Last Time

1. Last Time • Hypothesis Testing • 1-sided vs. 2-sided Paradox • Big Picture Goals • Hypothesis Testing • Margin of Error • Sample Size Calculations • Visualization • Histograms

2. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • Excel notation to avoid actual calculation • So no computers or calculators • Bring sheet of formulas, etc.

3. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • Excel notation to avoid actual calculation • So no computers or calculators • Bring sheet of formulas, etc. • No blue books needed

4. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • Excel notation to avoid actual calculation • So no computers or calculators • Bring sheet of formulas, etc. • No blue books needed (will just write on my printed version)

5. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • Material Covered: HW 1 – HW 5

6. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • Material Covered: HW 1 – HW 5 • Note: due Thursday, Feb. 19

7. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • Material Covered: HW 1 – HW 5 • Note: due Thursday, Feb. 19 • Will ask grader to return Mon. Feb. 23

8. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • Material Covered: HW 1 – HW 5 • Note: due Thursday, Feb. 19 • Will ask grader to return Mon. Feb. 23 • Can pickup in my office (Hanes 352)

9. Administrative Matters Midterm I, coming Tuesday, Feb. 24 • Material Covered: HW 1 – HW 5 • Note: due Thursday, Feb. 19 • Will ask grader to return Mon. Feb. 23 • Can pickup in my office (Hanes 352) • So today’s HW not included

10. Reading In Textbook Approximate Reading for Today’s Material: Pages 261-262, 9-14, 270-276, 30-34 Approximate Reading for Next Class: Pages 279-282, 34-43

11. Big Picture • Hypothesis Testing (Given dist’n, answer “yes-no”) • Margin of Error (Find dist’n, use to measure error) • Choose Sample Size (for given amount of error) Need better prob. tools

12. Big Picture • Margin of Error • Choose Sample Size Need better prob tools Start with visualizing probability distributions (key to “alternate representation”)

13. Histograms Idea: show rectangles, where area represents

14. Histograms Idea: show rectangles, where area represents: • Distributions: probabilities • Lists (of numbers): # of observations

15. Histograms Idea: show rectangles, where area represents: • Distributions: probabilities • Lists (of numbers): # of observations Note: will studies these in parallel for a while (several concepts apply to both)

16. Histograms Idea: show rectangles, where area represents: • Distributions: probabilities • Lists (of numbers): # of observations Caution: There are variations not based on areas, see bar graphs in text But eye perceives area, so sensible to use it

17. Histograms Steps for Constructing Histograms: • Pick class intervals that contain full dist’n

18. Histograms Steps for Constructing Histograms: • Pick class intervals that contain full dist’n a. Prob. dist’ns: If possible values are: x = 0, 1, … , n, get good picture from choice: [-½, ½), [½, 1.5), [1.5, 2.5), … , [n-½, n+½) where [1.5, 2.5) is “all #s ≥ 1.5 and < 2.5” (called a “half open interval”)

19. Histograms Steps for Constructing Histograms: • Pick class intervals that contain full dist’n a. Prob. dist’ns b. Lists: e.g. 2.3, 4.5, 4.7, 4.8, 5.1 Start with [1,3), [3,7) • As above use half open intervals (to break ties)

20. Histograms Steps for Constructing Histograms: • Pick class intervals that contain full dist’n a. Prob. dist’ns b. Lists: e.g. 2.3, 4.5, 4.7, 4.8, 5.1 Start with [1,3), [3,7) • Can use anything for class intervals • But some choices better than others…

21. Histograms Steps for Constructing Histograms: • Pick class intervals that contain full dist’n • Find “probabilities” or “relative frequencies” for each class (a) Probs: use f(x) for [x-½, x+½), etc. (b) Lists: [1,3): rel. freq. = 1/5 = 20% [3,7): rel. freq. = 4/5 = 80%

22. Histograms Steps for Constructing Histograms: • Pick class intervals that contain full dist’n • Find “probabilities” or “relative frequencies” for each class • Above each interval, draw rectangle where area represents class frequency

23. Histograms • Above each interval, draw rectangle where area represents class frequency (a) Probs: If width = 1, then area = width x height = height So get area = f(x), by taking height = f(x)

24. Histograms • Above each interval, draw rectangle where area represents class frequency (a) Probs: If width = 1, then area = width x height = height So get area = f(x), by taking height = f(x) E.g. Binomial Distribution

25. Binomial Prob. Histograms From Class Example 5 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg5.xls Construct Prob. Histo: • Create column of x values • Compute f(x) values • Make bar plot

26. Binomial Prob. Histograms • Make bar plot • “Insert” tab • Choose “Column” • Right Click – Select Data (Horizontal – x’s, “Add series”, Probs) • Resize, and move by dragging • Delete legend • Click and change title • Right Click on Bars, Format Data Series: • Border Color, Solid Line, Black • Series Options, Gap Width = 0

27. Binomial Prob. Histograms From Class Example 5 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg5.xls Construct Prob. Histo: • Create column of x values • Compute f(x) values • Make bar plot • Make several, for interesting comparison

28. Binomial Prob. Histograms From Class Example 5a

29. Binomial Prob. Histograms From Class Example 5a Compare Different p

30. Binomial Prob. Histograms From Class Example 5a Compare Different p: • Surprisingly similar “mound” shape

31. Binomial Prob. Histograms From Class Example 5a Compare Different p: • Surprisingly similar “mound” shape (will exploit this fact)

32. Binomial Prob. Histograms From Class Example 5a Compare Different p: • Centerpoint moves as p grows

33. Binomial Prob. Histograms From Class Example 5a Compare Different p: • Centerpoint moves as p grows (will quantify, and use this, too)

34. Binomial Prob. Histograms Important point: Binomial shows common shape across p

35. Binomial Prob. Histograms Important point: Binomial shows common shape across p Mound Shape (like dumping dirt out of a truck)

36. Binomial Prob. Histograms Important point: Binomial shows common shape across p Mound Shape (like dumping dirt out of a truck) What about n?

37. Binomial Prob. Histograms From Class Example 5b Compare Different n

38. Binomial Prob. Histograms From Class Example 5b Compare Different n: • Again very similar mound shape

39. Binomial Prob. Histograms From Class Example 5b Compare Different n: • Again very similar mound shape (will exploit this fact)

40. Binomial Prob. Histograms From Class Example 5b Compare Different n: • Center does not appear to move

41. Binomial Prob. Histograms From Class Example 5b Compare Different n: • Center does not appear to move, but check axes!

42. Binomial Prob. Histograms From Class Example 5b Compare Different n: • Center does not appear to move, but check axes! (will quantify, and use this, too)

43. Binomial Prob. Histograms From Class Example 5b Compare Different n: • But width of bump does seem to change

44. Binomial Prob. Histograms From Class Example 5b Compare Different n: • But width of bump does seem to change (will quantify, and use this, too)

45. Binomial Prob. Histograms Important point: Binomial shows common shape across p & n Mound Shape (like dumping dirt out of a truck)

46. Binomial Prob. Histograms Important point: Binomial shows common shape across p & n Mound Shape (like dumping dirt out of a truck) Question for later: How can we put this work?

47. And now for something (sort of) different Recall survey from first class meeting

48. And now for something (sort of) different Recall survey from first class meeting Display Results?

49. And now for something (sort of) different Recall survey from first class meeting Display Results? Use “bar graph”

50. And now for something (sort of) different Bar Graph from Survey, on major