HW4

hw4

Ken Docekal

Author

Ken Docekal

Published

November 14, 2022

Code

library(tidyverse)
library(AER)

Error in library(AER): there is no package called 'AER'

Code

library(stargazer)

Error in library(stargazer): there is no package called 'stargazer'

Code

knitr::opts_chunk$set(echo = TRUE)

Q1

A

Code

a <- 1240

b <- 18000

c <- -10536 + 53.8*a + 2.84*b 

c

[1] 107296

Code

145000 - 107296

[1] 37704

The predicted selling price is 107296 which makes the residual 37704. This means that our model’s estimated sale price of $107296, for a house size of 1240 square feet and lot size of 18000 square feet, is off from the actual observed price by $37704.

B

With fixed lot size each square-foot increase in home size increases price by $53.80. This is because the variable’s coefficient is 53.8 and will therefore only increase by this amount if no factors other than home size are considered by the model.

C

Code

53.8/ 2.84

[1] 18.94366

If we fix home size, lot size need to increase by 18.94 feet to have the equivalent impact as a one square foot increase in home size.

Q2

Code

library(alr4)

Loading required package: car

Loading required package: carData


Attaching package: 'car'

The following object is masked from 'package:dplyr':

    recode

The following object is masked from 'package:purrr':

    some

Loading required package: effects

lattice theme set by effectsTheme()
See ?effectsTheme for details.

Code

data('salary', package = 'alr4')

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.:27259  
 Max.   :38045

Code

tibble(salary)

# A tibble: 52 × 6
   degree  rank  sex     year ysdeg salary
   <fct>   <fct> <fct>  <int> <int>  <int>
 1 Masters Prof  Male      25    35  36350
 2 Masters Prof  Male      13    22  35350
 3 Masters Prof  Male      10    23  28200
 4 Masters Prof  Female     7    27  26775
 5 PhD     Prof  Male      19    30  33696
 6 Masters Prof  Male      16    21  28516
 7 PhD     Prof  Female     0    32  24900
 8 Masters Prof  Male      16    18  31909
 9 PhD     Prof  Male      13    30  31850
10 PhD     Prof  Male      13    31  32850
# … with 42 more rows

A

Code

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


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

Findings indicate a negative relationship of $3340 for females however this result is not significant at the 95% confidence level and the model has low predictive ability as adjusted R-squared is only .05. This is confirmed as p-value is the same because there is only one explanatory variable.

Code

model <- lm(salary ~ sex, data = salary)
coeftest(model, vcov. = vcovHC, type = "HC1")

Error in coeftest(model, vcov. = vcovHC, type = "HC1"): could not find function "coeftest"

Code

model <- lm(salary ~ sex, data = salary)
confint(model)

               2.5 %    97.5 %
(Intercept) 22812.81 26580.773
sexFemale   -6970.55   291.257

B

Code

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

confint(model)

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

95% confidence interval for sex controlling for other variables:

-697.81 to 3030.564

C

Code

 lm(salary ~ sex + degree + rank + year + ysdeg, data = salary) |> summary()


Call:
lm(formula = salary ~ sex + degree + rank + 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 ***
sexFemale    1166.37     925.57   1.260    0.214    
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 ***
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

sex The p-value of 0.214 indicates the effect of sex is insignificant at the 95% confidence interval; with all other variables held equal the coefficient indicates an increase of $1166.37 in salary for female sex with a positive slope.

degreePhD The p-value of 0.18 indicates the effect of a PhD degree is insignificant at the 95% confidence interval; with all other variables held equal the coefficient indicates an increase of $1388.61 in salary for PhD holders compared to Masters degree holders with a positive slope.

rank Rank values are in relation to the baseline of rankAsst. The p-values of both rankAssoc and rankProf are low enough to indicate significance at the 95% confidence interval; with all other variables held equal the coefficients indicate positive slopes with a salary increase of $5292.36 and $11,118.76 for Associate and Full Professors respectively compared to Assistant Professor salary.

year The p-value of less than .05 indicates the effect of years in current rank is significant at the 95% confidence interval; with all other variables held equal the coefficient indicates a positive slope with an increase of $476.31 in salary per increase in year.

ysdeg The p-value of 0.115 indicates the effect of ysdeg is insignificant at the 95% confidence interval; with all other variables held equal the coefficient indicates an decrease of $124.57 in salary per year since highest degree with a negative slope.

D

Code

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


Call:
lm(formula = salary ~ sex + degree + rank + 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) 21038.41    1109.12  18.969  < 2e-16 ***
sexFemale    1166.37     925.57   1.260    0.214    
degreePhD    1388.61    1018.75   1.363    0.180    
rankAsst    -5292.36    1145.40  -4.621 3.22e-05 ***
rankProf     5826.40    1012.93   5.752 7.28e-07 ***
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

When rank’s baseline is changed to Assc we are able to see the values for rankAsst. The coefficient of -5292.36 indicates a negative relationship with p-value indicating significance at the 95% confidence interval.

