Homework 4

hw4

homework4

abigailbalint

Author

Abigail Balint

Published

April 24, 2023

Code

library(tidyverse)
library(ggplot2)
library(dplyr)
library(readxl)
library(alr4)
library(smss)
knitr::opts_chunk$set(echo = TRUE)

Question 1

Equation: ŷ = −10,536 + 53.8x1 + 2.84x2 Subbing the size of home and lot size out for the variables in the equation: ŷ = −10,536 + 53.8(1240) + 2.84(18000) solving for y:

Code

a = -10536 + (53.8*1240) + (2.84*18000)
print(a)

[1] 107296

Code

a2=145000-a
print(a2)

[1] 37704

The predicted selling price is 107296. To find the residual, I am subtracting what it sold for from the predicted price. This means our prediction was low and the house sold for 37704 more than was predicted.

It is predicted to increase 53.8 dollars per square foot as this is the coefficient in the equation for home size square footage.
I think because the coefficient for home size is so much bigger than lot size, if we divide the coefficients by each other that would give the amount of home square feet we would need to have the same impact of one square foot in lot size, so 18.94 sq feet.

Code

c=53.8/2.84
print(c)

[1] 18.94366

Question 2

Code

summary(salary)

     degree      rank        sex          year            ysdeg      
 Masters:34   Asst :18   Male  :38   Min.   : 0.000   Min.   : 1.00  
 PhD    :18   Assoc:14   Female:14   1st Qu.: 3.000   1st Qu.: 6.75  
              Prof :20               Median : 7.000   Median :15.50  
                                     Mean   : 7.481   Mean   :16.12  
                                     3rd Qu.:11.000   3rd Qu.:23.25  
                                     Max.   :25.000   Max.   :35.00  
     salary     
 Min.   :15000  
 1st Qu.:18247  
 Median :23719  
 Mean   :23798  
 3rd Qu.:27258  
 Max.   :38045

The p value is .07 so we fail to reject the null hypothesis and are not able to say there is a significant difference in male vs female salaries.

Code

male <- salary %>%
  filter(sex == "Male")
female <- salary %>%
  filter(sex == "Female")
t.test(male$salary, female$salary, var.equal = TRUE)


    Two Sample t-test

data:  male$salary and female$salary
t = 1.8474, df = 50, p-value = 0.0706
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -291.257 6970.550
sample estimates:
mean of x mean of y 
 24696.79  21357.14

Fitting model and summary below:

Code

model1 <- lm(salary ~ degree + rank + sex + year + ysdeg, data = salary)
summary(model1)


Call:
lm(formula = salary ~ degree + rank + sex + year + ysdeg, 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

c: degree: Degree has a positive coefficient but not a significant p-value, indicating as degree increases salary does as well but not at a statistically significant level. rank: Rank has a very high positive coefficient as well as being very statistically significant, indicating a strong relationship between this variable and salary. sex: Similarly to degree, sex has a positive coefficient but not a significant p-value, indicating a positive slope here but not a statistically significant level. year: This variable is statistically significant with a positive slope indicating more years in a current rank may increase salary. ysdeg: This variable is not significant and actually has a negative slope/coefficient.

d: When I relevel rank I can see that it flips rank to have a negative slope/coefficient now instead of a positive one.

Code

levels(salary$rank)

[1] "Asst"  "Assoc" "Prof"

Code

rankrl <- relevel(salary$rank, ref="Prof")
model2 <- lm(salary ~ degree + rankrl + sex + year + ysdeg, data = salary)
summary(model2)


Call:
lm(formula = salary ~ degree + rankrl + sex + year + ysdeg, 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    
rankrlAsst  -11118.76    1351.77  -8.225 1.62e-10 ***
rankrlAssoc  -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

e.When I take rank out of the model, year and ysdeg now have a p-value that is statistically significant (a drastic change).

Code

model3 <- lm(salary ~ degree + sex + year + ysdeg, data = salary)
summary(model3)


Call:
lm(formula = salary ~ degree + sex + year + ysdeg, 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

To avoid multicollinearity, I don’t think we should use year and ysdeg both within the model since we now know there is an interaction due to the new dean 15 years ago that could cause these two variables to be related.

Below I am making a new variable that is if there is more than 15 years since highest degree they are assigned yes, otherwise no. I can see when fitting a model against salary this is very statistically significant, indicating the dean may have had an impact.

