Problem Set 5: Extending Multiple Linear Regression

Problem 1

auto<-read.csv("http://faculty.marshall.usc.edu/gareth-james/ISL/Auto.csv",
header=TRUE,
na.strings = "?")
auto=na.omit(auto)
auto<-auto[,-c(8:9)]
attach(auto)

(A)

modInt1<-lm(mpg~cylinders*displacement, data=auto)
summary(modInt1)

## 
## Call:
## lm(formula = mpg ~ cylinders * displacement, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -16.0432  -2.4308  -0.2263   2.2048  20.9051 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            48.22040    2.34712  20.545  < 2e-16 ***
## cylinders              -2.41838    0.53456  -4.524 8.08e-06 ***
## displacement           -0.13436    0.01615  -8.321 1.50e-15 ***
## cylinders:displacement  0.01182    0.00207   5.711 2.24e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.454 on 388 degrees of freedom
## Multiple R-squared:  0.6769, Adjusted R-squared:  0.6744 
## F-statistic:   271 on 3 and 388 DF,  p-value: < 2.2e-16

modInt2<-lm(mpg~cylinders:horsepower, data=auto)
summary(modInt2)

## 
## Call:
## lm(formula = mpg ~ cylinders:horsepower, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.2598  -3.4728  -0.4374   2.7793  17.8564 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          32.4981341  0.4473037   72.65   <2e-16 ***
## cylinders:horsepower -0.0144406  0.0005931  -24.35   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.923 on 390 degrees of freedom
## Multiple R-squared:  0.6031, Adjusted R-squared:  0.6021 
## F-statistic: 592.7 on 1 and 390 DF,  p-value: < 2.2e-16

modInt3<-lm(mpg~cylinders*weight, data=auto)
summary(modInt3)

## 
## Call:
## lm(formula = mpg ~ cylinders * weight, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -14.4916  -2.6225  -0.3927   1.7794  16.7087 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      65.3864559  3.7333137  17.514  < 2e-16 ***
## cylinders        -4.2097950  0.7238315  -5.816 1.26e-08 ***
## weight           -0.0128348  0.0013628  -9.418  < 2e-16 ***
## cylinders:weight  0.0010979  0.0002101   5.226 2.83e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.165 on 388 degrees of freedom
## Multiple R-squared:  0.7174, Adjusted R-squared:  0.7152 
## F-statistic: 328.3 on 3 and 388 DF,  p-value: < 2.2e-16

modInt3<-lm(mpg~cylinders:acceleration, data=auto)
summary(modInt3)

## 
## Call:
## lm(formula = mpg ~ cylinders:acceleration, data = auto)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -14.364  -4.090  -0.842   2.997  23.880 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            40.92789    1.24366   32.91   <2e-16 ***
## cylinders:acceleration -0.21146    0.01454  -14.54   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.293 on 390 degrees of freedom
## Multiple R-squared:  0.3515, Adjusted R-squared:  0.3499 
## F-statistic: 211.4 on 1 and 390 DF,  p-value: < 2.2e-16

modInt4<-lm(mpg~cylinders*year, data=auto)
summary(modInt4)

## 
## Call:
## lm(formula = mpg ~ cylinders * year, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.2164  -2.5792  -0.1558   2.2569  15.2532 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    -61.61775   15.10277  -4.080 5.47e-05 ***
## cylinders        5.51044    2.73705   2.013  0.04478 *  
## year             1.34054    0.19909   6.733 5.99e-11 ***
## cylinders:year  -0.11350    0.03647  -3.112  0.00199 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.131 on 388 degrees of freedom
## Multiple R-squared:  0.722,  Adjusted R-squared:  0.7199 
## F-statistic: 335.9 on 3 and 388 DF,  p-value: < 2.2e-16

modInt5<-lm(mpg~displacement:horsepower, data=auto)
summary(modInt5)

## 
## Call:
## lm(formula = mpg ~ displacement:horsepower, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.1917  -3.9460  -0.9919   3.0108  18.2170 
## 
## Coefficients:
##                           Estimate Std. Error t value Pr(>|t|)    
## (Intercept)              2.989e+01  3.901e-01   76.62   <2e-16 ***
## displacement:horsepower -2.694e-04  1.209e-05  -22.28   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.184 on 390 degrees of freedom
## Multiple R-squared:   0.56,  Adjusted R-squared:  0.5589 
## F-statistic: 496.4 on 1 and 390 DF,  p-value: < 2.2e-16

modInt6<-lm(mpg~displacement*weight, data=auto)
summary(modInt6)

