Mar. 5 Statistic for the day: U. S Mar. 5 Statistic for the day: U.S. rank, among 102 nations, of perceived corruption (according to Transparency International): 16th least corrupt. Least corrupt: Finland Most corrupt: Bangladesh Full list at http://www.transparency.org/cpi/2002/cpi2002.en.html Assignment: Have a great break These slides were created by Tom Hettmansperger and in some cases modified by David Hunter

Exercise 1, page 309 (sort of): Suppose you flip four coins. Which is more likely, HHHH or HTTH? Which is more likely, four total heads or two total heads? Note: These questions are not the same! One of these questions is often mistakenly answered due to belief in the “Law of small numbers” (also known as the Gambler’s Fallacy).

Flip a coin repeatedly. Which of the following is more likely? Your first seven flips are HHTHTTH Your first six flips are all heads (By the way, how do you calculate the exact probability of each of these events?)

Exercise 7, page 310 “In a high-risk population, virtually all people who test positive will truly be infected, but …for every infected person correctly identified in a low-risk population, an estimated ten noncarriers will test positive.” How can the low-risk statement be true, assuming that the test has high sensitivity and specificity?

Exercise 10, page 310 Suppose a rare disease occurs in about 1 out of 1000 people who are like you. A test for the disease has sensitivity of 95% and specificity of 90%. Compute the probability that you actually have the disease, given that your test results are positive. We’ve seen several such examples in lecture. A similar question is pretty much guaranteed on the midterm!

Tree diagrams: A possible tool for solving problems like the “rare disease” problem All people like you .001 .999 With disease Without disease .05 .95 .10 .90 Positive Negative Positive Negative .00095 .0999 Pr (Positive) = .00095+.0999 = .10085 Pr (Disease given Positive) = .00095/.10085 = .0094

Exercise 15, page 311. What’s the difference between these two statements? “I’m confident that there is at least one set of matching birthdays in this room” “I’m confident that there is at least one person in this room whose birthday matches my birthday” Which statement is more likely to be true? How many possible pairs of people are eligible for matching in each case? Assume 50 people are in the room.

With 50 people in the room… There are 49 possible pairs with me. There are 49+48+47+…+1 = 1225 total possible pairs. Pr (No match with my birthday) = (364/365)49=.874 Pr (No match at all) = .030 (and we can estimate by (364/365)1225=.035)

Randomized Response: A technique for asking sensitive questions Question 1: Have you ever smoked marijuana? Question 2: Is your mother’s birthday in Jan through May? If your father’s birthday is in July through Dec, answer question 1. Otherwise answer question 2.

Conditional Probabilities no yes Base rate Q1 1-p p 6/12 Q2 7/12 5/12 Unconditional Probabilities no yes Q1 .5(1-p) .5p Q2 .292 .208 .208+.5p Solve for p: .208+.5p = proportion of yeses

Stat100 survey Have you ever smoked marijuana? marijuana Count Percent No 97 41.45 Yes 137 58.55 N= 234 *= 3