Quoting from the document I suggested you read ( westfall/images/5349/practiceproblems_discussion.

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1 Spring 14, ISQS 5349 Midterm 1. Instructions: Closed book, notes and no electronic devices. Put all answers on scratch paper provided. Points (out of 100) are in parentheses. 1. (20) Define regression model in the most general terms. Do not mention anything about linearity, normality, or constant variance. Give an example from class of your own choosing to illustrate the definition. Solution: The regression model is a model for how the Y data are produced, given an X variable. In other words, it is a specification of the conditional distributions p(y X = x). For example, in the Toluca case, the model states that for a given lot size X, such as X = 100, the resulting workhours, Y, are produced from a distribution p(y X =100) of possible Y values that might occur when X = 100. When the model is more fully explicit, it will include details about what kinds of distributions these are, and how these distributions are linked through the X variable. Quoting from the document I suggested you read ( westfall/images/5349/practiceproblems_discussion.pdf): A statistical model is, by definition, a probability model that is assumed to produce data. A regression model is a special case of a statistical model, one that specifies the conditional distribution of the dependent variable, given fixed value(s) of the independent variable(s). The model usually contains unknown parameters, since the natural process that is modeled is not completely known or understood. These parameters are estimated using the data. 2. (20) A regression model for computer speed is Jobs/Hr = (Computer Memory) +. Observational data are collected on a sample of 100 students, many of whom happen to have computers with different memory. Explain what causation means in this context, and why it cannot be inferred from this model, given this data collection procedure. Solution: Here, causation refers to what would happen to Jobs/Hr if you change the computer s memory. If you intervene in Freedman s terminology, and change the computer to some higher RAM, then the difference between the two Jobs/Hr, one at the lower level of RAM, and one at the higher, is the causal effect of increasing RAM. But with this data collection procedure, we simply observe some students who have computers with lower RAM than others. If we compare Jobs/Hr for those two types of students, the difference is not the causal effect of intervention, because the difference could easily be due to other factors not included in the model. For example, students whose machines have smaller RAM also tend to have slower processing speeds (GhZ), so the effect observed in the study could be completely due to changes in GhZ rather than changes in RAM.

2 3. Here is some R code and output. gpa.sat = read.table(" reg = lm(gpa ~ SAT, data=gpa.sat) summary(reg) Output: Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-10 *** SAT e-05 *** A. (15) Interpret the in the Std. Error column in terms of repeated samples from the same process that produced these data. Solution: If we were to observe another sample just like this with the same sample size and specific SAT score data, but different GPAs (of course, because they are different students), we would of course get a coefficient of SAT that differs from If we were to observe many, many such samples, every one of them would give a different coefficient of SAT. The standard deviation of all these coefficients would be close to Of course, this assumes that all samples are produced by the same processes that produced the observed data. Quoting from the document I suggested you read ( westfall/images/5349/practiceproblems_discussion.pdf): the concept of standard deviation of b1 can be illustrated as discussed above in the simulation for problem 7. Simulate many samples, then find the sample standard deviation of the b1 estimates obtained from all the simulated samples; this sample standard deviation is an estimate of the true standard deviation of the b1 estimator. The sample standard deviation of the simulated b1 values will be closer to the true standard deviation as the number of simulated data sets tends to infinity. The standard errors obtained for each simulated sample are all estimates of this true standard deviation; and, as estimates of the true standard deviation, the standard errors will deviate from that value of the true standard deviation, sometimes higher, sometimes lower. 3.B. (15) Here is the output that results from 100*confint(reg)[2,]: 2.5 % 97.5 % Rounded off, these numbers are 0.054, Interpret the numbers 0.054, in terms of GPA and SAT.

3 Solution: By the Law of Large Numbers, the average GPA for a large number of students whose SAT is 1000 is approximately (1000), and the average GPA for a large number of students whose SAT is 900 is approximately (900). The difference between these averages is approximately If you multiply both endpoints of the confidence interval for 1 by 100, you have a confidence interval for Thus you can predict that the aforementioned difference between average GPAs will be in the range to (Grade points). Quoting from readings westfall/images/5349/conf_int_sig_test_pred.pdf: The 95% confidence interval for 1 is 2.85 < 1 < Thus we can predict that if many, many observations are produced by these two conditional distributions of work days, one of which has lotsize=81 and the other of which has lotsize = 80, then the average work-hours for the lotsize=81 distribution would be some number between 2.85 and 4.29 more than the average work-hours for the lotsize=80 distribution. By linearity, we could have picked (295,294) instead of (81,80). Note also that we can compare groups with (100, 50) for the two lot sizes, which, considering the range (20 to 120) of the lot size data, would be more relevant for this study. In this case the prediction is that the average work-hours for the lotsize=100 conditional distribution is 50 1 more than the average workhours for the lotsize=50 conditional distribution, and the confidence interval for this difference is 50(2.85) < 50( 1) < 50(4.29). 4. Explain how you would check the following assumptions using graphical methods only (no tests of hypotheses) for each one. Draw typical graphs to illustrate. 4.A. (15) Linearity Solution: I d draw the (x, y) scatterplot and overlay a LOESS curve and a least squares line. If the two reasonably coincide, then I d conclude that the linearity assumption is reasonable. If there are curvature patterns that make logical sense, and if these patterns are more pronounced than can easily be explained by data specific idiosyncrasies, than I d conclude there was curvature. Here is a graph to show a case where linearity is not a reasonable assumption.

4 4.B. (15) Constant variance Solution: I d draw the (x, resid ) scatterplot or the ( ŷ, resid ) scatterplot, where resid = y ŷ. I d overlay a LOESS curve and the flat line indicating the mean absolute residual. If the two reasonably coincide, then I d conclude that the constant variance assumption is reasonable. If absolute residuals tend to get systematically larger or systematically smaller, more than can easily be explained by data a specific idiosyncrasies, than I d conclude there was non constant variance. Here is a graph to show case where constant variance is not a reasonable assumption.

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