## 
## Call:
## lm(formula = mpg ~ displacement * weight, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -13.8664  -2.4801  -0.3355   1.8071  17.9429 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          5.372e+01  1.940e+00  27.697  < 2e-16 ***
## displacement        -7.831e-02  1.131e-02  -6.922 1.85e-11 ***
## weight              -8.931e-03  8.474e-04 -10.539  < 2e-16 ***
## displacement:weight  1.744e-05  2.789e-06   6.253 1.06e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.097 on 388 degrees of freedom
## Multiple R-squared:  0.7265, Adjusted R-squared:  0.7244 
## F-statistic: 343.6 on 3 and 388 DF,  p-value: < 2.2e-16

modInt7<-lm(mpg~displacement:acceleration, data=auto)
summary(modInt7)

## 
## Call:
## lm(formula = mpg ~ displacement:acceleration, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -14.4845  -2.8803  -0.6247   2.2305  22.0359 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               36.9335310  0.6079798   60.75   <2e-16 ***
## displacement:acceleration -0.0047080  0.0001936  -24.31   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.927 on 390 degrees of freedom
## Multiple R-squared:  0.6025, Adjusted R-squared:  0.6015 
## F-statistic: 591.2 on 1 and 390 DF,  p-value: < 2.2e-16

modInt8<-lm(mpg~displacement*year, data=auto)
summary(modInt8)

## 
## Call:
## lm(formula = mpg ~ displacement * year, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.8530  -2.4250  -0.2234   2.0823  16.9933 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       -7.288e+01  8.368e+00  -8.709  < 2e-16 ***
## displacement       2.523e-01  4.059e-02   6.216 1.32e-09 ***
## year               1.408e+00  1.102e-01  12.779  < 2e-16 ***
## displacement:year -4.080e-03  5.453e-04  -7.482 4.96e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.729 on 388 degrees of freedom
## Multiple R-squared:  0.7735, Adjusted R-squared:  0.7718 
## F-statistic: 441.7 on 3 and 388 DF,  p-value: < 2.2e-16

modInt9<-lm(mpg~horsepower:weight, data=auto)
summary(modInt9)

## 
## Call:
## lm(formula = mpg ~ horsepower:weight, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.5691  -3.3660  -0.6786   2.6173  17.5124 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        3.292e+01  4.471e-01   73.63   <2e-16 ***
## horsepower:weight -2.791e-05  1.106e-06  -25.25   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.815 on 390 degrees of freedom
## Multiple R-squared:  0.6204, Adjusted R-squared:  0.6194 
## F-statistic: 637.3 on 1 and 390 DF,  p-value: < 2.2e-16

modInt10<-lm(mpg~horsepower*acceleration, data=auto)
summary(modInt10)

## 
## Call:
## lm(formula = mpg ~ horsepower * acceleration, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -13.3442  -2.7324  -0.4049   2.4210  15.8840 
## 
## Coefficients:
##                          Estimate Std. Error t value Pr(>|t|)    
## (Intercept)             33.512440   3.420187   9.798  < 2e-16 ***
## horsepower               0.017590   0.027425   0.641 0.521664    
## acceleration             0.800296   0.211899   3.777 0.000184 ***
## horsepower:acceleration -0.015698   0.002003  -7.838 4.45e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.426 on 388 degrees of freedom
## Multiple R-squared:  0.6809, Adjusted R-squared:  0.6784 
## F-statistic: 275.9 on 3 and 388 DF,  p-value: < 2.2e-16

modInt11<-lm(mpg~horsepower:year, data=auto)
summary(modInt11)

## 
## Call:
## lm(formula = mpg ~ horsepower:year, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -13.7959  -3.5770  -0.4462   3.0817  17.3724 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      4.045e+01  8.016e-01   50.46   <2e-16 ***
## horsepower:year -2.158e-03  9.621e-05  -22.43   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.164 on 390 degrees of freedom
## Multiple R-squared:  0.5634, Adjusted R-squared:  0.5623 
## F-statistic: 503.3 on 1 and 390 DF,  p-value: < 2.2e-16

modInt12<-lm(mpg~weight*acceleration, data=auto)
summary(modInt12)

