You may use your calculator and a single page of notes.

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1 LAST NAME (Please Print): KEY FIRST NAME (Please Print): HONOR PLEDGE (Please Sign): Statistics 111 Midterm 4 This is a closed book exam. You may use your calculator and a single page of notes. The room is crowded. Please be careful to look only at your own exam. Try to sit one seat apart; the proctors may ask you to randomize your seating a bit. Report all numerical answers to at least two correct decimal places or (when appropriate) write them as a fraction. All question parts count for 1 point. 1

2 1. Consider an experiment in which you buy two loaves of bread from each of Durham s two bakeries on three different random days. For each loaf, you measure its density (in g/cl times 10). The data are as follows: Day Bakery Nov. 1 Nov. 6 Nov. 18 Able Baker 20, 22 26, 26 19, 20 Googlehupf 18, 19 21, 23 25, 26 Is this is a fixed-, random-, or mixed-effects design? Explain. (2 points) Mixed. The bakery is fixed but the day is random. The following table is derved from the data above: Source df SS MS F Bakery MS B F B Day MS D F D interaction MS I F I error MSerr total What is the sum of squares due to interaction? 2 What is the degrees of freedom for the Day effect? 6 What are the degrees of freedom for the error?.0029 What is the value of the statistic for testing the Bakery effect? MS B /MS BD = /[58.167/2] What is the value of the statistic for testing the interaction effect? MS BD /MS err = [58.167/2]/[5.499/6]. 2

3 5.14 From the table, what critical value do you use for testing the Day effect (at the.05 level)? F 2,6 Yes Does density differ according Bakery at the.05 level? Because the interaction is significant, the Bakery effect must be significant. 2. Suppose you have done a one-way ANOVA that compares the GPAs of four different majors. The ANOVA test is significant at the 0.05 level. The mean squared error was 0.14, and there were three observations on each major. If the average GPA of stats majors was 3.7, the average for econ majors was 3.5, the average for lit majors was 3.1, and the average for bio majors was 2.8, then list all pairs of majors that are significantly different at the 0.05 level. Use Fisher s LSD. The test statistic is X 1 X 2 = ( X 1 X 2 )/( /3). mse/n1 + mse/n 2 The critical value comes from a t-table with 8 degrees of freedom (the number of df on the error term), at the level (to correspond with the 0.05 overall level from the ANOVA). This critical value is 2.306, so the difference is significant whenever X 1 X 2 is greater than * (2 0.14)/3 = Stats majors are significantly better than bio majors, but no other groups are significantly different. 3. When is a one-way ANOVA design more powerful than an RCBD? When the Block effect is not significant. 4. Suppose that the lifespan of a laptop has cumulative distribution function x 3 /27 for 0 x 3. What is the survival function of a laptop? 1 x3 27 3

4 What is the hazard function of a laptop? 3x 2 /(27 x 3 ) Grader: Be alert that this fraction might be represented in serveral different ways. f(x)/[1 F(x)] = (x 2 /9)/[1 (x 3 /27)] = 3x 2 /(27 x 3 ). increasing What kind of failure rate does a laptop have? (We discussed four kinds in class.) b 5. Which is better for a laptop: (a) the Cox proportional hazards model or (b) a competing risks model? The typical failure modes are getting dropped, spilling a can of soda, getting a virus, and so forth. Actual mechanical degradation is minor. So mostly the lifespan of a laptop is a horse race between a number of bad events, each of which are independent. 6. You fit a Cox proportional hazards model to the lifetime of pairs of shoes. The covariates are x 1, how many miles a person walks per day; x 2, a variable that indicates whether the shoe is a dress shoe (1) or a sneaker (0); and x 3, how much the shoe cost. You maximize the partial likelihood in order to estimate the coefficients, finding ˆβ 1 = 3, ˆβ 2 = 2 and ˆβ 3 = 1. If the baseline lifespan of a shoe follows a Rayleigh distribution with parameters θ 0 = 1 and θ 1 = 2, then what is the hazard function under the proportional hazards model? λ(t) = exp(3x 1 2x 2 x 3 )[1 + 2t)] 2.72 Suppose Goldmund walks 15 miles per day in sneakers that cost $60, while Narcissus walks 12 miles per day in dress shoes that cost $50. What is the hazard ratio for Goldmund compared to Narcissus? (Goldmund is in the numerator.) exp( )/exp( ) = Among 50 randomly chosen students who major in economics, 20 are Republican and 10 are Independent; the rest are Democrats. Among 100 randomly chosen English 4

