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Stat 155, Section 2, Last Time

Stat 155, Section 2, Last Time. Continuous Random Variables Probabilities modeled with areas Normal Curve Calculate in Excel: NORMDIST & NORMINV Means, i.e. Expected Values Useful for “average over many plays” Independence of Random Variables. Reading In Textbook.

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Stat 155, Section 2, Last Time

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  1. Stat 155, Section 2, Last Time • Continuous Random Variables • Probabilities modeled with areas • Normal Curve • Calculate in Excel: NORMDIST & NORMINV • Means, i.e. Expected Values • Useful for “average over many plays” • Independence of Random Variables

  2. Reading In Textbook Approximate Reading for Today’s Material: Pages 277-286, 291-305 Approximate Reading for Next Class: Pages 291-305, 334-351

  3. Midterm I - Results Preliminary comments: • Circled numbers are points taken off • Total for each problem in brackets • Points evenly divided among parts • Page total in lower right corner • Check those sum to total on front • Overall score out of 100 points

  4. Midterm I - Results Interpretation of Scores: • Too early for letter grades • These will change a lot: • Some with good grades will relax • Some with bad grades will wake up • Don’t believe “A & C” average to “B”

  5. Midterm I - Results Too early for letter Grades: Recall Previous scatterplot

  6. Midterm I - Results Interpretation of Scores: • 85 – 100 Very Pleased

  7. Midterm I - Results Interpretation of Scores: • 85 – 100 Very Pleased • 65 – 84 OK

  8. Midterm I - Results Interpretation of Scores: • 85 – 100 Very Pleased • 65 – 84 OK • 0 – 64 Recommend Drop Course (if not, let’s talk personally…)

  9. Midterm I - Results Histogram of Results: Overall I’m very pleased relative to other courses

  10. Variance of Random Variables Again consider discrete random variables: Where distribution is summarized by a table,

  11. Variance of Random Variables Again connect via frequentist approach:

  12. Variance of Random Variables Again connect via frequentist approach:

  13. Variance of Random Variables So define: Variance of a distribution As: random variable

  14. Variance of Random Variables E. g. above game: =(1/2)*5^2+(1/6)*1^2+(1/3)*8^2 Note: one acceptable Excel form, e.g. for exam (but there are many)

  15. Standard Deviation Recall standard deviation is square root of variance (same units as data) E. g. above game: Standard Deviation =sqrt((1/2)*5^2+(1/6)*1^2+(1/3)*8^2)

  16. Variance of Random Variables HW: C16: Find the variance and standard deviation of the distribution in 4.59. (0.752, 0.867)

  17. Properties of Variance • Linear transformation I.e. “ignore shifts” var( ) = var ( ) (makes sense) And scales come through squared (recall s.d. on scale of data, var is square)

  18. Properties of Variance ii. For X and Y independent (important!) I. e. Variance of sum is sum of variances Here is where variance is “more natural” than standard deviation:

  19. Properties of Variance E. g. above game: Recall “double the stakes”, gave same mean, as “play twice”, but seems different Doubling: Play twice, independently: Note: playing more reduces uncertainty (var quantifies this idea, will do more later)

  20. Variance of Random Variables HW: C17: Suppose that the random variable X models winter daily maximum temperatures, and that X has mean 5o C and standard deviation 10o C. Let Y be the temp. in degrees Fahrenheit (a) What is the mean of Y? (41oF) Hint: Recall the conversion: C=(5/9)(F-32)

  21. Variance of Random Variables HW: C17: (cont.) (b) What is the standard deviation of Y? (18oF)

  22. And now for something completely different Recall Distribution of majors of students in this course:

  23. And now for something completely different Couldn’t Find Any Great Jokes, So…

  24. And now for something completely different An Interesting and Relevant Issue: • “Places Rated” • Rankings Published by Several… • We’ve been #1? • Are we great ot what? Will take a careful look later

  25. Chapter 5 Sampling Distributions Idea: Extend probability tools to distributions we care about: • Counts in Political Polls • Measurement Error

  26. Counts in Political Polls Useful model: Binomial Distribution Setting: n independent trials of an experiment with outcomes “Success” and “Failure”, with P{S} = p. Say X = #S’s has a “Binomial(n,p) distribution”, and write “X ~ Bi(n,p)” (parameters, like for Normal dist.)

