Applied Data Analysis. Fall 2015

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1 Applied Data Analysis Fall 2015

2 Course information: Labs Anna Walsdorff Tues AM Mary Clare Roche Mon. 2-4 PM

3 Lecture outline 1. Practice questions 2. Inference and regression

4 Question 1 For women age in the U.S. in 2005, with full-time jobs, the relationship between education (years of schooling completed) and personal income (dollars) can be summarized as follows: Education Income Mean ,000 St. Dev ,000 Estimate the average income of those women who have finished high school, but have not gone on to college (12 years of education). The correlation is 0.34.

personal income (dollars) can be summarized as follows: Education Income Mean 14.0 32,000 St. Dev. 2.

5 z-table

6 Question 1 answer =

7 Question 1 answer = = 0.28

8 Question 1 answer = = $26, $32, 000 = $24, 720

9 Question 2 For the first-year students at a certain university, the correlation between SAT scores and first-year GPA was The scatter diagram is football-shaped. Predict the percentile rank for the first-year GPA for a student whose percentile rank on the SAT was 1. 90% 2. 30% 3. 50% 4. unknown

10 z-table

11 Question 2 answer 1. On the z-table, 90% translates to = Going back to the z-table, gives us about 78%.

12 Question 2 answer 1. On the z-table, 90% translates to = Going back to the z-table, gives us about 78%. 2. On the z-table, 30% translates to = Going back to the z-table, gives us about 38%.

13 Question 2 answer 1. On the z-table, 90% translates to = Going back to the z-table, gives us about 78%. 2. On the z-table, 30% translates to = Going back to the z-table, gives us about 38% %

2. On the z-table, 30% translates to -0.52. 0.52 0.6 = 0.312.

14 Question 2 answer 1. On the z-table, 90% translates to = Going back to the z-table, gives us about 78%. 2. On the z-table, 30% translates to = Going back to the z-table, gives us about 38% % 4. 50%

On the z-table, 30% translates to -0.52. 0.52 0.6 = 0.312.

15 Question 3 As part of their training, air force pilots make two practice landings with instructors and are rated on performance. The instructors discuss the ratings with the pilots after each landing. Statistical analysis shows that pilots who make poor landings the first time tend to do better the second time. Conversely, pilots who make good landings the first time tend to do worse the second time. The conclusion: criticism helps the pilots while praise makes them perform worse. As a result, instructors were ordered to criticize all landings, good or bad. Was this policy warranted by the facts?

Statistical analysis shows that pilots who make poor landings the first time tend to do better the second time.

16 Question 3 answer No, the air force is guilty of making the regression fallacy. The results are probably due to the regression effect.

17 Question 4 An admissions officer is trying to choose between two methods of predicting first-year scores. One method has an r.m.s. error of 12. The other has an r.m.s. error of 7. Other things being equal, which should she choose? Why?

One method has an r.m.s. error of 12. The other has an r.m.s. error of 7.

18 Question 4 answer The one with the smaller r.m.s. error because it will be more accurate.

19 Question 5 At a certain college, the first-year GPAs average about 3.0, with a SD of about 0.5; they are correlated at about 0.6 with high-school GPA. Person A predicts first-year GPAs just using the average. Person B predicts first-year GPAs by regression, using the high-school GPAs. Which person makes the smaller r.m.s. error? Smaller by what factor?

Person A predicts first-year GPAs just using the average.

20 Question 5 answer Person B, who uses more information.

21 Question 5 answer Person B, who uses more information. The r.m.s. will be smaller by a factor of 1 r 2 = = 0.8

22 Question 6 Pearson and Lee obtained the following results for about 1,000 families: r = 0.25 Husband height Wife height Mean St. Dev What percentage of the women were over 5 8? 2. Of the women who were married to men of height 6 feet, what percentage were over 5 8?

23 z-table

24 Question 6 answer = %.

25 Question 6 answer = % = = = = %

26 Inference Up to this point, I have not been clear between the difference between the true regression line and the estimated regression line. This is just like the difference between µ x and x. The true regression line is y i = β 0 + β 1 x i + ɛ i

27 Inference Up to this point, I have not been clear between the difference between the true regression line and the estimated regression line. This is just like the difference between µ x and x. The true regression line is y i = β 0 + β 1 x i + ɛ i The estimated regression line is ŷ i = a + bx i = ˆβ 0 + ˆβ 1 x i

28 Return of the hypothesis test b is an estimator so it must have a sampling distribution and a standard error. If it has those, we can perform hypothesis tests. H 0 : β = 0 H 1 : β 0

29 Return of the hypothesis test b is an estimator so it must have a sampling distribution and a standard error. If it has those, we can perform hypothesis tests. H 0 : β = 0 H 1 : β 0 Now we just need to calculate a standard error.

30 Have to take my word for it The standard error for the regression coefficient is s b = b 1 r 2 r n 2 = s y 1 r 2 s x n 2

31 The test statistic is distributed as a t with n 2 degrees of freedom. b s b t n 2

32 Is the effect of income on contacts real? Contacts Income Mean St. Dev b = = a = (4.32) = 1.3

33 The test s b = b 1 r 2 r n 2 = = 0.151

34 The test s b = b 1 r 2 r n 2 = = t = = 3.47

35 The test s b = b 1 r 2 r n 2 = = t = = 3.47 The p-value is approximately 0, and we reject the null hypothesis.

36 What did we learn? If you had majored in international relations, you would not need this class.

. 58 58 60 62 64 66 68 70 72 74 76 78 Father s height (inches)

. 58 58 60 62 64 66 68 70 72 74 76 78 Father s height (inches) PEARSON S FATHER-SON DATA The following scatter diagram shows the heights of 1,0 fathers and their full-grown sons, in England, circa 1900 There is one dot for each father-son pair Heights of fathers and