## 
## Call:
## lm(formula = mpg ~ weight * acceleration, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.5823  -2.6411  -0.3517   2.2611  15.6704 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          2.814e+01  4.872e+00   5.776 1.57e-08 ***
## weight              -3.168e-03  1.461e-03  -2.168  0.03076 *  
## acceleration         1.117e+00  3.097e-01   3.608  0.00035 ***
## weight:acceleration -2.787e-04  9.694e-05  -2.875  0.00426 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.249 on 388 degrees of freedom
## Multiple R-squared:  0.706,  Adjusted R-squared:  0.7037 
## F-statistic: 310.5 on 3 and 388 DF,  p-value: < 2.2e-16

modInt13<-lm(mpg~weight*year, data=auto)
summary(modInt13)

## 
## Call:
## lm(formula = mpg ~ weight * year, data = auto)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.0397 -1.9956 -0.0983  1.6525 12.9896 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.105e+02  1.295e+01  -8.531 3.30e-16 ***
## weight       2.755e-02  4.413e-03   6.242 1.14e-09 ***
## year         2.040e+00  1.718e-01  11.876  < 2e-16 ***
## weight:year -4.579e-04  5.907e-05  -7.752 8.02e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.193 on 388 degrees of freedom
## Multiple R-squared:  0.8339, Adjusted R-squared:  0.8326 
## F-statistic: 649.3 on 3 and 388 DF,  p-value: < 2.2e-16

modInt14<-lm(mpg~acceleration*year, data=auto)
summary(modInt14)

## 
## Call:
## lm(formula = mpg ~ acceleration * year, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.9341  -4.9339  -0.6187   4.7066  18.0828 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)   
## (Intercept)       -89.41449   34.07514  -2.624  0.00903 **
## acceleration        2.09675    2.17707   0.963  0.33609   
## year                1.32728    0.45386   2.924  0.00365 **
## acceleration:year  -0.01738    0.02885  -0.602  0.54727   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.026 on 388 degrees of freedom
## Multiple R-squared:  0.4085, Adjusted R-squared:  0.4039 
## F-statistic: 89.31 on 3 and 388 DF,  p-value: < 2.2e-16

Interactions that appear significant at the 0.0 level: cylinders with displacment, horsepower, weight, then displacment with horsepower, weight, acceleration, year, then horsepower with weight, acceleration, year, and then weight with year.

Interactions that appear significant at the 0.001 level: cylinders with year, and then weight with acceleration.

The interaction between accleration and year was not significant at the 0.05 level.

(B)

Taking some of the interactions that were significant at the 0.0 level;

modInt1<-lm(mpg~cylinders*log(displacement), data=auto)
summary(modInt1)

## 
## Call:
## lm(formula = mpg ~ cylinders * log(displacement), data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -15.9898  -2.5737  -0.4187   2.1946  19.9768 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                  84.8717    11.4743   7.397 8.71e-13 ***
## cylinders                     0.6668     2.5000   0.267    0.790    
## log(displacement)           -12.1710     2.2511  -5.407 1.12e-07 ***
## cylinders:log(displacement)  -0.0921     0.4337  -0.212    0.832    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.387 on 388 degrees of freedom
## Multiple R-squared:  0.6865, Adjusted R-squared:  0.6841 
## F-statistic: 283.2 on 3 and 388 DF,  p-value: < 2.2e-16

When taking the log of displacment in the interaction between cylinders and dispalcment, the interaction was no longer significant at any level.

modInt2<-lm(mpg~(cylinders)^2:horsepower, data=auto)
summary(modInt2)

## 
## Call:
## lm(formula = mpg ~ (cylinders)^2:horsepower, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.2598  -3.4728  -0.4374   2.7793  17.8564 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          32.4981341  0.4473037   72.65   <2e-16 ***
## cylinders:horsepower -0.0144406  0.0005931  -24.35   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.923 on 390 degrees of freedom
## Multiple R-squared:  0.6031, Adjusted R-squared:  0.6021 
## F-statistic: 592.7 on 1 and 390 DF,  p-value: < 2.2e-16

When squaring cylinders in the interaction between cylinders and horsepower, the interaction remained significant at the 0.0 level with a p less than 2e-16

modInt3<-lm(mpg~sqrt(cylinders)*weight, data=auto)
summary(modInt3)

