1. Mar. 15 Statistics for the day: Highest Temp ever recorded in State College: 102 degrees (July 9, 1936 and July 17, 1988) Lowest temp ever recorded in State College: -18 degrees (January 19-20, 1994) These slides were created by Tom Hettmansperger and in some cases modified by David Hunter Review Exam Friday, March 19 Chapters 10, 11, 12, 15, 16, 17 Source: http://pasc.met.psu.edu

3. Best fitting line through the data: called the REGRESSION LINE Strength of relationship: measured by CORRELATON

4. calories = -10 + 60(serving size in oz) ------------------------------------------------- For example if you have a 6 oz sandwich on the average you expect to get about: -10 + 60(6) = -10 + 360 = 350 calories -------------------------------------------------- For a 10 oz sandwich: -10 + 60(10) = -10 +600 = 590

5. calories = -10 + 60(serving size in oz) -10 is called the intercept 60 is called the slope One way to interpret slope: For every extra oz of serving you get an increase of 60 calories

6. Facts about correlation, measured for two quantitative variables +1 means perfect increasing linear relationship +1 means perfect increasing linear relationship -1 means perfect decreasing linear relationship -1 means perfect decreasing linear relationship 0 means no linear relationship 0 means no linear relationship + means one increases as the other increases + means one increases as the other increases - means one increases as the other decreases - means one increases as the other decreases

7. Outliers Outliers are data that are not compatible with the bulk of the data. They show up in graphical displays as detached or stray points. Sometimes they indicate errors in data input. Experts estimate that roughly 5% of all data entered is in error. Sometimes they are the most important data points.

8. Example

10. A bad outlier:

11. Another bad outlier:

12. The Moral: There can be good outliers: Election fraud. We use them to identify important parts of the data. Or in analyzing put options for extreme cases. More often the outliers are bad. They can depress the correlation and make you think the relationship is weaker than it really is. They can increase the correlation and make it appear that the relationship is stronger than it really is. IMPORTANT: Always look at a scatter plot as well as compute the correlation.

13. Another problem: Sometimes we see strong relationship in absurd examples. Two seemingly unrelated variables have a high correlation. This signals the presence of a third variable that is highly correlated with the other two. (Confounding or interaction)

14. A third variable: vocabulary vs shoe size

15. How can we have such high correlation between shoe size and vocabulary? Easy: Both increase with age and hence age is a hidden variable. Age is positively correlated with both shoe size and with vocabulary.

16. Two categorical variables: Explanatory variable: Gender Response variable: Body Pierced or Not Survey question: Have you pierced any other part of your body? (Except for ears) Research Question: Is there a significant difference between women and men in terms of body pierces?

17. Data: NoYes Women8451135 Men96399 18054234 Pierced? Gender? Explanatory Response From Stat 100.2, spring 2004 (missing responses omitted)

18. Response: body pierced? no yes All female 62.22 37.78 100.00 male 96.97 3.03 100.00 All 76.92 23.08 100.00 Percentages Research question: Is there a significant difference Between women and men? (i.e., between 62.22% and 96.97%) 62.22 = 84/135 96.97 = 96/99

19. The Debate: The research advocate claims that there is a significant difference. The skeptic claims there is no real difference. The data differences simply happen by chance.

20. The strategy for determining statistical significance: First, figure out what you expect to see if there is no difference between females and males First, figure out what you expect to see if there is no difference between females and males Second, figure out how far the data is from what is expected. Second, figure out how far the data is from what is expected. Third, decide if the distance in the second step is large. Third, decide if the distance in the second step is large. Fourth, if large then claim there is a statistically significant difference. Fourth, if large then claim there is a statistically significant difference.

21. Research Advocate: OK. Suppose there is really no difference in the population as you, the Skeptic, claim. We will compare what you, The Skeptic, expect to see and what you actually do see in the data. Skeptic: How do we figure out what we expect to see?

22. NoYes Women135 Men99 18054234

23. Rows: gender Columns: body pierces top lines of numbers are observed bottom lines are expected (by skeptic) no yes All female 84 51 135 103.85 31.15 135.00 male 96 3 99 76.15 22.85 99.00 All 180 54 234 180.00 54.00 234.00

24. How to measure the distance between what the research advocate observes in the table and what the skeptic expects: Add up the following for each cell: Now how do we decide if 38.85 is large or not? If it is large enough the skeptic concedes to the research advocate and agrees there is a statistically significant difference. How large is enough?

25. But our chi-squared is 38.85 so the research advocate easily wins! There is a statistically significant difference between men and women. Chi-squared distribution with 1 degree of freedom: If chi-squared statistic is larger than 3.84, it is declared large and the research advocate wins.

26. Why 1 degree of freedom? NoYes Women136 Men101 26211237 Note that black box is the ONLY one we can fill arbitrarily. Once that box is filled, all others are determined by margins!

