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There are 3 problems for a total of 30 points.
## Warning: package ’alr4’ was built under R version 4.0.4
## Warning: package ’car’ was built under R version 4.0.3
## Warning: package ’carData’ was built under R version 4.0.3
## Warning: package ’effects’ was built under R version 4.0.4
The data file salary (in the alr4 library) concerns salary and other characteristics of all faculty in a small Midwestern college collected in the early 1980s for presentation in legal proceedings for which discrimination against women in salary was at issue. All persons in the data hold tenured or tenure track positions; temporary faculty are not included. The variables include degree, a factor with levels PhD and MS; rank, a factor with levels Asst, Assoc, and Prof; sex, a factor with levels Male and Female (Male is the reference level); year, years in current rank; ysdeg, years since highest degree, and salary, academic year salary in dollars.
data(salary) ## import the data: "salary" which stored in alr4
(a) (2 points) Fit a simple linear regression: regress salary on sex and write down the estimated equation. Interpret the estimated coefficient of sex.
(b) (2 points) Fit a multiple regression: regress salary on sex and year. Interpret again the estimated coefficient of sex. Do you have the same interpretation as in (a) and can you explain the difference if there is any?
(c) (2 points) Now regress salary on sex, year, degree, rank, and ysdeg. Based on this model, report the test statistic, p-value, decision, and the final conclusion for testing the hypothesis that the mean salary for men and women is the same.
(d) (2 points) Using the regression in (c), obtain and interpret a 99% confidence interval for the difference in salary between males and females.
(e) (2 points) Regress salary on sex, year and their interactions. Provide an interpretation for the estimated coefficient of the interaction term.
The data file Highway (in the alr4 library) contains the automobile accident rate in 39 sections of large highways in the state of Minnesota in 1973. The variables are described below.
1. rate: Accident rate per million vehicle miles
2. len: Length of the segment in miles
3. adt: Average daily traffic count in thousands
4. trks: Truck volume as a percentage of the total volume
5. slim: Speed limit
6. shld: Shoulder width in feet of outer shoulder on the roadway
7. sigs: Number of signalized interchanges per mile in the segment data("Highway")
The variable sigs is 0 for freeway-type road segments but can be well over 2 for other segments. We can’t use logarithms because of the 0 values. The variable sigs is a rate per mile, so we add the constant to the number of signals in the segment, and then recompute a rate.
Highway$sigs1 <- with(Highway, (sigs * len + 1)/len)
We consider the logarithms of len, adt, trks, and sigs1. Now, we have eight predictors after transformation: log(len), shld, log(adt), log(trks), lane, slim, lwid, itg, log(sigs1), acpt, and htype. Because there are too many potential preditors, we would like to conduct variable selection first before using the model.
(a) (2 points) With the Highway data, start with the full model
log(rate) = log(len)+shld+log(adt)+log(trks)+lane+slim+lwid+itg +log(sigs1)+acpt+htype+ϵ
and do a backward elimination using BIC as the criterion. Write down the final estimated model.
(b) (2 points) With the Highway data, start with a model without any predictor and do a forward addition using BIC as the criterion. Write down the final estimated model.
(c) (2 points) Based on models from part (a) and (b), compare their adjusted R2 and MSE, which model is better?
(d) (2 points) Using the model you think is better based on part (c), make the residual vs fitted values plot. Do you see any problems?
(e) (2 points) Using the same model as part (d), plot the quantile-quantile plot to check normality. Do you see any problems?
Breakdowns of machines that produce steel cans are very costly. The more breakdowns, the fewer cans produced, and the smaller the company’s profits. To help anticipate profit loss, the owners of a can company would like to find a model that will predict the number of breakdowns on the assembly line. The model proposed by the company’s statisticians is the following:
y = β0 + β1x1 + β2x2 + β3x3 + β4x4 + ϵ
where is the number of breakdowns per 8-hour shift, = 1, afternoon shift
0, otherwise
, = 1, midnight shift
0, otherwise
x3 is the temperature of the plant (◦F),and x4 is the number of inexperienced personnel working on the assembly line. After the model is fit using the least squares procedure, the residuals are plotted against yˆ, as shown in the accompanying figure.
(a) (2 points) Do you detect a pattern in the residual plot? What does this suggest about the least squares assumptions?
(b) (2 points) Given the nature of the response variable y and the pattern detected in part a, what model adjustments would you recommend.
The regression analysis for the transformed model (where y∗ is the transformed variable)
y∗ = √y = β0 + β1x1 + β2x2 + β3x3 + β4x+ϵ
produces the prediction equation
yˆ∗ = 1.3 + 0.008x1 − 0.13x2 + 0.0025x3 + 0.26x4
(c) (2 points) Use the equation above to interpret the estimated coefficient of x1 and x2.
(d) (2 points) Use the equation to predict the number of breakdowns during the midnight shift if the temperature of the plant at that time is 87◦F and if there is only one inexperienced worker on the assembly line.
(e) (2 points) A 95% prediction interval for y∗ when x1 = 0, x2 = 0, x3 = 90◦F , and x4 = 2 is (1.965, 2.125). For those same values of the independent variables, find a 95% prediction interval for y, the number of breakdowns per 8-hour shift.
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