Problem 1

In this problem, we will use mtcars dataset, and three variables: mgp (miles per gallon), hp (horsepower) and wt (weight).

(The mtcars dataset is a classic,built-in dataset in R, derived from 1974 Motor Trend magazine. It contains data on fuel consumption and 10 aspects of automobile design and performance for 32 car models from the early 1970s).

plot

data(mtcars)
pairs(mtcars[, c("mpg", "hp", "wt")])

mpg vs. wt (bottom-left and top-right panels) - There’s a clear negative relationship: as car weight (wt) increases, miles per gallon (mpg) decreases.

mpg vs. hp (top-middle and bottom-left panels) - There’s also a negative relationship: cars with more horsepower (hp) tend to have lower fuel efficiency (mpg).The relationship looks a bit more scattered than mpg vs wt, suggesting hp is a weaker predictor of mpg than wt.

wt vs. hp (middle-right and bottom-middle panels) There’s a positive relationship: heavier cars also tend to have more horsepower.This correlation is important - it means wt and hp are not independent, and multicollinearity could matter if both are used in a multiple regression.

a simple linear regression

We run the following simple linear regression model

model.mpgvswt <- lm(mpg ~ wt,data=mtcars)    
summary(model.mpgvswt)
## 
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5432 -2.3647 -0.1252  1.4096  6.8727 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
## wt           -5.3445     0.5591  -9.559 1.29e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.046 on 30 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7446 
## F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

The slope is -5.3445. That means that for every one unit increase in weight, the mpg decreases by 5.3445 miles per gallon.

Problem 2

A <- matrix(c(1:6),nrow=3,ncol=2)
A
##      [,1] [,2]
## [1,]    1    4
## [2,]    2    5
## [3,]    3    6
B <- matrix(c(4,2,5,8),nrow=2,ncol=2)
B
##      [,1] [,2]
## [1,]    4    5
## [2,]    2    8
A%*%B
##      [,1] [,2]
## [1,]   12   37
## [2,]   18   50
## [3,]   24   63