27. How many degrees of freedom? AlwaysSometimesNever Women One df Two df 136 Men101 10610526237 Degrees of freedom (df) always equal (Number of rows – 1) times (Number of columns – 1)

28. Health studies and risk Research question: Do strong electromagnetic fields cause cancer? 50 dogs randomly split into two groups: no field, yes field The response is whether they get lymphoma. Rows: mag field Columns: cancer no yes All no 20 5 25 yes 10 15 25 All 30 20 50

29. Rows: mag field Columns: cancer observed above the expected no yes All no 20 5 25 15.00 10.00 25.00 yes 10 15 25 15.00 10.00 25.00 All 30 20 50 30.00 20.00 50.00 Chi-Square = 8.333 (compare to 3.84) Research advocate wins!

30. 1.Identify the ‘bad’ response category: yes cancer 2.Risk for categories of explanatory variable Identify treatment category Identify baseline (control) category 3.Treatment risk: 15/25 or.60 or 60% 4.Baseline risk: 5/25 or.20 or 20% 5.Relative risk: Treatment risk over Baseline risk =.60/.20=3 So risk due to mag field is 3 times higher than baseline risk. One more on the next page: Terminology and jargon:

31. Increased risk (percentage change in risk): So the percentage change is 200% A 200% increase in treatment risk over baseline risk for getting cancer.

32. Final note: When the chi-squared test is statistically significant then it makes sense to compute the various risk statements. If there is no statistical significance then the skeptic wins. There is no evidence in the data for differences in risk for the categories of the explanatory variable.

33. Research question: Is ghost sighting related to age? Do young and old people differ in ghost sighting? The skeptic responds by saying he doesn’t believe that there is any difference between the age groups. We need to see the data to resolve the debate. Then we can consider assessing the risk. Exercise 9, p219 of the text.

34. Expected counts are printed below observed yes no Total young 212 1313 1525 174.9 1350.1 old 465 3913 4378 502.1 3875.9 Total 677 5226 5903 Chi-Sq = 7.870 + 1.020 + 2.742 + 0.355 = 11.987 The research advocate wins and skeptic loses. There is evidence in the data that there are differences in the population.

35. The percent of young who saw a ghost: 212/1525 =.139 Answer: 13.9% The proportion of old who saw a ghost: 465/4378 =.106 Answer:.106 The risk of young seeing ghost: Answer: 212/1525 or.139 or 13.9% Odds ratio?

36. Odds The odds of something happening are given by a ratio: The odds of something happening are given by a ratio: For example, if you flip a fair coin, the odds of heads are 1 (or sometimes “1 to 1”). For example, if you flip a fair coin, the odds of heads are 1 (or sometimes “1 to 1”). An odds ratio is the ratio of two odds! An odds ratio is the ratio of two odds!

37. The odds that a young person saw a ghost: 212/1313 =.161 The odds that an older person saw a ghost: 465/3912 =.119 The odds ratio: Answer:.161/.106 = 1.35

38. Relative risk of young person seeing a ghost compared to older person: Answer:.139/.106 = 1.31 We would say that the risk that a younger person sees a ghost is 1.31 times higher than the risk that an older person sees a ghost. The increased risk that a young person sees a ghost over that of an older person: Answer: (.139 -.106)/.106 =.31 Hence we would say that young people have a 31% higher risk of seeing a ghost than older people.

39. Statistical significance Statistical significance is related to the size of the sample. But that makes sense. More data, more information, more precise inference. So statistical significance is related to two things: 1.The size of the difference between the percentages. Big differences are more likely to show stat. significance. 2. The size of the sample. Bigger samples are more likely to show statistical significance irrespective of the size of the difference in percentages.

40. Practical significance Even if the difference in percentages is uninteresting and of no practical interest, the difference may be statistically significant because we have a large sample. Hence, in the interpretation of statistical significance, we must also address the issue of practical significance. In other words, you must answer the skeptic’s second question: WHO CARES?

41. Probability Relative Frequency Personal Opinion Experiment Repeated Sampling Experience Non-repeatable Event Physical World Assumptions Estimate Probability Repeated Sampling Check by Repeated Sampling

42. Rules: For combining probabilities 0 < Probability < 1 1.If there are only two possible outcomes, then their probabilities must sum to 1. 2.If two events cannot happen at the same time, they are called mutually exclusive. The probability of at least one happening (one or the other) is the sum of their probabilities. [1. is a special case of this.] 3.If two events do not influence each other, they are called independent. The probability that they happen at the same time is the product of their probabilities. 4.If the occurrence of one event forces the occurrence of another event, then the probability of the second event is always at least as large as the probability of the first event.

43. Are mutually exclusive events independent or dependent? Remember the tests: 1.Two events are mutually exclusive if they cannot happen at the same time. 2.Two events are independent if the occurrence of one does not alter the probability of the other occurring. Or, another way, if the probability of the occurrence of one event changes when we find out whether the other event occurred or not.

44. New Rule: Suppose we are considering a series of events. The probability of at least one of the events occurring is: Pr( at least one ) = 1 – Pr( none ) This follows directly from Rule 1 since ‘at least one’ or ‘none’ has to occur.

