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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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**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**Approximate Reading for Today’s Material: Pages 277-286, 291-305 Approximate Reading for Next Class: Pages 291-305, 334-351**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**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”**Midterm I - Results**Too early for letter Grades: Recall Previous scatterplot**Midterm I - Results**Interpretation of Scores: • 85 – 100 Very Pleased**Midterm I - Results**Interpretation of Scores: • 85 – 100 Very Pleased • 65 – 84 OK**Midterm I - Results**Interpretation of Scores: • 85 – 100 Very Pleased • 65 – 84 OK • 0 – 64 Recommend Drop Course (if not, let’s talk personally…)**Midterm I - Results**Histogram of Results: Overall I’m very pleased relative to other courses**Variance of Random Variables**Again consider discrete random variables: Where distribution is summarized by a table,**Variance of Random Variables**Again connect via frequentist approach:**Variance of Random Variables**Again connect via frequentist approach:**Variance of Random Variables**So define: Variance of a distribution As: random variable**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)**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)**Variance of Random Variables**HW: C16: Find the variance and standard deviation of the distribution in 4.59. (0.752, 0.867)**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)**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:**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)**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)**Variance of Random Variables**HW: C17: (cont.) (b) What is the standard deviation of Y? (18oF)**And now for something completely different**Recall Distribution of majors of students in this course:**And now for something completely different**Couldn’t Find Any Great Jokes, So…**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**Chapter 5**Sampling Distributions Idea: Extend probability tools to distributions we care about: • Counts in Political Polls • Measurement Error**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.)**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)**Binomial Distributions**HW on Binomial Assumptions: 5.1, 5.2 (a. no, n?, b. yes, c. yes)**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**Binomial Probs in EXCEL**To compute P{X=x}, for X ~ Bi(n,p): x n p**Binomial Probs in EXCEL**To compute P{X=x}, for X ~ Bi(n,p): Cumulative: P{X=x}: false P{X<=x}: true**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**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…**Binomial Probs in EXCEL**HW: 5.3 5.4 (0.194) Rework, using the Binomial Distribution: 4.52c,d**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)**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”**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)**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**Binomial Distribution**HW on Mean and Variance: 5.5**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”?**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”?**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”…**Binomial Distribution**HW: 5.13 5.19**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**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**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”?**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”?**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”?**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**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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