall definition of independence:
P{A | B} = P{A}
Counts analog of P{A|B}???
Recall:
So equivalent condition is:

29. Independence in 2-Way Tables
Counts analog of P{A|B}???
Equivalent condition for independence is:
So for counts, look for:
Table Prop’n = Row Marg’l Prop’n x Col’n Marg’l Prop’n
i.e. Entry = Product of Marginals

30. Independence in 2-Way Tables
Visualize Product of Marginals for:
Class Example 31 (Wine & Music), Part 4
Shows same structure
as marginals
But not match between
music & wine
Good null hypothesis

31. Independence in 2-Way Tables
Independent model appears different
But is it really different?
Or could difference be simply explained by natural sampling variation?
Check for statistical significance…

32. Independence in 2-Way Tables
Approach:
Measure “distance between tables”
Use Chi Square Statistic
Has known probability distribution when table is independent
Assess significance using P-value
Set up as: H0: Indep. HA: Dependent
P-value = P{what saw or m.c. | Indep.}

33. Independence in 2-Way Tables
Chi-square statistic: Based on:
Observed Counts (raw data),
Expected Counts (under indep.),
Notes:
Small for only random variation
Large for significant departure from indep.

34. Independence in 2-Way Tables
Chi-square statistic calculation:
Class example 31, Part 5:
Calculate term by term
Then sum
Is X2 = “big” or “small”?

35. Independence in 2-Way Tables
H0 distribution of the X2 statistic:
“Chi Squared” (another Greek letter )
Parameter: “degrees of freedom”
(similar to T distribution)
Excel Computation:
CHIDIST (given cutoff, find area = prob.)
CHIINV (given prob = area, find cutoff)

36. Independence in 2-Way Tables
Explore the distribution:
Applet from Webster West (U. So. Carolina)
Right Skewed Distribution
Nearly Gaussian for more d.f.

37. Independence in 2-Way Tables
For test of independence, use:
degrees of freedom =
= (#rows – 1) x (#cols – 1)
E.g. Wine and Music:
d.f. = (3 – 1) x (3 – 1) = 4

38. Independence in 2-Way Tables
E.g. Wine and Music:
P-value = P{Observed X2 or m.c. | Indep.} =
= P{X2 = 18.3 of m.c. | Indep.} =
= P{X2 >= 18.3 | d.f. = 4} = =
Also see Class Example 31, Part 5

39. Independence in 2-Way Tables
E.g. Wine and Music:
P-value = 0.001
Yes-No: Very strong evidence against independence, conclude music has a statistically significant effect
Gray-Level: Also very strong evidence

40. Independence in 2-Way Tables
Excel shortcut:
CHITEST
Avoids the (obs-exp)^2 / exp calculat’n
Automatically computes d.f.
Returns P-value