#problem 10 This question should be answered using the Carseats data set.

library(ISLR2)
attach(Carseats)
head(Carseats)
summary(Carseats)
##      Sales          CompPrice       Income        Advertising    
##  Min.   : 0.000   Min.   : 77   Min.   : 21.00   Min.   : 0.000  
##  1st Qu.: 5.390   1st Qu.:115   1st Qu.: 42.75   1st Qu.: 0.000  
##  Median : 7.490   Median :125   Median : 69.00   Median : 5.000  
##  Mean   : 7.496   Mean   :125   Mean   : 68.66   Mean   : 6.635  
##  3rd Qu.: 9.320   3rd Qu.:135   3rd Qu.: 91.00   3rd Qu.:12.000  
##  Max.   :16.270   Max.   :175   Max.   :120.00   Max.   :29.000  
##    Population        Price        ShelveLoc        Age          Education   
##  Min.   : 10.0   Min.   : 24.0   Bad   : 96   Min.   :25.00   Min.   :10.0  
##  1st Qu.:139.0   1st Qu.:100.0   Good  : 85   1st Qu.:39.75   1st Qu.:12.0  
##  Median :272.0   Median :117.0   Medium:219   Median :54.50   Median :14.0  
##  Mean   :264.8   Mean   :115.8                Mean   :53.32   Mean   :13.9  
##  3rd Qu.:398.5   3rd Qu.:131.0                3rd Qu.:66.00   3rd Qu.:16.0  
##  Max.   :509.0   Max.   :191.0                Max.   :80.00   Max.   :18.0  
##  Urban       US     
##  No :118   No :142  
##  Yes:282   Yes:258  
##                     
##                     
##                     
##

(a) Fit a multiple regression model to predict Sales using Price, Urban, and US.

fit<-lm(Sales ~ Price + Urban + US, data = Carseats)
summary(fit)
## 
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
## 
## 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) Provide an interpretation of each coefficient in the model. Be careful—some of the variables in the model are qualitative!

price coeffecient for price is -0.054459. Means for every dollar increase and the proce of the carseat, the sotre sale decreases by $54 on average.

The coefficient for US = Yes is 1.200573 which means comapred to stores outsides the US, the carseats are sold on average $1,200 more comparewd to stores outside the US.

(c) Write out the model in equation form, being careful to handle the qualitative variables properly. $ Sales = 13.04 - 0.05Price -0.022Urban + 1.2US$

(d) For which of the predictors can you reject the null hypothesis _H0 : βj = 0?__

see part (B) for interpretation. Price and US = yes are significant, we can reject the null hypothesis $H_0: $

(e) On the basis of your response to the previous question, fit a smaller model that only uses the predictors for which there is evidence of association with the outcome.

fit<-lm(Sales ~ Price + Urban + US, data = Carseats)
summary(fit)
## 
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
## 
## 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

(f) How well do the models in (a) and (e) fit the data?

Not good: adjusted R squared is 0.2335 for part a, and adjusted r squared is 0.2354 for part e. > 0.7 is preferred (g) Using the model from (e), obtain 95 % confidence intervals for the coefficient(s).

confint(fit)
##                   2.5 %      97.5 %
## (Intercept) 11.76359670 14.32334118
## Price       -0.06476419 -0.04415351
## UrbanYes    -0.55597316  0.51214085
## USYes        0.69130419  1.70984121

(h) Is there evidence of outliers or high leverage observations in the model from (e)?

par(mfrow=c(2,2))
plot(fit)

1. Residuals vs Fitted Plot The residuals show to be randomely scattered around zerom indicating no major non-linearity or heteroscedasticity, few residuals seem to be larger then -+ 5, indicating potential outliers

2. Q-Q Plot of Residuals Majority of the points lie close to the diagonal line, while some deviate at the right tail, which indicates outlier

3. Scale-Location Plot The data points are evenly spread. Some variance increases at the higher fitted values

4. Residuals vs Leverage Plot The majority of points have low leverage