Code

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


Call:
lm(formula = salary ~ sex + degree + rank + 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 ***
sexFemale     1166.37     925.57   1.260    0.214    
degreePhD     1388.61    1018.75   1.363    0.180    
rankAssoc    -5826.40    1012.93  -5.752 7.28e-07 ***
rankAsst    -11118.76    1351.77  -8.225 1.62e-10 ***
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

When we change the baseline to Prof the model tells us that with other variables equal professor with rank Asst would receive $11,118.76 less salary and a professor with rank Assoc would receive $5826.40 less salary. Both these results are significant at the 95% confidence interval.

E

Rank would be a biased estimator?? multicollinear?

Code

lm(salary ~ sex + degree + year + ysdeg, data = salary) |> summary()


Call:
lm(formula = salary ~ sex + degree + 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 ***
sexFemale   -1286.54    1313.09  -0.980 0.332209    
degreePhD   -3299.35    1302.52  -2.533 0.014704 *  
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

Comparing the rank excluded model with a non-excluded model we can see that rank is not multicollinear as each rank variable outcome is significant according to its individual p-value and adjusted R-squared decreases when rank is removed from the model. The rank-included model has relatively greater predictive power.

F

We can turn ysdeg into a dummy variable dean with 1 if hired after the dean, 0 if hired before the dean.

Code

salary$dean <- ifelse(salary$ysdeg <=15, 1, 0)
tibble(salary)

# A tibble: 52 × 7
   degree  rank  sex     year ysdeg salary  dean
   <fct>   <fct> <fct>  <int> <int>  <int> <dbl>
 1 Masters Prof  Male      25    35  36350     0
 2 Masters Prof  Male      13    22  35350     0
 3 Masters Prof  Male      10    23  28200     0
 4 Masters Prof  Female     7    27  26775     0
 5 PhD     Prof  Male      19    30  33696     0
 6 Masters Prof  Male      16    21  28516     0
 7 PhD     Prof  Female     0    32  24900     0
 8 Masters Prof  Male      16    18  31909     0
 9 PhD     Prof  Male      13    30  31850     0
10 PhD     Prof  Male      13    31  32850     0
# … with 42 more rows

Multicollinearity check:

Code

cor(salary$ysdeg, salary$year)

[1] 0.6387763

Code

cor(salary$ysdeg, salary$dean)

[1] -0.8434239

Code

cor(salary$year, salary$dean)

[1] -0.5394452

Code

lm(salary ~ sex + degree + year + rank + ysdeg + dean, data = salary) |> summary()


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

Residuals:
    Min      1Q  Median      3Q     Max 
-3621.2 -1336.8  -271.6   530.1  9247.6 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  25179.14    1901.59  13.241  < 2e-16 ***
sexFemale     1084.09     921.49   1.176    0.246    
degreePhD     1135.00    1031.16   1.101    0.277    
year           460.35      95.09   4.841 1.63e-05 ***
rankAssoc    -6177.44    1043.04  -5.923 4.39e-07 ***
rankAsst    -11411.45    1362.02  -8.378 1.16e-10 ***
ysdeg          -47.86      97.71  -0.490    0.627    
dean          1749.09    1372.83   1.274    0.209    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2382 on 44 degrees of freedom
Multiple R-squared:  0.8602,    Adjusted R-squared:  0.838 
F-statistic: 38.68 on 7 and 44 DF,  p-value: < 2.2e-16

Code

lm(salary ~ sex + degree + rank + dean, data = salary) |> summary()


Call:
lm(formula = salary ~ sex + degree + rank + dean, data = salary)

Residuals:
    Min      1Q  Median      3Q     Max 
-6187.5 -1750.9  -438.9  1719.5  9362.9 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  29511.3      784.0  37.640  < 2e-16 ***
sexFemale     -829.2      997.6  -0.831    0.410    
degreePhD     1126.2     1018.4   1.106    0.275    
rankAssoc    -7100.4     1297.0  -5.474 1.76e-06 ***
rankAsst    -11925.7     1512.4  -7.885 4.37e-10 ***
dean           319.0     1303.8   0.245    0.808    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3023 on 46 degrees of freedom
Multiple R-squared:  0.7645,    Adjusted R-squared:  0.7389 
F-statistic: 29.87 on 5 and 46 DF,  p-value: 2.192e-13

Selected model excluding ysdeg:

Code

lm(salary ~ sex + degree + year + rank + dean, data = salary) |> summary()


