Assignment 2 Linear Regression
Alejandro Perez
2023-02-09
R Markdown
###2 The KNN classifyer is considered a method that is used to solve prolbems with a qualitative response. This method compares training data with the test observation variant(X0). Similar data points are then estimated for conditional probability for a set class(j). In contrast to the KNN regression method that measures quantitative response problems. The KNN regresion method uses the observation variant to estimate the average of all observation responses.
###9
**(9a)
library(MASS)
library(ISLR2)## Warning: package 'ISLR2' was built under R version 4.1.3##
## Attaching package: 'ISLR2'## The following object is masked from 'package:MASS':
##
## Bostonplot(Auto)
**(9b)
Auto$new_horsepower<-as.numeric(ifelse(Auto$horsepower=="NA",NA,Auto$horsepower))**(9c)
lm.fit=lm(mpg~cylinders+displacement+horsepower+weight+acceleration+year+origin, data=Auto)
summary(lm.fit)##
## Call:
## lm(formula = mpg ~ cylinders + displacement + horsepower + weight +
## acceleration + year + origin, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.5903 -2.1565 -0.1169 1.8690 13.0604
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -17.218435 4.644294 -3.707 0.00024 ***
## cylinders -0.493376 0.323282 -1.526 0.12780
## displacement 0.019896 0.007515 2.647 0.00844 **
## horsepower -0.016951 0.013787 -1.230 0.21963
## weight -0.006474 0.000652 -9.929 < 2e-16 ***
## acceleration 0.080576 0.098845 0.815 0.41548
## year 0.750773 0.050973 14.729 < 2e-16 ***
## origin 1.426141 0.278136 5.127 4.67e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.328 on 384 degrees of freedom
## Multiple R-squared: 0.8215, Adjusted R-squared: 0.8182
## F-statistic: 252.4 on 7 and 384 DF, p-value: < 2.2e-16**(9d)
par(mfrow = c(2, 2))
plot(lm.fit)
**(9e)
summary (lm(mpg~cylinders*displacement*horsepower*weight*acceleration*year*origin ,data=Auto[, 1:8]))##
## Call:
## lm(formula = mpg ~ cylinders * displacement * horsepower * weight *
## acceleration * year * origin, data = Auto[, 1:8])
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.4533 -1.0707 0.0000 0.9962 9.1994
##
## Coefficients: (15 not defined because of singularities)
## Estimate
## (Intercept) 6.785e+04
## cylinders -1.585e+04
## displacement -1.419e+03
## horsepower -1.300e+03
## weight 4.373e+01
## acceleration -8.400e+03
## year 6.232e+02
## origin -5.789e+04
## cylinders:displacement 3.404e+02
## cylinders:horsepower 3.118e+02
## displacement:horsepower 8.962e+00
## cylinders:weight -1.158e+01
## displacement:weight 2.313e-01
## horsepower:weight 4.055e-01
## cylinders:acceleration 2.009e+03
## displacement:acceleration 1.682e+01
## horsepower:acceleration 1.076e+01
## weight:acceleration 3.748e-01
## cylinders:year -1.748e+02
## displacement:year 8.341e+00
## horsepower:year 1.607e+00
## weight:year -9.176e-01
## acceleration:year 9.485e+01
## cylinders:origin 1.319e+04
## displacement:origin 1.351e+03
## horsepower:origin 1.241e+03
## weight:origin -4.564e+01
## acceleration:origin 7.439e+03
## year:origin -7.745e+02
## cylinders:displacement:horsepower -2.075e+00
## cylinders:displacement:weight -4.967e-02
## cylinders:horsepower:weight -9.443e-02
## displacement:horsepower:weight -2.805e-03
## cylinders:displacement:acceleration -3.446e+00
## cylinders:horsepower:acceleration -1.650e+00
## displacement:horsepower:acceleration -6.798e-02
## cylinders:weight:acceleration -3.977e-02
## displacement:weight:acceleration -3.174e-03
## horsepower:weight:acceleration -3.717e-03
## cylinders:displacement:year -1.871e+00
