Stat 230 – Intermediate Statistics – Section 01 Final Exam, December 18, 2016

Name: Stat 230 – Intermediate Statistics – Section 01 Final Exam, December 18, 2016 Problem 1 2 3 4 5 Total 16 14 17 10 18 75 Points Earned Possible Points 1. We are interested in understanding the relationship between the average wine consumption rates (in liters per person) and the number of ischemic heart disease deaths (per 1000 men aged 55-64 years old) in industrialized countries. The variables are provided as Wine and Mortality. Use the provided output to address the questions that follow. mod1=lm(Mortality~log(Wine)) summary(mod1) Call: lm(formula = Mortality ~ log(Wine)) Residuals: Min 1Q Median -2.2559 -1.0849 -0.1914 3Q 0.7326 Max 2.5687 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 10.2795 0.8316 12.360 1.34e-09 *** log(Wine) -1.7712 0.3468 -5.108 0.000105 *** Residual standard error: 1.498 on 16 degrees of freedom Multiple R-squared: 0.6199, Adjusted R-squared: 0.5961 F-statistic: 26.09 on 1 and 16 DF, p-value: 0.0001054 a. (1 point) How many countries were in the data set? b. (2 points) Describe the relationship between Mortality and log(Wine). c. (3 points) Describe how well the model fits the data using 2 unrelated statistics as support for your response. d. (2 points) Assuming the conditions for inference are met, the test statistic of _______ and p-value of __________ indicate that ___________________ is a significant predictor of _____________________. (Even if you feel the randomness and independence conditions are not met, we could still assess significance for this data set). e. (4 points) Interpret the p-value you used in d. for someone who does not have a statistics background. In other words, explain what that p-value is in layman's terms. f. (4 points) A plot used to check a condition for inference in this setting is provided at the right. Clearly state the condition the plot is used to check AND comment on whether you believe the condition is met here. 2. In the late 1990s, a series of lawsuits were filed related to a potential relationship between failures of Firestone tires used on Ford Explorers (a particular type of tire on a certain make/model of car) under heavy loads and fatal accidents. Data from 1995 through 1999 on fatal car accidents involving these types of vehicles can be examined to see if there is a relationship. Our data set contains the following variables: Make - whether or not the car is a Ford VehicleAge - as a proxy for tire age, in years Passengers - number of passengers (not including the driver) Cause - whether the cause of the accident was the tires or not The observations in the data set were carefully chosen. These are all fatal single-car accidents without alcohol involved according to the official police reports. Here is a summary of the data: summary(data) Make VehicleAge Ford : 522 Min. :0.000 Other:1799 1st Qu.:1.000 Median :2.000 Mean :2.372 3rd Qu.:3.000 Max. :5.000 Passengers Min. : 0.0000 1st Qu.: 0.0000 Median : 0.0000 Mean : 0.9405 3rd Qu.: 1.0000 Max. :11.0000 Cause NotTire:2294 Tire : 27 tally(Cause~Make,data) Make Cause Ford Other NotTire 500 1794 Tire 22 5 a. (3 points) i. What are the odds of a tire being the cause of the accident for Ford vehicles? ii. What are the odds of a tire being the cause of the accident for Other vehicles? iii. What is the odds ratio of a tire being the cause of the accident for Ford vehicles relative to Other vehicles? b. (2 points) If you wanted to fit a simple logistic regression model to predict Cause (treating Tire as the success) using Make as the predictor, would you need to examine an empirical logit plot to check the linearity condition? Explain your response. The following model was fit: logm = glm(Cause ~ Make+VehicleAge+Passengers, family=binomial(logit)) summary(logm) Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -6.6612 0.7329 -9.088 < 2e-16 MakeOther -2.8614 0.5189 -5.515 3.49e-08 VehicleAge 0.8499 0.1784 4.765 1.89e-06 Passengers 0.6277 0.1042 6.025 1.69e-09 --- *** *** *** *** (Dispersion parameter for binomial family taken to be 1) Null deviance: 294.2 Residual deviance: 195.6 AIC: 203.6 on 2320 on 2317 degrees of freedom degrees of freedom Number of Fisher Scoring iterations: 9 c. (2 points) Report the fitted model for Ford Vehicles in logit form. d. (3 points) Interpret the slope coefficient for "MakeOther" in context. e. (4 points) Assume that the conditions for inference are met. If we wanted to determine if the overall model - logm - was useful in predicting Cause, we would use a _____________________________ test, with a test statistic here of _____________________. The associated p-value would be found by using a ________________________ distribution (fully specify the distribution). R reports the p-value is < 2.2e-16 (not shown), so we conclude (fill in conclusion) ... 3. There is a general concern in education about the performance of female versus male students, especially in STEM disciplines. Here, we examine data from 861 students where their sex, ACT Math score, and course preparation (which sequence of courses they took in mathematics) are recorded. A summary of the data set is provided. Note that the maximum possible ACT math score is 36, and that the course background is provided with later letters indicating more advanced mathematics preparation (c is more preparation than b, which is more preparation than a). summary(data) Sex Background Score female:523 a:130 Min. : 0.00 male :338 b:610 1st Qu.:11.00 c:121 Median :15.00 Mean :15.33 3rd Qu.:19.00 Max. :36.00 We want to use the data to examine whether or not there are differences in average ACT math score across the sexes and course backgrounds. a. (1 point) What is the name of the appropriate statistical procedure to apply to address the question of interest? (Be specific). Source Df SS MS Sex Background Residuals 14704.7 857 24.4 F p-value 58.617 5.164e-14 301.273 < 2.2e-16 - - b. (2 points) Complete the partial table provided based on the R output. (Space provided for scratch work). c. (1 point) What is your best estimate of the common variance assumed in this procedure? Provide notation and a numeric value. d. (3 points) Assume the conditions for inference are met. Does it appear that there is a difference in average ACT Math scores for the sexes after accounting for background? Explain, with statistical support for your response. 