1.     t value  Pr(>|t|)

(Intercept) -2.601

speed 9.464 \(1.49e^{-12}\)

Residual standard error: 15.38 on 48 degrees of freedom

Multiple R-squared: 0.6511

F-statistic: 89.57 on 1 and 49 DF, p-value: \(1.49e^{-12}\)

anova(mod)

          Mean Sq  F value  Pr(>F)

speed 21186 89.5656 \(1.49e^{-12}\)

Residuals 236.54

Auto <- read.table("http://faculty.marshall.usc.edu/gareth-james/ISL/Auto.data", 
                   header=TRUE,
                   na.strings = c("?","NA"))
model <- lm(mpg ~ horsepower, data = Auto)
summary(model)
## 
## Call:
## lm(formula = mpg ~ horsepower, data = Auto)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -13.5710  -3.2592  -0.3435   2.7630  16.9240 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 39.935861   0.717499   55.66   <2e-16 ***
## horsepower  -0.157845   0.006446  -24.49   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.906 on 390 degrees of freedom
##   (5 observations deleted due to missingness)
## Multiple R-squared:  0.6059, Adjusted R-squared:  0.6049 
## F-statistic: 599.7 on 1 and 390 DF,  p-value: < 2.2e-16

ai. There is a strong relationship between the predictor and the response, as evident by the p value being almost 0.

aii. Since the p value is <2e-16, we can say the relationsip between the predictor and the response is very strong.

aiii. The relationship between the two is negative, as evident by the estimate for the slope being about -0.15.

aiv.

newdata = data.frame(horsepower = 98)
predict(model, newdata, interval = "confidence")
##        fit      lwr      upr
## 1 24.46708 23.97308 24.96108
predict(model, newdata, interval = "prediction")
##        fit     lwr      upr
## 1 24.46708 14.8094 34.12476

Therefore the predicted mpg is 24.46, and the associated confidence and prediction intervals are listed above.

plot(Auto$horsepower, Auto$mpg)
abline(model)

plot(model)

The linear fit does not seem like a good fit for the data. As we can see from the residuals plot, there is a clear pattern that indicates non-linearity, and the mean appears to be above 0. Additionally, the qqplot has many values that do not fall on the line, indicating non-linearity.