HW4

hw4

Liam Tucksmith

Author

Liam Tucksmith

Published

May 12, 2023

Code

library(smss)
library(alr4)
knitr::opts_chunk$set(echo = TRUE)

For recent data in Jacksonville, Florida, on y = selling price of home (in dollars), x1 = size of home(in square feet), and x2 = lot size (in square feet), the prediction equation is y = ???10,536 + 53.8x1 + 2.84x2.

A. A particular home of 1240 square feet on a lot of 18,000 square feet sold for $145,000. Find the predicted selling price and the residual, and interpret.

Code

x1 <- 1240
x2 <- 18000
y <- 145000

predicted <- (-10536 + 53.8*(x1) + 2.84*(x2))
residual <- y - predicted
print(predicted)

[1] 107296

Code

print(residual)

[1] 37704

The predicted value is lower than the actual value, indicating that the equation is undervaluing the homes.

B. For fixed lot size, how much is the house selling price predicted to increase for each square- foot increase in home size? Why?

Code

x1 <- 1240
x2 <- 18000
y <- 145000

fixed_price <- (-10536 + 53.8*(x1) + 2.84*(x2))

x1 <- 1241
fixed_price_sqft_inc <- (-10536 + 53.8*(x1) + 2.84*(x2))

print(fixed_price_sqft_inc - fixed_price)

[1] 53.8

Using the numbers from part A, we can see that a 1 sqft increase results in a $53.80 predicted sell price increase when the lot size is fixed.

C. According to this prediction equation, for fixed home size, how much would lot size need to increase to have the same impact as a one-square-foot increase in home size?

Code

home_size_inc <- fixed_price_sqft_inc - fixed_price

x1 <- 1240
x2 <- 18000
y <- 145000

fixed_price_c <- (-10536 + 53.8*(x1) + 2.84*(x2))

lot_inc_pred <- fixed_price_c + home_size_inc
part1 <- (-10536 + 53.8*(x1))

part1 + 2.84((x2)+z) = lot_inc_pred (lot_inc_pred - part1)= (((x2)+z)2.84)/2.84 (lot_inc_pred - part1)/2.84 = (x2)+z

Code

z <- (lot_inc_pred - part1)/2.84 - x2
print(z)

[1] 18.94366

To have an impact of a 1 sqft increase in home size, lot size would have to increase by 18.9 sqft.

2 (Data file: salary in alr4 R package). The data file 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; Year, years in current rank; ysdeg, years since highest degree, and salary, academic year salary in dollars.

A. Test the hypothesis that the mean salary for men and women is the same, without regard to any other variable but sex. Explain your findings.

Code

summary(lm(salary ~ sex, data = salary))


Call:
lm(formula = salary ~ sex, data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-8602.8 -4296.6  -100.8  3513.1 16687.9 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)    24697        938  26.330   <2e-16 ***
sexFemale      -3340       1808  -1.847   0.0706 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5782 on 50 degrees of freedom
Multiple R-squared:  0.0639,    Adjusted R-squared:  0.04518 
F-statistic: 3.413 on 1 and 50 DF,  p-value: 0.0706

On average, the women make $3340 less than the men. The p-value of 0.07 indicates that this relationship is not statistically significant at the .05 level.

B. Run a multiple linear regression with salary as the outcome variable and everything else as predictors, including sex. Assuming no interactions between sex and the other predictors, obtain a 95% confidence interval for the difference in salary between males and females.

Code

summary(lm(salary ~ ., data = salary))


Call:
lm(formula = salary ~ ., data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-4045.2 -1094.7  -361.5   813.2  9193.1 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 15746.05     800.18  19.678  < 2e-16 ***
degreePhD    1388.61    1018.75   1.363    0.180    
rankAssoc    5292.36    1145.40   4.621 3.22e-05 ***
rankProf    11118.76    1351.77   8.225 1.62e-10 ***
sexFemale    1166.37     925.57   1.260    0.214    
year          476.31      94.91   5.018 8.65e-06 ***
ysdeg        -124.57      77.49  -1.608    0.115    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2398 on 45 degrees of freedom
Multiple R-squared:  0.855, Adjusted R-squared:  0.8357 
F-statistic: 44.24 on 6 and 45 DF,  p-value: < 2.2e-16

Code

confint(lm(salary ~ ., data = salary))

                 2.5 %      97.5 %
(Intercept) 14134.4059 17357.68946
degreePhD    -663.2482  3440.47485
rankAssoc    2985.4107  7599.31080
rankProf     8396.1546 13841.37340
sexFemale    -697.8183  3030.56452
year          285.1433   667.47476
ysdeg        -280.6397    31.49105

C. Interpret your finding for each predictor variable; discuss (a) statistical significance, (b) interpretation of the coefficient / slope in relation to the outcome variable and other variables

degreePhD - Not statistically significant. The coefficient of $1388 suggests that the faculty with phDs have, on avergage, a salary $1388 higher than faculty with masters.

