hw

# Load necessary library and data
library(ISLR)
data("Auto")

# Check the structure of the Auto dataset
str(Auto)

## 'data.frame':    392 obs. of  9 variables:
##  $ mpg         : num  18 15 18 16 17 15 14 14 14 15 ...
##  $ cylinders   : num  8 8 8 8 8 8 8 8 8 8 ...
##  $ displacement: num  307 350 318 304 302 429 454 440 455 390 ...
##  $ horsepower  : num  130 165 150 150 140 198 220 215 225 190 ...
##  $ weight      : num  3504 3693 3436 3433 3449 ...
##  $ acceleration: num  12 11.5 11 12 10.5 10 9 8.5 10 8.5 ...
##  $ year        : num  70 70 70 70 70 70 70 70 70 70 ...
##  $ origin      : num  1 1 1 1 1 1 1 1 1 1 ...
##  $ name        : Factor w/ 304 levels "amc ambassador brougham",..: 49 36 231 14 161 141 54 223 241 2 ...

# View the first few rows of the Auto dataset
head(Auto)

##   mpg cylinders displacement horsepower weight acceleration year origin
## 1  18         8          307        130   3504         12.0   70      1
## 2  15         8          350        165   3693         11.5   70      1
## 3  18         8          318        150   3436         11.0   70      1
## 4  16         8          304        150   3433         12.0   70      1
## 5  17         8          302        140   3449         10.5   70      1
## 6  15         8          429        198   4341         10.0   70      1
##                        name
## 1 chevrolet chevelle malibu
## 2         buick skylark 320
## 3        plymouth satellite
## 4             amc rebel sst
## 5               ford torino
## 6          ford galaxie 500

# a. Perform simple linear regression with mpg as the response and horsepower as the predictor
lm_model <- lm(mpg ~ horsepower, data = Auto)

# Print the summary of the linear regression model
summary(lm_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
## Multiple R-squared:  0.6059, Adjusted R-squared:  0.6049 
## F-statistic: 599.7 on 1 and 390 DF,  p-value: < 2.2e-16

# i. Is there a relationship between the predictor and the response?
# Answer: To determine if there's a relationship, check the p-value associated with the predictor (horsepower) coefficient.
# If the p-value is less than the significance level (e.g., 0.05), then there's evidence of a relationship.
# Look for the "Pr(>|t|)" value for horsepower in the summary output.

# ii. How strong is the relationship between the predictor and the response?
# Answer: Look for the R-squared value in the summary output. 
# The R-squared value indicates the proportion of variance in the response variable (mpg) explained by the predictor variable (horsepower).
# Higher R-squared values indicate stronger relationships.

# iii. Is the relationship between the predictor and the response positive or negative?
# Answer: Look at the sign of the coefficient for horsepower in the summary output.
# A positive coefficient indicates a positive relationship, while a negative coefficient indicates a negative relationship.

# b. Plot the response and the predictor using a scatterplot
plot(Auto$horsepower, Auto$mpg, xlab = "Horsepower", ylab = "MPG", main = "Scatterplot of MPG vs Horsepower")
abline(lm_model, col = "red")  # Add least squares regression line