## 
## Call:
## lm(formula = mpg ~ sqrt(cylinders) * weight, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -14.8453  -2.6167  -0.3847   1.7669  16.7109 
## 
## Coefficients:
##                          Estimate Std. Error t value Pr(>|t|)    
## (Intercept)             86.718766   7.533994  11.510  < 2e-16 ***
## sqrt(cylinders)        -19.061380   3.393991  -5.616 3.73e-08 ***
## weight                  -0.018523   0.002544  -7.281 1.86e-12 ***
## sqrt(cylinders):weight   0.005025   0.001018   4.935 1.19e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.176 on 388 degrees of freedom
## Multiple R-squared:  0.716,  Adjusted R-squared:  0.7138 
## F-statistic:   326 on 3 and 388 DF,  p-value: < 2.2e-16

When squaring cylinders, the interaction between cylinders and weight is still significant but the p value was larger.

modInt5<-lm(mpg~displacement:log(horsepower), data=auto)
summary(modInt5)

## 
## Call:
## lm(formula = mpg ~ displacement:log(horsepower), data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -12.1952  -3.1592  -0.4499   2.5358  17.2179 
## 
## Coefficients:
##                              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                  33.68761    0.45431   74.15   <2e-16 ***
## displacement:log(horsepower) -0.01109    0.00042  -26.40   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.681 on 390 degrees of freedom
## Multiple R-squared:  0.6412, Adjusted R-squared:  0.6403 
## F-statistic:   697 on 1 and 390 DF,  p-value: < 2.2e-16

When taking the log of horsepower in the interaction between displacement and horsepower, the relationship was still significant at the 0.0 level.

modInt6<-lm(mpg~cos(displacement)*weight, data=auto)
summary(modInt6)

## 
## Call:
## lm(formula = mpg ~ cos(displacement) * weight, data = auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.3082  -2.8682  -0.3057   2.0703  16.3998 
## 
## Coefficients:
##                            Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               4.581e+01  8.871e-01  51.638   <2e-16 ***
## cos(displacement)        -7.756e-01  1.224e+00  -0.634    0.527    
## weight                   -7.538e-03  2.817e-04 -26.755   <2e-16 ***
## cos(displacement):weight  1.089e-06  4.088e-04   0.003    0.998    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.312 on 388 degrees of freedom
## Multiple R-squared:  0.6971, Adjusted R-squared:  0.6948 
## F-statistic: 297.7 on 3 and 388 DF,  p-value: < 2.2e-16

When taking the cosine of displacement in the interaction between displacement and weight, the interaction was no longer significant at any level.

modInt7<-lm(mpg~displacement:tan(acceleration), data=auto)
summary(modInt7)

## 
## Call:
## lm(formula = mpg ~ displacement:tan(acceleration), data = auto)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -14.613  -6.066  -0.626   5.372  22.988 
## 
## Coefficients:
##                                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                    2.363e+01  3.925e-01  60.192  < 2e-16 ***
## displacement:tan(acceleration) 1.147e-04  3.364e-05   3.411 0.000714 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.701 on 390 degrees of freedom
## Multiple R-squared:  0.02897,    Adjusted R-squared:  0.02648 
## F-statistic: 11.64 on 1 and 390 DF,  p-value: 0.0007141

When taking the tan() of acceleration in the interaction between dispalcment and acceleration, the interaction was still significant at the 0.0 level, but the P value was much larger.

Problem 2

#install.packages("ISLR")
library(ISLR)
data(Carseats)
names(Carseats)

##  [1] "Sales"       "CompPrice"   "Income"      "Advertising" "Population" 
##  [6] "Price"       "ShelveLoc"   "Age"         "Education"   "Urban"      
## [11] "US"

attach(Carseats)

(A)

mod=lm(Sales~Price+Urban+US)
summary(mod)

## 
## Call:
## lm(formula = Sales ~ Price + Urban + US)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.9206 -1.6220 -0.0564  1.5786  7.0581 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 13.043469   0.651012  20.036  < 2e-16 ***
## Price       -0.054459   0.005242 -10.389  < 2e-16 ***
## UrbanYes    -0.021916   0.271650  -0.081    0.936    
## USYes        1.200573   0.259042   4.635 4.86e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared:  0.2393, Adjusted R-squared:  0.2335 
## F-statistic: 41.52 on 3 and 396 DF,  p-value: < 2.2e-16