rankAssoc - Statistically significant. The coefficient of $5292 suggests that the faculty with associate rank have, on avergage, a salary $5292 higher than faculty with assistant rank.

rankProf - Statistically significant. The coefficient of $11118 suggests that the faculty with phDs have, on avergage, a salary $11118 higher than faculty with assistant rank.

sexFemale - Not statistically significant. The coefficient of $1166 suggests that the female faculty have, on avergage, a salary $1166 higher than male faculty.

year - Statistically significant. The coefficient of $476 suggests that the faculty gain $476 in salary each additional year worked.

ysdeg - Not statistically significant. The coefficient of -$124 suggests that the faculty with phDs have, on avergage, a salary -$124 lower with each passing year since high school graduation.

D. Change the baseline category for the rank variable. Interpret the coefficients related to rank again.

Code

salary$rank <- relevel(salary$rank, ref = 'Prof')
summary(lm(salary ~ ., data = salary))


Call:
lm(formula = salary ~ ., data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-4045.2 -1094.7  -361.5   813.2  9193.1 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  26864.81    1375.29  19.534  < 2e-16 ***
degreePhD     1388.61    1018.75   1.363    0.180    
rankAsst    -11118.76    1351.77  -8.225 1.62e-10 ***
rankAssoc    -5826.40    1012.93  -5.752 7.28e-07 ***
sexFemale     1166.37     925.57   1.260    0.214    
year           476.31      94.91   5.018 8.65e-06 ***
ysdeg         -124.57      77.49  -1.608    0.115    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2398 on 45 degrees of freedom
Multiple R-squared:  0.855, Adjusted R-squared:  0.8357 
F-statistic: 44.24 on 6 and 45 DF,  p-value: < 2.2e-16

Code

confint(lm(salary ~ ., data = salary))

                  2.5 %      97.5 %
(Intercept)  24094.8394 29634.78404
degreePhD     -663.2482  3440.47485
rankAsst    -13841.3734 -8396.15463
rankAssoc    -7866.5550 -3786.25144
sexFemale     -697.8183  3030.56452
year           285.1433   667.47476
ysdeg         -280.6397    31.49105

Nothing changed, the model displays the same information, just with different variables. Here, the coefficients tell us that assistant faculty members make $11118 less than professors, and associate faculty members make $7866 less than professors.

E. Finkelstein (1980), in a discussion of the use of regression in discrimination cases, wrote, “[a] variable may reflect a position or status bestowed by the employer, in which case if there is discrimination in the award of the position or status, the variable may be ‘tainted.’” Thus, for example, if discrimination is at work in promotion of faculty to higher ranks, using rank to adjust salaries before comparing the sexes may not be acceptable to the courts. Exclude the variable rank, refit, and summarize how your findings changed, if they did.

Code

summary(lm(salary ~ . -rank, data = salary))


Call:
lm(formula = salary ~ . - rank, data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-8146.9 -2186.9  -491.5  2279.1 11186.6 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 17183.57    1147.94  14.969  < 2e-16 ***
degreePhD   -3299.35    1302.52  -2.533 0.014704 *  
sexFemale   -1286.54    1313.09  -0.980 0.332209    
year          351.97     142.48   2.470 0.017185 *  
ysdeg         339.40      80.62   4.210 0.000114 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3744 on 47 degrees of freedom
Multiple R-squared:  0.6312,    Adjusted R-squared:  0.5998 
F-statistic: 20.11 on 4 and 47 DF,  p-value: 1.048e-09

The sex coefficient flipped to show that females make -$1286 less than males, but it’s still not statistically signigicant.

F. Everyone in this dataset was hired the year they earned their highest degree. It is also known that a new Dean was appointed 15 years ago, and everyone in the dataset who earned their highest degree 15 years ago or less than that has been hired by the new Dean. Some people have argued that the new Dean has been making offers that are a lot more generous to newly hired faculty than the previous one and that this might explain some of the variation in Salary.

Create a new variable that would allow you to test this hypothesis and run another multiple regression model to test this. Select variables carefully to make sure there is no multicollinearity. Explain why multicollinearity would be a concern in this case and how you avoided it. Do you find support for the hypothesis that the people hired by the new Dean are making higher than those that were not?

