STAT3032_004_HW6_S2023_Solution (Shared)

STAT 3032 Regression and Correlated Data Homework 6 (Solution) Please show your work on each problem for full credit. A correct answer, unsupported by the necessary explanation , R code or output will receive very little if any credit. Your work needs to be organized in a reasonably neat and coherent way, and submitted as a pdf file on Canvas. Please do not share this handout outside the class. Problem 1 The Current Population Survey (CPS) is used to supplement census information between census years. The data file cps1985.csv contains a random sample of 534 persons from the CPS data collected in 1985, with information on wages and other characteristics of the workers. The variables we will use in the analyses are listed below: wage Wage (dollars per hour). age The age of the worker union Whether the worker has union membership. The possible values are “Yes” and “No”. Download the cps1985.csv data file from Canvas. Import the dataset into R and answer the following questions. (a)_[1 pt] Fit Model A ( wage ~ 1 + age + union) and provide the model summary. Solution: > modA = lm(wage~1+age+union,data = cps1985) > summary(modA) Call: lm(formula = wage ~ 1 + age + union, data = cps1985) Residuals: Min 1Q Median 3Q Max -8.862 -3.352 -1.187 1.977 36.929 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.09967 0.71624 8.516 < 2e-16 *** age 0.07009 0.01866 3.757 0.000191 *** unionYes 1.90745 0.56921 3.351 0.000862 *** ---

STAT 3032 Regression and Correlated Data Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 5.015 on 531 degrees of freedom Multiple R-squared: 0.05138, Adjusted R-squared: 0.04781 F-statistic: 14.38 on 2 and 531 DF, p-value: 8.282e-07 (b)_[2 pts] Interpret the slope of age in Model A in context. Solution: Holding the union membership constant, for every 1 year increase in age, the wage increases by 0.07009 dollars per hour on average. (c)_[2 pts] Interpret the slope of unionYes in Model A in context. Solution: Holding age constant, when compared to those without the union membership, the workers who have union membership earn 1.90745 dollars per hour more on average. Alternatively , Holding the age constant, the wage increases by 1.90745 dollars per hour on average when switching from not being in the union to being in the union. (d)_[1 pt] Fit the Model B ( wage ~ 1 + age + union + age:union) and provide the model summary. Solution: > modB = lm(wage~1+age+union+age:union,data = cps1985) > summary(m2) Call: lm(formula = wage ~ 1 + age + union + age:union, data = cps1985) Residuals: Min 1Q Median 3Q Max -8.123 -3.342 -1.167 1.879 37.137 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 5.60183 0.78091 7.173 2.48e-12 *** age 0.08385 0.02055 4.081 5.18e-05 *** unionYes 4.93788 1.99130 2.480 0.0135 * age:unionYes -0.07736 0.04872 -1.588 0.1129 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 5.008 on 530 degrees of freedom

STAT 3032 Regression and Correlated Data lm(formula = wage ~ 1 + exper, data = cps1985) Residuals: Min 1Q Median 3Q Max -8.247 -3.601 -1.111 2.332 36.084 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 8.37997 0.38895 21.545 <2e-16 *** exper 0.03614 0.01793 2.016 0.0443 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 5.124 on 532 degrees of freedom Multiple R-squared: 0.007579, Adjusted R-squared: 0.005714 F-statistic: 4.063 on 1 and 532 DF, p-value: 0.04433 (b)_[1 pt] Fit Model D that uses exper and sector (main effects only) to predict wage and provide the model summary. Solution: > modD = lm(wage~1+exper+sector,data = cps1985) > summary(modD) Call: lm(formula = wage ~ 1 + exper + sector, data = cps1985) Residuals: Min 1Q Median 3Q Max -12.011 -3.001 -0.945 1.924 32.679 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.51341 0.55243 11.791 < 2e-16 *** exper 0.05172 0.01643 3.147 0.00174 ** sectorconst 1.87394 1.14080 1.643 0.10105 sectormanag 5.25580 0.78286 6.714 4.94e-11 *** sectormanuf 0.49955 0.73441 0.680 0.49667 sectorother 1.19459 0.73445 1.627 0.10444 sectorprof 4.63452 0.65406 7.086 4.47e-12 *** sectorsales 0.12505 0.88767 0.141 0.88802 sectorservice -1.02039 0.69479 -1.469 0.14253 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 4.638 on 525 degrees of freedom Multiple R-squared: 0.1978, Adjusted R-squared: 0.1856 F-statistic: 16.18 on 8 and 525 DF, p-value: < 2.2e-16

STAT 3032 Regression and Correlated Data (c)_[1 pt] What would Model D look like in space? Please explain. Solution: Model D would be 8 parallel lines. Since the model contains one quantitative variable ( exper ), the shape will be a line. Since there is one categorical predictor variable ( sector ) with 8 levels, there will be 8 lines. Since there is no interaction, the lines are parallel to each other. (d)_[1 pt] What is the difference between the degrees of freedom of RSS in Model C and Model D? Your answer should be a number. Briefly explain why this is not 1. Solution: The difference between the degrees of freedom is 532 − 525 = 7 degrees of freedom. This is because the additional predictor variable in Model D ( sector ) requires 7 dummy variables and we need to use 7 degrees of freedom to estimate their slopes. (e)_[3 pts] Use the Partial F test to compare Model C and Model D. (i) [1 pt] What are the null and alternative hypotheses? Please define the parameters. (ii) [1 pt] What is the test statistic value? (iii) [1 pt] Based on the test result, do you prefer Model C and Model D? Please explain. You may use 0.05 as the significance level. Solution: (i) The hypotheses are H 0 : β 2 = β 3 = β 4 = β 5 = β 6 = β 7 = β 8 = 0 H A : at least 1 β j ≠ 0 for j = 2,3 ,…, 7,8 Where β 2 , ... , β 8 are the coefficients of the dummy variables for sector . OR: H 0 : wage~1 + exper H A : wage ~ 1 + exper + sector OR: H 0 : E ( wage )= β 0 + β 1 exper H A : E ( wage )= β 0 + β 1 exper + β 2 sectorconst + β 3 sectormanag + β 4 sectormanuf + β 5 sectorother + β 6 sectorprof + β 7 sectorsales + β 8 sectorservice (ii) From R, the anova output is > anova(modC,modD) Analysis of Variance Table