Problem 10

(a) Fit a multiple regression model to predict sales using price, urban, and US.

library(ISLR2)
attach(Carseats)
fit<-lm(Sales~Price+Urban+US)
summary(fit)

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

The table above

  1. \(Sales = \beta_0 + \beta_1 * Price + \beta_2 * uRBAN + \beta_3 * US + \epilson\)

  2. ‘Price’ and ‘US’

fit<-lm(Sales~Price+US)
summary(fit)

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

  1. terrible, each model explains around 23% of the variance in saloes

  1. evidence of outliers from (e)?
summary(influence.measures(fit))

Potentially influential observations of
     lm(formula = Sales ~ Price + US) :

    dfb.1_ dfb.Pric dfb.USYs dffit   cov.r   cook.d hat    
26   0.24  -0.18    -0.17     0.28_*  0.97_*  0.03   0.01  
29  -0.10   0.10    -0.10    -0.18    0.97_*  0.01   0.01  
43  -0.11   0.10     0.03    -0.11    1.05_*  0.00   0.04_*
50  -0.10   0.17    -0.17     0.26_*  0.98    0.02   0.01  
51  -0.05   0.05    -0.11    -0.18    0.95_*  0.01   0.00  
58  -0.05  -0.02     0.16    -0.20    0.97_*  0.01   0.01  
69  -0.09   0.10     0.09     0.19    0.96_*  0.01   0.01  
126 -0.07   0.06     0.03    -0.07    1.03_*  0.00   0.03_*
160  0.00   0.00     0.00     0.01    1.02_*  0.00   0.02  
166  0.21  -0.23    -0.04    -0.24    1.02    0.02   0.03_*
172  0.06  -0.07     0.02     0.08    1.03_*  0.00   0.02  
175  0.14  -0.19     0.09    -0.21    1.03_*  0.02   0.03_*
210 -0.14   0.15    -0.10    -0.22    0.97_*  0.02   0.01  
270 -0.03   0.05    -0.03     0.06    1.03_*  0.00   0.02  
298 -0.06   0.06    -0.09    -0.15    0.97_*  0.01   0.00  
314 -0.05   0.04     0.02    -0.05    1.03_*  0.00   0.02_*
353 -0.02   0.03     0.09     0.15    0.97_*  0.01   0.00  
357  0.02  -0.02     0.02    -0.03    1.03_*  0.00   0.02  
368  0.26  -0.23    -0.11     0.27_*  1.01    0.02   0.02_*
377  0.14  -0.15     0.12     0.24    0.95_*  0.02   0.01  
384  0.00   0.00     0.00     0.00    1.02_*  0.00   0.02  
387 -0.03   0.04    -0.03     0.05    1.02_*  0.00   0.02  
396 -0.05   0.05     0.08     0.14    0.98_*  0.01   0.00  
outlying.obs<-c(26,29,43,50,51,58)
carseats.small<-Carseats[-outlying.obs,]
                 )
26   0.24  -0.18    -0.17     0.28_*  0.97_*  0.03   0.01  
29  -0.10   0.10    -0.10    -0.18    0.97_*  0.01   0.01  
43  -0.11   0.10     0.03    -0.11    1.05_*  0.00   0.04_*
50  -0.10   0.17    -0.17     0.26_*  0.98    0.02   0.01  
51  -0.05   0.05    -0.11    -0.18    0.95_*  0.01   0.00  
58  -0.05  -0.02     0.16    -0.20    0.97_*  0.01   0.01  
69  -0.09   0.10     0.09     0.19    0.96_*  0.01   0.01  
126 -0.07   0.06     0.03    -0.07    1.03_*  0.00   0.03_*
160  0.00   0.00     0.00     0.01    1.02_*  0.00   0.02  
166  0.21  -0.23    -0.04    -0.24    1.02    0.02   0.03_*
172  0.06  -0.07     0.02     0.08    1.03_*  0.00   0.02  
175  0.14  -0.19     0.09    -0.21    1.03_*  0.02   0.03_*
210 -0.14   0.15    -0.10    -0.22    0.97_*  0.02   0.01  
270 -0.03   0.05    -0.03     0.06    1.03_*  0.00   0.02  
298 -0.06   0.06    -0.09    -0.15    0.97_*  0.01   0.00  
314 -0.05   0.04     0.02    -0.05    1.03_*  0.00   0.02_*
353 -0.02   0.03     0.09     0.15    0.97_*  0.01   0.00  
357
368 
377 
384  
387
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