Code

salary$yrs_since_dean <- ifelse(salary$ysdeg <= 15, 1, 0)
cor.test(salary$yrs_since_dean, salary$ysdeg)


    Pearson's product-moment correlation

data:  salary$yrs_since_dean and salary$ysdeg
t = -11.101, df = 50, p-value = 4.263e-15
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.9074548 -0.7411040
sample estimates:
       cor 
-0.8434239

Code

summary(lm(salary ~ . -ysdeg, data = salary))


Call:
lm(formula = salary ~ . - ysdeg, data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-3403.3 -1387.0  -167.0   528.2  9233.8 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)     24425.32    1107.52  22.054  < 2e-16 ***
degreePhD         818.93     797.48   1.027   0.3100    
rankAsst       -11096.95    1191.00  -9.317 4.54e-12 ***
rankAssoc       -6124.28    1028.58  -5.954 3.65e-07 ***
sexFemale         907.14     840.54   1.079   0.2862    
year              434.85      78.89   5.512 1.65e-06 ***
yrs_since_dean   2163.46    1072.04   2.018   0.0496 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2362 on 45 degrees of freedom
Multiple R-squared:  0.8594,    Adjusted R-squared:  0.8407 
F-statistic: 45.86 on 6 and 45 DF,  p-value: < 2.2e-16

Looking at the yrs_since_dean variable, the p-value is statistically significant and the positive coefficient, both supporting the hypothesis that the dean has been offering higher salaries to newer faculty.

(Data file: house.selling.price in smss R package)

A. Using the house.selling.price data, run and report regression results modeling y = selling price (in dollars) in terms of size of home (in square feet) and whether the home is new (1 = yes; 0 = no). In particular, for each variable; discuss statistical significance and interpret the meaning of the coefficient.

Code

data("house.selling.price")
summary(lm(Price ~ Size + New, data = house.selling.price))


Call:
lm(formula = Price ~ Size + New, data = house.selling.price)

Residuals:
    Min      1Q  Median      3Q     Max 
-205102  -34374   -5778   18929  163866 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -40230.867  14696.140  -2.738  0.00737 ** 
Size           116.132      8.795  13.204  < 2e-16 ***
New          57736.283  18653.041   3.095  0.00257 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 53880 on 97 degrees of freedom
Multiple R-squared:  0.7226,    Adjusted R-squared:  0.7169 
F-statistic: 126.3 on 2 and 97 DF,  p-value: < 2.2e-16

Both Size and New are statistically significant in this model, and both have positive coefficients. For New, this indicates that new houses are, on average, $57736 more expensive than non-new. And for Size, a 1 sqft increase in size indicates a $116 increase in sell price.

B. Report and interpret the prediction equation, and form separate equations relating selling price to size for new and for not new homes.

selling price prediction = -40230 + 116(size) + 57736(new)

The prediction equation indicates that while controlling for size and whether or not the size of the car is new, the prediction on average is $40230 lower than the actual selling price.

New: selling price prediction = -40230 + 116(size) + 57736

Not new: selling price prediction = -40230 + 116(size)

C. Find the predicted selling price for a home of 3000 square feet that is (i) new, (ii) not new.

Code

new_prediction <- -40230 + 116*(3000) + 57736
not_new_prediction <- -40230 + 116*(3000)

print(new_prediction)

[1] 365506

Code

print(not_new_prediction)

[1] 307770

D. Fit another model, this time with an interaction term allowing interaction between size and new, and report the regression results

Code

summary(lm(Price ~ Size + New + (Size * New), data = house.selling.price))


Call:
lm(formula = Price ~ Size + New + (Size * New), data = house.selling.price)

Residuals:
    Min      1Q  Median      3Q     Max 
-175748  -28979   -6260   14693  192519 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -22227.808  15521.110  -1.432  0.15536    
Size           104.438      9.424  11.082  < 2e-16 ***
New         -78527.502  51007.642  -1.540  0.12697    
Size:New        61.916     21.686   2.855  0.00527 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 52000 on 96 degrees of freedom
Multiple R-squared:  0.7443,    Adjusted R-squared:  0.7363 
F-statistic: 93.15 on 3 and 96 DF,  p-value: < 2.2e-16

E. Report the lines relating the predicted selling price to the size for homes that are (i) new, (ii) not new.

New: selling price prediction = -22228 + 104(size) - 78528 + 62(size) = -100756 + 168(size)

Not new: selling price prediction = -22228 + 104(size)

F. Find the predicted selling price for a home of 3000 square feet that is (i) new, (ii) not new.

Code

new_prediction <- -100756 + 168*(3000)
not_new_prediction <- 22228 + 104*(3000)

print(new_prediction)

[1] 403244

Code

print(not_new_prediction)

[1] 334228

G. Find the predicted selling price for a home of 1500 square feet that is (i) new, (ii) not new.

Code

new_prediction <- -100756 + 168*(1500)
not_new_prediction <- 22228 + 104*(1500)

print(new_prediction)

[1] 151244

Code

print(not_new_prediction)

[1] 178228

Comparing to (F), explain how the difference in predicted selling prices changes as the size of home increases.

As the size of the home increase, the not new homes increase in price at a lower rate than the new homes. This is shown by the not new homes being predicted as more selling for more on average than the new homes for the 1500 sqft home, and the opposite being true for the 3000 sqft home predictions.

H. Do you think the model with interaction or the one without it represents the relationship of size and new to the outcome price? What makes you prefer one model over another?

I prefer the model without the interaction term, because Size and New are both statistically significant and therefore all the variables in the model are statistically significant, which isn’t the case for the model with the interaction term.