## cylinders:horsepower:year -1.675e-01
## displacement:horsepower:year -1.594e-02
## cylinders:weight:year 2.400e-01
## displacement:weight:year -7.017e-04
## horsepower:weight:year -6.825e-04
## cylinders:acceleration:year -2.225e+01
## displacement:acceleration:year -1.068e-01
## horsepower:acceleration:year -1.653e-01
## weight:acceleration:year -6.359e-03
## cylinders:displacement:origin -3.266e+02
## cylinders:horsepower:origin -2.947e+02
## displacement:horsepower:origin -8.446e+00
## cylinders:weight:origin 1.216e+01
## displacement:weight:origin -2.103e-01
## horsepower:weight:origin -3.992e-01
## cylinders:acceleration:origin -1.777e+03
## displacement:acceleration:origin -1.175e+01
## horsepower:acceleration:origin -3.670e+00
## weight:acceleration:origin -1.515e-01
## cylinders:year:origin 2.131e+02
## displacement:year:origin -7.372e+00
## horsepower:year:origin -6.129e-01
## weight:year:origin 9.512e-01
## acceleration:year:origin -8.096e+01
## cylinders:displacement:horsepower:weight 6.153e-04
## cylinders:displacement:horsepower:acceleration 8.053e-03
## cylinders:displacement:weight:acceleration 3.041e-04
## cylinders:horsepower:weight:acceleration 3.603e-04
## displacement:horsepower:weight:acceleration 2.971e-05
## cylinders:displacement:horsepower:year 1.436e-03
## cylinders:displacement:weight:year 5.645e-05
## cylinders:horsepower:weight:year 5.134e-05
## displacement:horsepower:weight:year 6.861e-06
## cylinders:displacement:acceleration:year 1.517e-02
## cylinders:horsepower:acceleration:year 2.417e-02
## displacement:horsepower:acceleration:year 1.006e-03
## cylinders:weight:acceleration:year 7.543e-04
## displacement:weight:acceleration:year 4.571e-05
## horsepower:weight:acceleration:year 5.842e-05
## cylinders:displacement:horsepower:origin 1.974e+00
## cylinders:displacement:weight:origin 4.576e-02
## cylinders:horsepower:weight:origin 9.146e-02
## displacement:horsepower:weight:origin 2.646e-03
## cylinders:displacement:acceleration:origin 2.354e+00
## cylinders:horsepower:acceleration:origin -5.449e-02
## displacement:horsepower:acceleration:origin 2.922e-02
## cylinders:weight:acceleration:origin -1.186e-02
## displacement:weight:acceleration:origin 1.539e-03
## horsepower:weight:acceleration:origin 2.184e-03
## cylinders:displacement:year:origin 1.676e+00
## cylinders:horsepower:year:origin -8.713e-02
## displacement:horsepower:year:origin 8.343e-03
## cylinders:weight:year:origin -2.488e-01
## displacement:weight:year:origin 3.907e-04
## horsepower:weight:year:origin 5.097e-04
## cylinders:acceleration:year:origin 1.900e+01
## displacement:acceleration:year:origin 3.522e-02
## horsepower:acceleration:year:origin 6.027e-02
## weight:acceleration:year:origin 2.916e-03
## cylinders:displacement:horsepower:weight:acceleration -2.295e-06
## cylinders:displacement:horsepower:weight:year -4.178e-07
## cylinders:displacement:horsepower:acceleration:year -1.124e-04
## cylinders:displacement:weight:acceleration:year -4.299e-06
## cylinders:horsepower:weight:acceleration:year -5.449e-06
## displacement:horsepower:weight:acceleration:year -4.413e-07
## cylinders:displacement:horsepower:weight:origin -5.872e-04
## cylinders:displacement:horsepower:acceleration:origin NA
## cylinders:displacement:weight:acceleration:origin NA
## cylinders:horsepower:weight:acceleration:origin NA
## displacement:horsepower:weight:acceleration:origin -1.684e-05
## cylinders:displacement:horsepower:year:origin NA
## cylinders:displacement:weight:year:origin NA
## cylinders:horsepower:weight:year:origin NA