  27. Binomial Distributions Models much more than political polls: E.g. Coin tossing (recall saw “independence” was good) E.g. Shooting free throws (in basketball) • Is p always the same? • Really independent? (turns out to be OK)

  28. Binomial Distributions HW on Binomial Assumptions: 5.1, 5.2 (a. no, n?, b. yes, c. yes)

  29. Binomial Distributions Could work out a formula for Binomial Probs, but results are summarized in Excel function: BINOMDIST Example of Use: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg19.xls

  30. Binomial Probs in EXCEL To compute P{X=x}, for X ~ Bi(n,p): x n p

  31. Binomial Probs in EXCEL To compute P{X=x}, for X ~ Bi(n,p): Cumulative: P{X=x}: false P{X<=x}: true

  32. Binomial Probs in EXCEL http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg19.xls Check this spreadsheet for details of other parts, and some important variations

  33. Binomial Probs in EXCEL Next time: More slides on BINOMDIST, And illustrate things like P{X < 3} = P{X <= 2}, etc. Using a number line, and filled in dots…

  34. Binomial Probs in EXCEL HW: 5.3 5.4 (0.194) Rework, using the Binomial Distribution: 4.52c,d

  35. Binomial Distribution “Shape” of Binomial Distribution: Use Probability Histogram Just a bar graph, where heights are probabilities Note: connected to previous histogram by frequentist view (via histogram of repeated samples)

  36. Binomial Distribution Study Distribution Shapes using Excel http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg20.xls Part I: different p, note several ranges of p are shown Part II: different n, note really “live in different areas”

  37. Binomial Distribution A look under the hood http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg20.xls Create probability histograms by: • Create Column of xs (e.g. B9:B29) • Create Probs (using BINOMDIST, C9:J29) • Plot with Chart Wizard Click Chart & Chart Wizard Follow steps, check “series” carefully)

  38. Binomial Distribution With some calculation, can show: For X ~ Bi(n,p): Mean: (# trials x P{S}) Variance: S. D.: Relate to (center & spread) of each histo: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg20.xls

  39. Binomial Distribution HW on Mean and Variance: 5.5

  40. Binomial Distribution E.g.: Class HW on %Males at UNC: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg17.xls Note Theoretical Means in E115:H115, Compare to Sample Means in E110:H110: Q1: Sample Mean smaller – course not representative Q2: Sample Mean bigger – bias toward males Q3: Sample Mean bigger – bias toward males Q4: Sample Mean close Which differences are “significant”?

  41. Binomial Distribution E.g.: Class HW on %Males at UNC: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg17.xls Note Theoretical SDs in E116:H115,6 Compare to Sample SDs in E112:H112: Q1: Sample SDs smaller – course population smaller Q2: Sample SDs bigger – variety of doors (different p) Q3: Sample SDs bigger – variety of choices (diff. p?) Q4: Sample SDs close Which differences are “significant”?

  42. Binomial Distribution E.g.: Class HW on %Males at UNC: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg17.xls Probability Histograms (see 3rd column of plots), Good view of above ideas (for samples): Q1: mean too small, not enough spread Q2: mean too big, too spread Q3: mean too big, too spread Q4: looks “about right”…

  43. Binomial Distribution HW: 5.13 5.19

  44. And now for something completely different An Interesting and Relevant Issue: • “Places Rated” • Rankings Published by Several… • We’ve been #1? • Are we great ot what? Will take a careful look now

  45. And now for something completely different Interesting Article: Analysis of Data from the Places Rated Almanac By: Richard A. Becker; Lorraine Denby; Robert McGill; Allan R. Wilks Published in: The American Statistician, Vol. 41, No. 3. (Aug., 1987), pp. 169-186. Hyperlink to JSTOR

  46. And now for something completely different Main Ideas: • For data base used in ratings • Did careful analysis • In an unbiased way • Studied several aspects • An interesting issue: Who was “best”?

  47. And now for something completely different Who was “best”? • Data base had 8 factors • How should we weight them? • Evenly? • Other choices? • Just choose some? (typical approach) • Can we make our city “best”?

  48. And now for something completely different Who was “best”? • Approach: Consider all possible ratings (i.e. all sets of weights) • Which places can be #1? • Which places can be “worst”?

  49. And now for something completely different Which places can be #1? • 134 cities are “best” • Including Raleigh Durham area Which places can be “worst”? • Even longer list here • But Raleigh Durham not here

  50. And now for something completely different Which places can be #1? Which places can be “worst”? Interesting fact: Several cities on both lists!

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