(B)

An intercept of 13.04 suggests that we would explect 13.04 Sales when all explanatory variables are = 0 or there are no explanatory variables. The Price coefficient of -0.054 suggests that for every one unit increase in price, we would expect sales to decrease by 0.054 units. The Urban coefficient of -0.0219 suggests that if the Sale is made in an Urban setting (hence, UrbanYes) then we would expect sales to decrease by 0.0219 units. The US coefficient of 1.2 suggests that if the Sale is made in the US then we would expect sales to go up by 1.2 unites.

(C)

UrbanNo & USNo: Yi = 13.043469 - 0.054459*Xi1

UrbanYes & USNO: Yi = 13.043469 - 0.021916 - 0.054459*Xi1

UrbanNo & USYes: Yi = 13.043469 + 1.200573 - 0.054459*Xi1

Urban Yes & USYes: Yi = 13.043469 - 0.021916 + 1.200573 - 0.054459*Xi1

(D)

We can reject the null hypothesis for UrbanYes. This suggests that we would not expect the mean value for Yi to change/be different from other mean values of Yi due to a change in UrbanYes other than because of chance.

(E)

mod2=lm(Sales~Price+US)
summary(mod2)

## 
## Call:
## lm(formula = Sales ~ Price + US)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.9269 -1.6286 -0.0574  1.5766  7.0515 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 13.03079    0.63098  20.652  < 2e-16 ***
## Price       -0.05448    0.00523 -10.416  < 2e-16 ***
## USYes        1.19964    0.25846   4.641 4.71e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.469 on 397 degrees of freedom
## Multiple R-squared:  0.2393, Adjusted R-squared:  0.2354 
## F-statistic: 62.43 on 2 and 397 DF,  p-value: < 2.2e-16

(F)

Because the second model that excludes ’Urban" has a lower residual standard error (2.469 < 2.472) and a greater adjusted R-squared value (0.2354 > 0.2335) we might conlude that the second smaller model that excludes Urban fits our data better. When observing the different plots of the models below it is difficult to determine what model fits the data better graphically.

plot(mod)

plot(mod2)

(G)

confint(mod2)

##                   2.5 %      97.5 %
## (Intercept) 11.79032020 14.27126531
## Price       -0.06475984 -0.04419543
## USYes        0.69151957  1.70776632

Problem Set #5

Ross Enlow

10/9/2019

Problem Set 5: Extending Multiple Linear Regression

Problem 1

(A)

Interactions that appear significant at the 0.0 level: cylinders with displacment, horsepower, weight, then displacment with horsepower, weight, acceleration, year, then horsepower with weight, acceleration, year, and then weight with year.

Interactions that appear significant at the 0.001 level: cylinders with year, and then weight with acceleration.

The interaction between accleration and year was not significant at the 0.05 level.

(B)

Taking some of the interactions that were significant at the 0.0 level;

When taking the log of displacment in the interaction between cylinders and dispalcment, the interaction was no longer significant at any level.

When squaring cylinders in the interaction between cylinders and horsepower, the interaction remained significant at the 0.0 level with a p less than 2e-16

When squaring cylinders, the interaction between cylinders and weight is still significant but the p value was larger.

When taking the log of horsepower in the interaction between displacement and horsepower, the relationship was still significant at the 0.0 level.

When taking the cosine of displacement in the interaction between displacement and weight, the interaction was no longer significant at any level.

When taking the tan() of acceleration in the interaction between dispalcment and acceleration, the interaction was still significant at the 0.0 level, but the P value was much larger.

Problem 2

(A)

(B)

(C)

UrbanNo & USNo: Yi = 13.043469 - 0.054459*Xi1

UrbanYes & USNO: Yi = 13.043469 - 0.021916 - 0.054459*Xi1

UrbanNo & USYes: Yi = 13.043469 + 1.200573 - 0.054459*Xi1

Urban Yes & USYes: Yi = 13.043469 - 0.021916 + 1.200573 - 0.054459*Xi1

(D)

We can reject the null hypothesis for UrbanYes. This suggests that we would not expect the mean value for Yi to change/be different from other mean values of Yi due to a change in UrbanYes other than because of chance.

(E)

(F)

(G)