Problem 1

a

There do not appear to be any noteworthy features in these plots. They each appear to follow a different distribution.

b

Y seems to correlate with X3 and X4 strongly, with some correlation shown for X1 and X2. There does not seem to be an issue with multicollinearity.

##            Y        X1        X2        X3        X4
## Y  1.0000000 0.5144107 0.4970057 0.8970645 0.8693865
## X1 0.5144107 1.0000000 0.1022689 0.1807692 0.3266632
## X2 0.4970057 0.1022689 1.0000000 0.5190448 0.3967101
## X3 0.8970645 0.1807692 0.5190448 1.0000000 0.7820385
## X4 0.8693865 0.3266632 0.3967101 0.7820385 1.0000000

c

Parameters are displayed in output below:

## 
## Call:
## lm(formula = Y ~ X1 + X2 + X3 + X4, data = data)
## 
## Coefficients:
## (Intercept)           X1           X2           X3           X4  
##  -124.38182      0.29573      0.04829      1.30601      0.51982
##   (Intercept)            X1            X2            X3            X4 
## -124.38182058    0.29572537    0.04828772    1.30601100    0.51981909
## 
## Call:
## lm(formula = Y ~ X1 + X2 + X3 + X4, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.9779 -3.4506  0.0941  2.4749  5.9959 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -124.38182    9.94106 -12.512 6.48e-11 ***
## X1             0.29573    0.04397   6.725 1.52e-06 ***
## X2             0.04829    0.05662   0.853  0.40383    
## X3             1.30601    0.16409   7.959 1.26e-07 ***
## X4             0.51982    0.13194   3.940  0.00081 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.099 on 20 degrees of freedom
## Multiple R-squared:  0.9629, Adjusted R-squared:  0.9555 
## F-statistic: 129.7 on 4 and 20 DF,  p-value: 5.262e-14
## (Intercept)          X1          X2          X3          X4 
##  9.94106316  0.04397141  0.05661745  0.16409129  0.13194285
##                  Estimate Std. Error     t value     Pr(>|t|)
## (Intercept) -124.38182058 9.94106316 -12.5119234 6.475642e-11
## X1             0.29572537 0.04397141   6.7254011 1.523710e-06
## X2             0.04828772 0.05661745   0.8528769 4.038262e-01
## X3             1.30601100 0.16409129   7.9590514 1.261726e-07
## X4             0.51981909 0.13194285   3.9397290 8.099714e-04

d

It appears that all variables should be retained except for X2. X2 has the best case for deletion because its p-value is high.

Problem 2

a

The fitted model seems very reasonable. It follows the observed data very well.

c

Assumptoin of normality somewhat holds. It would be wise to do deeper analysis. One could interpret the data either way in this case.

## [1] 0.9772966

Problem 3

a

The best model is the model containing X1, X3, and X4.

##                  Estimate Std. Error     t value     Pr(>|t|)
## (Intercept) -124.38182058 9.94106316 -12.5119234 6.475642e-11
## X1             0.29572537 0.04397141   6.7254011 1.523710e-06
## X2             0.04828772 0.05661745   0.8528769 4.038262e-01
## X3             1.30601100 0.16409129   7.9590514 1.261726e-07
## X4             0.51981909 0.13194285   3.9397290 8.099714e-04
##                Estimate  Std. Error   t value     Pr(>|t|)
## (Intercept) -78.5413917 16.14208680 -4.865628 8.236882e-05
## X1            0.2523577  0.08692488  2.903170 8.503442e-03
## X2            0.2123617  0.10505114  2.021508 5.615531e-02
## X4            1.2884924  0.17909367  7.194517 4.321123e-07
##               Estimate Std. Error    t value    Pr(>|t|)
## (Intercept) -7.9666688 23.3136535 -0.3417169 0.735806546
## X1           0.4482956  0.1501227  2.9861937 0.006808895
## X2           0.5044119  0.1762068  2.8626125 0.009048073
##              Estimate Std. Error  t value    Pr(>|t|)
## (Intercept) 41.321561 18.0098510 2.294387 0.031231211
## X1           0.492245  0.1711054 2.876852 0.008517216
## [1] "Model 1 AIC: 147.901137560136  BIC: 155.214392509345"
## [1] "Model 2 AIC: 181.582996063248  BIC: 187.677375187589"
## [1] "Model 3 AIC: 210.649462021124  BIC: 215.524965320596"
## [1] "Model 4 AIC: 216.564937232918  BIC: 220.221564707523"
## [1] "Model 5 AIC: 222.249079238426  BIC: 224.686830888163"
## (Intercept)          X1          X2          X4 
## -78.5413917   0.2523577   0.2123617   1.2884924
## (Intercept)          X1          X2          X4 
## 16.14208680  0.08692488  0.10505114  0.17909367
## [1] 0.8453581
## [1] 0.8232664

b

## [1] "Y = X3   AIC :  -36.834 BIC :  -34.396"
## [1] "Y = X4   AIC :  -31.248 BIC :  -28.81"
## [1] "Y = X1   AIC :  -3.684 BIC :  -1.246"
## [1] "Y = X2   AIC :  -3.093 BIC :  -0.655"
## [1] "Y = X1+X3   AIC :  -61.575 BIC :  -57.918"
## [1] "Y = X3+X4   AIC :  -46.442 BIC :  -42.785"
## [1] "Y = X1+X4   AIC :  -36.221 BIC :  -32.564"
## [1] "Y = X2+X3   AIC :  -35.007 BIC :  -31.35"
## [1] "Y = X2+X4   AIC :  -32.23 BIC :  -28.574"
## [1] "Y = X1+X2   AIC :  -9.6 BIC :  -5.943"
## [1] "Y = X1+X3+X4   AIC :  -73.455 BIC :  -68.579"
## [1] "Y = X1+X2+X3   AIC :  -59.988 BIC :  -55.112"
## [1] "Y = X2+X3+X4   AIC :  -44.793 BIC :  -39.917"