hw1

The dataset teengamb concerns a study of teenage gambling in Britain. Fit a regression model with the expenditure on gambling as the response and the sex, status, income and verbal score as predictors. Present the output.

library(faraway)

## Warning: 套件 'faraway' 是用 R 版本 4.3.1 來建造的

data(teengamb)
r<-lm(gamble~sex+status+income+verbal, data=teengamb)
summary(r)

## 
## Call:
## lm(formula = gamble ~ sex + status + income + verbal, data = teengamb)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -51.082 -11.320  -1.451   9.452  94.252 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  22.55565   17.19680   1.312   0.1968    
## sex         -22.11833    8.21111  -2.694   0.0101 *  
## status        0.05223    0.28111   0.186   0.8535    
## income        4.96198    1.02539   4.839 1.79e-05 ***
## verbal       -2.95949    2.17215  -1.362   0.1803    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 22.69 on 42 degrees of freedom
## Multiple R-squared:  0.5267, Adjusted R-squared:  0.4816 
## F-statistic: 11.69 on 4 and 42 DF,  p-value: 1.815e-06

(a) What percentage of variation in the response is explained by these predictors?

\(R^2 = 0.5267\). The percentage of variation in the response is 52.7%.

(b) Which observation has the largest (positive) residual? Give the case number.

r.res<-r$residuals
max(r.res)

## [1] 94.25222

which.max(r.res)

## 24 
## 24

The observationcase with the case number 24 has the largest positive residual (94.25).

(c) Computet the mean and median of the residuals.

mean(r.res)

## [1] -1.556914e-16

median(r.res)

## [1] -1.451392

The mean of the residuals = 0. The median of the residuals = -1.451.

(d) Compute the correlation of the residuals with the fitted values.

cor(r$fitted.values, r.res)

## [1] -6.215823e-17

The correlation of the residuals with the fitted values = 0.

(e) Compute the correlation of the residuals with the income.

cor(teengamb$gamble, r.res)

## [1] 0.687951

The correlation of the residuals with the income = 0.688.

(f) For all other predictors held constant, what would be the difference in predicted expenditure on gambling for a male comapred to a female?

male: sex = 0

female: sex = 1

Therefore, for all other predictors held constant, the difference in predicted expenditure on gampling for a male compared to a female is gamle(sex=0) - gamle(sex=1) = 22.118.