45. Long Run Behavior We CANNOT predict individual outcomes. BUT We CAN predict quite accurately long run behavior. -------------------------------------------------------------------- Standard example: We cannot predict the outcome of a single toss of a coin very precisely: Pr(head) =.50 But in the long run we expect about 50% heads and tails.

47. Two laws (only one of them valid): Law of large numbers: Over the long haul, we expect about 50% heads (this is true). Law of large numbers: Over the long haul, we expect about 50% heads (this is true). “Law of small numbers”: If we’ve seen a lot of tails in a row, we’re more likely to see heads on the next flip (this is completely bogus). “Law of small numbers”: If we’ve seen a lot of tails in a row, we’re more likely to see heads on the next flip (this is completely bogus). Remember: The law of large numbers OVERWHELMS; it does not COMPENSATE.

48. When will it happen? (p264 text) Odd Man Consider the odd man game. Three people toss a coin. The odd man has to pay for the drinks. You are the odd man if you get a head and the other two have tails or if you get a tail and the other two have heads. Pr(no odd man) = Pr(HHH or TTT) = Pr(HHH) + Pr(TTT) Rule 2 = (1/2) 3 + (1/2) 3 Rule 3 =1/8 + 1/8 =1/4 =.25 Pr( odd man ) = 1 – Pr(no odd man) = 1 -.25 =.75 Rule 1

49. Pr( odd man occurs on the third try) = Pr(miss, miss, hit) = Pr(miss)Pr(miss)Pr(hit) Rule 3 =[Pr(miss)] 2 Pr(hit) =[.25] 2.75 =.047

50. Expectation Insurance Example 14 p267 extended. Suppose my insurance company has 10,000 policy holders and they are all skateboarders. I collect a $500 premium each year. I pay off $1500 for a claim of a skate board accident. From past experience I know 10% ( ie. 1000) will file a claim. How much do I expect to make per customer?

51. Pr(claim) =.10 loss is $1500 - $500 = $1000 recorded as -$1000 Pr(no claim) =.90 gain is $500 -------------------------------------------------------------------------- Expected value =.10x(-1000) +.90x(500) = -100 + 450 = 350 dollars per customer -------------------------------------------------------------------------- Expected value for the 10,000 customers = 10,000x350 = 3,500,000 dollars per year

52. 333333D 662222C 555111B 444400A Side value Pr( B beats A ) = 2/3 Pr( C beats B ) = 2/3 Pr( D beats C ) = 2/3 Pr (A beats D ) = 2/3 Hence, there is NO best die! You can always pick a winner. Efron Dice

53. Cancer testing: confusion of the inverse Suppose we have a cancer test for a certain type of cancer. Sensitivity of the test: If you have cancer then the probability of a positive test is.98. Pr(+ given you have C) =.98 Specificity of the test: If you do not have cancer then the probability of a negative test is.95. Pr(- given you do not have C) =.95 Base rate: The percent of the population who has the cancer. This is the probability that someone has C. Suppose for our example it is 1%. Hence, Pr(C) =.01.

54. +Positive-Negative C(Cancer).98.02.01 no C (no Cancer).05.95.99 Sensitivity Specificity Base Rate Percent table Suppose you go in for a test and it comes back positive. What is the probability that you have cancer? false positivefalse negative

55. Count table from a percent table+- C.98.02.01 no C.05.95.99 +-C982100 49594059,900 593940710,000 Pr(C given a + test) = 98/593 =.165

56. All people like you Tree diagrams: A possible tool for solving problems like the “rare disease” problem With diseaseWithout disease PositiveNegativePositiveNegative.01.02.98.05.95.99.0495.0098 Pr (Positive) =.0098+.0495 =.0593 Pr (Disease given Positive) =.0098/.0593 =.165

57. Recall earlier quiz we didn’t have: Mary likes earrings and spends time at festivals shopping for jewelry. Her boy friend and several of her close girl friends have tattoos. They have encouraged her to also get a tattoo. Unknown to you, Mary will be sitting next to you in the next stat100.2 class. Which of the following do you think is more likely and why? A.Mary is a physics major. B.Mary is a physics major with pierced ears.

58. A possible cause of this fallacy is the Representative heuristic: leads people to assign higher probabilities than are warranted to scenarios that are representative of how we imagine things would happen. An answer of B (Mary is a physics major with pierced ears) is impossible and illustrates the Conjunction fallacy: assigning higher probability to a detailed scenario involving the conjunction of events than to one of the simple events that make up the conjunction.

59. 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).

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

61. 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 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” “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.

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

63. 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. Randomized Response: A technique for asking sensitive questions

64. ConditionalProbabilitiesnoyes Base rate Q11-pp6/12 Q27/125/126/12 UnconditionalProbabilitiesnoyesQ1.5(1-p).5p Q2.292.208.208+.5p Solve for p:.208+.5p = proportion of observed yeses in sample