1)

p=pt(9.464259928, 48,lower.tail=FALSE)
2*p
## [1] 1.488495e-12

intercept t value: -2.60107 speed t value: 9.464259928 speed pr: 1.488e^(-12)

Mutliple R: 0.65107

F stat: 89.565 on 1 and 48 p val:1.490127e-12

degrees of freedom: 48

Anova speed:Ms=21186 F=89.566 p= 1.490127e-12 residual:Ms=236.541

auto=read.csv("auto.csv", header = TRUE, na.strings = "?")
auto=na.omit(auto)


mod=lm(auto$mpg~auto$horsepower)

summary(mod)
## 
## Call:
## lm(formula = auto$mpg ~ auto$horsepower)
## 
## 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 ***
## auto$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

2a i) yes

 ii) There is a decently strong relationship since the R^2 ~ .6
 
 iii)The relationship is negative
 
 iv) mpg=-0.157845(98)+39.935861=24.467051

2b)

plot(auto$horsepower,auto$mpg)+abline(a=39.935861, b=-.157845, col = "blue")

## integer(0)

#2c The relationship between horspoer and mpg seems to not be completley linear, so using the slr fit to predict values would proude error.

plot(mod)

#3a

auto<-auto[,-c(8:9)]
attach(auto)

pairs(auto)

#b

cor(auto)
##                     mpg  cylinders displacement horsepower     weight
## mpg           1.0000000 -0.7776175   -0.8051269 -0.7784268 -0.8322442
## cylinders    -0.7776175  1.0000000    0.9508233  0.8429834  0.8975273
## displacement -0.8051269  0.9508233    1.0000000  0.8972570  0.9329944
## horsepower   -0.7784268  0.8429834    0.8972570  1.0000000  0.8645377
## weight       -0.8322442  0.8975273    0.9329944  0.8645377  1.0000000
## acceleration  0.4233285 -0.5046834   -0.5438005 -0.6891955 -0.4168392
## year          0.5805410 -0.3456474   -0.3698552 -0.4163615 -0.3091199
##              acceleration       year
## mpg             0.4233285  0.5805410
## cylinders      -0.5046834 -0.3456474
## displacement   -0.5438005 -0.3698552
## horsepower     -0.6891955 -0.4163615
## weight         -0.4168392 -0.3091199
## acceleration    1.0000000  0.2903161
## year            0.2903161  1.0000000

#c

mlr_mod=lm(mpg ~ cylinders+displacement+horsepower+weight+acceleration+year, data=auto)


summary(mlr_mod)
## 
## Call:
## lm(formula = mpg ~ cylinders + displacement + horsepower + weight + 
##     acceleration + year, data = auto)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.6927 -2.3864 -0.0801  2.0291 14.3607 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -1.454e+01  4.764e+00  -3.051  0.00244 ** 
## cylinders    -3.299e-01  3.321e-01  -0.993  0.32122    
## displacement  7.678e-03  7.358e-03   1.044  0.29733    
## horsepower   -3.914e-04  1.384e-02  -0.028  0.97745    
## weight       -6.795e-03  6.700e-04 -10.141  < 2e-16 ***
## acceleration  8.527e-02  1.020e-01   0.836  0.40383    
## year          7.534e-01  5.262e-02  14.318  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.435 on 385 degrees of freedom
## Multiple R-squared:  0.8093, Adjusted R-squared:  0.8063 
## F-statistic: 272.2 on 6 and 385 DF,  p-value: < 2.2e-16
  1. yes 2)Weight and year seem to have a significant relationship to mpg. 3)There is a positive rletaionship of .7 between year and mpg.

#d

Y<-as.matrix(mpg)
head(Y)
##      [,1]
## [1,]   18
## [2,]   15
## [3,]   18
## [4,]   16
## [5,]   17
## [6,]   15
dim(Y)
## [1] 392   1
n=dim(Y)[1]

X<-matrix(c(rep(1, n),
            cylinders, 
            displacement,horsepower,weight,acceleration,year), 
          ncol=7, 
          byrow=FALSE)
head(X)
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,]    1    8  307  130 3504 12.0   70
## [2,]    1    8  350  165 3693 11.5   70
## [3,]    1    8  318  150 3436 11.0   70
## [4,]    1    8  304  150 3433 12.0   70
## [5,]    1    8  302  140 3449 10.5   70
## [6,]    1    8  429  198 4341 10.0   70
dim(X)
## [1] 392   7
betaHat<-solve(t(X)%*%X)%*%t(X)%*%Y
betaHat
##               [,1]
## [1,] -1.453525e+01
## [2,] -3.298591e-01
## [3,]  7.678430e-03
## [4,] -3.913556e-04
## [5,] -6.794618e-03
## [6,]  8.527325e-02
## [7,]  7.533672e-01

#E

plot(mlr_mod)

There are values in the y direction that get quite large as the grapgh goes to 35, which show how they impact relationship significantly.