3. e. (5 points) The following output was generated after examining the results above. Summarize what this output reveals to you, being sure to describe what multiple comparisons are and what a family-wise confidence level is. TukeyHSD(mod2) Tukey multiple comparisons of means 95% family-wise confidence level Fit: aov(formula = x) $Sex diff lwr upr p adj male-female 2.639591 1.962905 3.316277 0 $Background diff lwr upr p adj b-a 4.604987 3.484593 5.725382 0 c-a 14.633561 13.168471 16.098650 0 c-b 10.028573 8.874342 11.182805 0 3. f. (5 points) The following plot was generated from the data. Address the following questions in a wellconstructed response: What is the plot called? What is it used for? Comment on what you learn from this plot in this setting. 4. Suppose we are considering designing a study to examine differences between sexes and course backgrounds for average ACT Reading scores (another of the four subscores) at a particular high school. Assume that you can randomly sample students from the school as they register freshman year, that no student drops out, and that all students take the ACT. For the course backgrounds, assume there are 4 different preparations (which can again be ordered from least preparation to most preparation), based on what courses the students take during their time in high school. a. (3 points) What are the factor(s) in this proposed study? If possible, identify them as observational or experimental based on the description provided. b. (3 points) An intrepid researcher is thinking about randomly assigning the students to course backgrounds when they enter the high school. i. This approach is treating the course backgrounds as an _________________________ factor. ii. Discuss any concerns you have about this aspect of the proposed design. c. (4 points) (General) Describe what blocking is and why it is useful to consider in experimental design. 5. Do wage earnings differ by race? We examine data from the Current Population Survey from 2011 to compare weekly wage earnings for males between 18 and 70 years of age who reported race as either "only white" or "only black". A variable summary shows the variables in the data set. Most are fairly self-explanatory. EducationCode is a numeric value where higher values indicate more education. Use the output provided to address the questions that follow: summary(data) Region Midwest :1261 Northeast:1019 South :1563 West :1109 EducationCode Min. :31.00 1st Qu.:39.00 Median :40.00 Mean :40.68 3rd Qu.:43.00 Max. :46.00 MetropolitanStatus Age Metropolitan :3997 Min. :16.00 Not Identified : 35 1st Qu.:33.00 Not Metropolitan: 920 Median :43.00 Mean :42.96 3rd Qu.:52.00 Max. :85.00 Race WeeklyEarnings Black: 407 Min. : 0.46 White:4545 1st Qu.: 575.00 Median : 870.00 Mean :1054.56 3rd Qu.:1346.15 Max. :2884.61 mod4=lm(WeeklyEarnings~Race+Age+EducationCode+Region,data) summary(mod4) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -4091.0982 130.3802 -31.378 < 2e-16 *** RaceWhite 173.3394 30.0898 5.761 8.88e-09 *** Age 10.1655 0.6584 15.441 < 2e-16 *** EducationCode 110.7038 3.1211 35.470 < 2e-16 *** RegionNortheast 119.7601 24.1342 4.962 7.20e-07 *** RegionSouth 15.4524 21.8401 0.708 0.47927 RegionWest 74.8462 23.5923 3.172 0.00152 ** Residual standard error: 572.4 on 4945 degrees of freedom Multiple R-squared: 0.2589, Adjusted R-squared: 0.258 F-statistic: 287.9 on 6 and 4945 DF, p-value: < 2.2e-16 mod5=lm(WeeklyEarnings~Race+Age,data) summary(mod5) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 297.0058 44.6616 6.65 3.24e-11 *** RaceWhite 250.5075 33.2700 7.53 6.02e-14 *** Age 12.2829 0.7358 16.69 < 2e-16 *** Residual standard error: 643 on 4949 degrees of freedom Multiple R-squared: 0.06394, Adjusted R-squared: 0.06356 F-statistic: 169 on 2 and 4949 DF, p-value: < 2.2e-16 anova(mod5,mod4) Analysis of Variance Table Model 1: WeeklyEarnings ~ Race + Age Model 2: WeeklyEarnings ~ Race + Age + EducationCode + Region Res.Df RSS Df Sum of Sq F Pr(>F) 1 4949 2046079573 2 4945 1619984843 4 426094730 325.16 < 2.2e-16 *** a. (3 points) Describe how Region is being incorporated in mod4. For example, what is the RegionSouth variable and why is there no line for RegionMidwest in the output? b. (2 points) Is mod5 a parallel slopes model or not? Explain how you can tell. c. (6 points) Assume the conditions for inference are met. We want to know if mod4 is an improvement over mod5 in terms of predicting weekly wage earnings. Perform an appropriate hypothesis test, clearly stating what variables are of interest. (Hypotheses can be in words or notation). This test procedure is called a ___________________________________________. Null Hypothesis: Alternative Hypothesis: (Again, assume the conditions for inference are met .) Test statistic: Conclusion (in context): p-value: d. (4 points) Explain what adjusted R-squared is and compare the models (mod4 and mod5) based on that statistic. Which model would you prefer for predicting weekly wage earnings? e. (3 points) The intern who made the models informs you that mod4 was found via forward selection. Describe forward selection in your own words.