5 majors, 25 are Republican, 25 are Independent, and the rest are Democrats. You want to decide whether one s major and politics are related. In words specific to this context, what is your null hypothesis? There is no relationship between major and political party If politics and major are unrelated, what is the expected number of Independent econ majors that you expect to see? 50*35/150 = What is the value of your test statistic? Make the contingency table; follow the formula. χ 2 2 Which table do you use for this test? (Include degrees of freedom, if appropriate.) 0.2 > P-value > 0.15 What is the significance probability? (Give a range, if appropriate.) At the.05 level, what conclusion do you reach? Express this in the specific context of the problem. We fail to reject the null there is no evidence that major is related to political viewpoint. 8. What does it mean for a social network to have a scale-free (or power) law? It means that the probability of having k friends is proportional to k α for some constant α. 9. Consider a system of the following form: 5

6 0.74 Assume m = 3 and n = 2. Assume that all components in subsystem 1 have λ = 1 and all components in subsystem 2 have λ = 2. What is the probability that the system fails in less than one time unit? The probability that the first subsystem fails before x = 1 is the probability that all three components fail, which is (1 exp( 1)) 3 = a. Similarly, the probability that the second system fails before x = 1 is (1 exp( 2)) 3 = b. The probability that one or more of the subsystems fail before time x = 1 is 1 minus the probability that neither fails, or 1 (1 a)(1 b). Putting this together gives a = , b = , and the answer is Suppose that when a component fails, the load is shifted to other components in the same subsystem. Qualitatively, how would this change your previous calculation? The lifespan would decrease; a failure in one subsystem increases the chance that other components in that system will also fail. 10. Suppose that the average number of hits that your start-up s website gets in a day has a Poisson distribution with λ = (number of linking sites). 6

7 MadAds claims that if you sign with them, the number of sites that link to yours will be 0 with probability.1, 1 with probability.2, 2 with probability.3, and otherwise is 3. During the one-day free trial period, you get 6 hits. The strategy is to make a table similar to the one in class for the RU486 example. Let x be the true number of linking sites; then there are four models, with x = 0, 1, 2, 3. The prior probabilities for each of these models, as given above, are.1,.2,.3, and.4. The probability of the data (i.e., 6 hits) given each of the models is found by evaluating the Poisson probability λ 6 exp λ/6! for the λ equal to 1.25, 2.25, 3.25, and 4.25, respectively, for x equal to 0, 1, 2 and 3. The table is as follows: Model Prior P[data model] Product Posterior 0 (λ = 1.25) (λ = 2.25) (λ = 3.25) (λ = 4.25) total If MadAds is correct, what is the probability that 3 sites link to yours? From the table If you decide to retain MadAds, what is the expected number of hits you will have tomorrow? (0.0029)(1.25) + (0.0546)(2.25) + (0.2737)(3.25) + (0.6692)(4.25) = If you retain MadAds, what is the probability of 0 hits tomorrow? The probability of 0 hits is exp( λ). So the formula is exp( 1.25) exp( 2.35) exp( 3.25) exp( 4.25) = List all true, and only the true, statements. D, F, G, H A. Triadic relations in social networks are the basis for social capital. B. In the Jefferson High data on sexual relationships, race, gender, and smoking were sufficient to give good fit. 7

8 C. In a two-way ANOVA with random effects, the test statistic for a main effect uses the mean squared error term in the denominator. D. The strategy in ANOVA is to partition the total variation in the model in to the parts attributable to specific effects. E. In a two-way ANOVA, if a factor effect is not significant, we conclude that there is no effect due to that factor. F. If a hazard function is constant, then the lifetime is exponential. G. Sir Ronald Fisher worked at the Rothamsted Experimental Research Station. H. In the analysis of the Trayvon Martin blog posts, n-grams represented sets of words that formed common phrases. I have really enjoyed this class. I hope you liked it too. DLB 8