Call:
lm(formula = salary ~ sex + degree + year + rank + dean, 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 ***
sexFemale      907.14     840.54   1.079   0.2862    
degreePhD      818.93     797.48   1.027   0.3100    
year           434.85      78.89   5.512 1.65e-06 ***
rankAssoc    -6124.28    1028.58  -5.954 3.65e-07 ***
rankAsst    -11096.95    1191.00  -9.317 4.54e-12 ***
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

The new variable dummy dean is derived from ysdeg so we exclude ysdeg from the model. Controlling for all other variables, dean’s coefficient of 2163.46 represents a positive relationship between dean hiring and salary however this relationship is not statistically significant. Based on the selected model we are not able to reject the null hypothesis that people hired by the new Dean are making a higher salary at the 95% confidence level due to the p-value of .05.

Q3

Code

library(smss)

Warning: package 'smss' was built under R version 4.2.2

Code

data('house.selling.price', package = 'smss')

Code

summary(house.selling.price)

      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

Code

tibble(house.selling.price)

# A tibble: 100 × 7
    case Taxes  Beds Baths   New  Price  Size
   <int> <int> <int> <int> <int>  <int> <int>
 1     1  3104     4     2     0 279900  2048
 2     2  1173     2     1     0 146500   912
 3     3  3076     4     2     0 237700  1654
 4     4  1608     3     2     0 200000  2068
 5     5  1454     3     3     0 159900  1477
 6     6  2997     3     2     1 499900  3153
 7     7  4054     3     2     0 265500  1355
 8     8  3002     3     2     1 289900  2075
 9     9  6627     5     4     0 587000  3990
10    10   320     3     2     0  70000  1160
# … with 90 more rows

A

Code

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


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

The model prediciting home selling price in terms of size and newness is statistically significant overall and accounts for .72 of observed prices. Individually, both Size and New variables are significant at the 95% confidence interval; the Size coefficient indicates an of $116.13 per additional square feet while the New coefficient indicates new homes sell for $57,736.28 more than old homes.

B

Looking only at new homes, the mean selling price for new homes is predicted to be $290964.

Code

prediction1 <- predict(lm(Price ~ Size + New, data = subset(house.selling.price, New=="1"))) 

summary(prediction1)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  98870  184542  251916  290964  386247  556178

Looking only at non-new homes, the mean selling price for non-new homes is predicted to be $138,567.

Code

prediction0 <- predict(lm(Price ~ Size + New, data = subset(house.selling.price, New=="0"))) 

summary(prediction0)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  38347   98921  125030  138567  158451  400748

C

Code

n <- lm(Price ~ Size + New, data = house.selling.price)

The predicted selling price for a new home of 3000 square feet is $366,016.30

Code

predict(n, data.frame(Size = 3000) + (New = 1))

       1 
366016.3

The predicted selling price for a non-new home of 3000 square feet is $308,163.90

Code

predict(n, data.frame(Size = 3000) + (New = 0))

       1 
308163.9

D

The model with an interaction term between size and new indicates overall model significance at the 95% confidence interval due to the low model p-value. With other variables held equal however, the variable New is no longer statistically significant.

Code

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


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

The regression line for homes hat are new and not new show the same positive linear relationship and a very similar slope. This indicates that the degree of newness’ effect on selling price is similar for new and not new homes.

Code

new0 <- subset(house.selling.price, New=="0")

ggplot(new0, aes(y=Size, x=Price)) + 
  geom_point(colour = "red") + 
   geom_smooth(method = "lm", se=FALSE)

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

Code

new1 <- subset(house.selling.price, New=="1")

ggplot(new1, aes(y=Size, x=Price)) + 
  geom_point(colour = "blue") + 
   geom_smooth(method = "lm", se=FALSE)

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

F

The predicted selling price with Size*New

Code

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

The predicted selling price for a new home of 3000 square feet is $398,473.90.

Code

predict(nI, data.frame(Size = 3000) + (New = 1))

       1 
398473.9

The predicted selling price for a non-new home of 3000 square feet is $291,087.40.

Code

predict(nI, data.frame(Size = 3000) + (New = 0))

       1 
291087.4

G

Code

predict(nI, data.frame(Size = 1500) + (New = 1))

       1 
148942.5

Code

predict(nI, data.frame(Size = 1500) + (New = 0))

       1 
134429.8

As house size increase the difference in selling price also increases as for a home of 1500 square feet the different between new and non-new is predicted to be $14,512.70, compared to a difference of $107,386.50 for a 3000 square foot home.

H

Code

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


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

Code

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


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

The model with the interaction term is preferable compared to without the interaction term as the adjusted R-squared value is higher while both models are statistically significant indicating that model is more representative of the relationship of Size and New to Price.