Code

salary2 <- salary  %>%
  mutate(ysdeg15 = ifelse(ysdeg > 15, "yes", "no"))
model4 <- lm(salary ~ ysdeg15, data = salary2)
summary(model4)


Call:
lm(formula = salary ~ ysdeg15, data = salary2)

Residuals:
   Min     1Q Median     3Q    Max 
 -8294  -3486  -1772   3829  10576 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  20125.9      913.4  22.033  < 2e-16 ***
ysdeg15yes    7343.5     1291.8   5.685 6.73e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4658 on 50 degrees of freedom
Multiple R-squared:  0.3926,    Adjusted R-squared:  0.3804 
F-statistic: 32.32 on 1 and 50 DF,  p-value: 6.734e-07

Question 3

Code

library(smss)
data("house.selling.price")
house <- house.selling.price
summary(house)

      case            Taxes           Beds       Baths           New      
 Min.   :  1.00   Min.   :  20   Min.   :2   Min.   :1.00   Min.   :0.00  
 1st Qu.: 25.75   1st Qu.:1178   1st Qu.:3   1st Qu.:2.00   1st Qu.:0.00  
 Median : 50.50   Median :1614   Median :3   Median :2.00   Median :0.00  
 Mean   : 50.50   Mean   :1908   Mean   :3   Mean   :1.96   Mean   :0.11  
 3rd Qu.: 75.25   3rd Qu.:2238   3rd Qu.:3   3rd Qu.:2.00   3rd Qu.:0.00  
 Max.   :100.00   Max.   :6627   Max.   :5   Max.   :4.00   Max.   :1.00  
     Price             Size     
 Min.   : 21000   Min.   : 580  
 1st Qu.: 93225   1st Qu.:1215  
 Median :132600   Median :1474  
 Mean   :155331   Mean   :1629  
 3rd Qu.:169625   3rd Qu.:1865  
 Max.   :587000   Max.   :4050

Based on this model, both whether it is new or not and the size are statistically significant variables in predicting price, although size is much more significant with a much smaller p value, the positive coefficient for whether the home is new or not is much higher.

Code

model5 <- lm(Price ~ New + Size, data = house)
summary(model5)


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

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 ** 
New          57736.283  18653.041   3.095  0.00257 ** 
Size           116.132      8.795  13.204  < 2e-16 ***
---
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

Prediction equation for new: y=-40230.867 + 57736.283new + 116.132size
3000 square feet for new vs not:

Code

new <- -40230.867 + 57736.283*1 + 116.132*3000
print (new)

[1] 365901.4

Code

notnew <- -40230.867 + 57736.283*0 + 116.132*3000
print(notnew)

[1] 308165.1

Selling price of new: 365901.4 Selling price of not new: 308165.1

Fitting model with interaction:

Code

model6 <- lm(Price ~ New*Size, data = house)
summary(model6)


Call:
lm(formula = Price ~ New * Size, data = house)

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    
New         -78527.502  51007.642  -1.540  0.12697    
Size           104.438      9.424  11.082  < 2e-16 ***
New:Size        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

Plot for new:

Code

housenew <- house %>%
  filter(New==1)
ggplot(data = housenew, aes(x = Size, y = Price)) +
  geom_point() +
  labs(x = "Home size", y = "Selling price") +
  geom_smooth(method = "lm", se = FALSE)

`geom_smooth()` using formula 'y ~ x'

Plot for not new:

Code

housenotnew <- house %>%
  filter(New==0)
ggplot(data = housenotnew, aes(x = Size, y = Price)) +
  geom_point() +
  labs(x = "Home size", y = "Selling price") +
  geom_smooth(method = "lm", se = FALSE)

`geom_smooth()` using formula 'y ~ x'

Same answer as C I think.
1. 300 square feet for new vs not was: Selling price of new: 365901.4 Selling price of not new: 308165.1

Now doing this for 1500 sf I got the same result. Is thi s because the new vs not new coefficient holds so much weight here?

Code

new2 <- -40230.867 + 57736.283*1 + 116.132*1500
print (new)

[1] 365901.4

Code

notnew2 <- -40230.867 + 57736.283*0 + 116.132*1500
print(notnew)

[1] 308165.1

I would use the model without interaction here because since the new vs not is so strong I feel like it overpowers the size variable and it makes more sense to look at them separately.