## displacement:horsepower:weight:year:origin -4.410e-06
## cylinders:displacement:acceleration:year:origin NA
## cylinders:horsepower:acceleration:year:origin NA
## displacement:horsepower:acceleration:year:origin -4.537e-04
## cylinders:weight:acceleration:year:origin NA
## displacement:weight:acceleration:year:origin -2.222e-05
## horsepower:weight:acceleration:year:origin -3.306e-05
## cylinders:displacement:horsepower:weight:acceleration:year 3.293e-08
## cylinders:displacement:horsepower:weight:acceleration:origin NA
## cylinders:displacement:horsepower:weight:year:origin NA
## cylinders:displacement:horsepower:acceleration:year:origin NA
## cylinders:displacement:weight:acceleration:year:origin NA
## cylinders:horsepower:weight:acceleration:year:origin NA
## displacement:horsepower:weight:acceleration:year:origin 2.522e-07
## cylinders:displacement:horsepower:weight:acceleration:year:origin NA
## Std. Error
## (Intercept) 5.058e+04
## cylinders 9.813e+03
## displacement 9.892e+02
## horsepower 9.564e+02
## weight 3.998e+01
## acceleration 5.946e+03
## year 7.248e+02
## origin 4.376e+04
## cylinders:displacement 2.325e+02
## cylinders:horsepower 2.133e+02
## displacement:horsepower 6.394e+00
## cylinders:weight 9.975e+00
## displacement:weight 1.695e-01
## horsepower:weight 3.201e-01
## cylinders:acceleration 1.380e+03
## displacement:acceleration 1.593e+01
## horsepower:acceleration 2.313e+01
## weight:acceleration 7.925e-01
## cylinders:year 1.665e+02
## displacement:year 6.084e+00
## horsepower:year 4.964e+00
## weight:year 7.729e-01
## acceleration:year 6.703e+01
## cylinders:origin 9.455e+03
## displacement:origin 9.883e+02
## horsepower:origin 9.258e+02
## weight:origin 3.692e+01
## acceleration:origin 5.873e+03
## year:origin 5.774e+02
## cylinders:displacement:horsepower 1.370e+00
## cylinders:displacement:weight 3.278e-02
## cylinders:horsepower:weight 6.523e-02
## displacement:horsepower:weight 2.095e-03
## cylinders:displacement:acceleration 1.925e+00
## cylinders:horsepower:acceleration 2.294e+00
## displacement:horsepower:acceleration 1.616e-01
## cylinders:weight:acceleration 8.196e-02
## displacement:weight:acceleration 4.950e-03
## horsepower:weight:acceleration 9.754e-03
## cylinders:displacement:year 1.240e+00
## cylinders:horsepower:year 4.792e-01
## displacement:horsepower:year 3.453e-02
## cylinders:weight:year 1.956e-01
## displacement:weight:year 1.063e-03
## horsepower:weight:year 2.087e-03
## cylinders:acceleration:year 1.523e+01
## displacement:acceleration:year 1.814e-01
## horsepower:acceleration:year 2.979e-01
## weight:acceleration:year 1.018e-02
## cylinders:displacement:origin 2.342e+02
## cylinders:horsepower:origin 2.146e+02
## displacement:horsepower:origin 6.208e+00
## cylinders:weight:origin 9.713e+00
## displacement:weight:origin 1.620e-01
## horsepower:weight:origin 2.979e-01
## cylinders:acceleration:origin 1.391e+03
## displacement:acceleration:origin 1.128e+01
## horsepower:acceleration:origin 1.032e+01
## weight:acceleration:origin 3.466e-01
## cylinders:year:origin 1.533e+02
## displacement:year:origin 5.763e+00
## horsepower:year:origin 2.320e+00
## weight:year:origin 7.427e-01
## acceleration:year:origin 6.551e+01
## cylinders:displacement:horsepower:weight 4.125e-04
## cylinders:displacement:horsepower:acceleration 1.331e-02
## cylinders:displacement:weight:acceleration 4.129e-04
## cylinders:horsepower:weight:acceleration 8.587e-04
## displacement:horsepower:weight:acceleration 5.895e-05
## cylinders:displacement:horsepower:year 2.859e-03
## cylinders:displacement:weight:year 8.960e-05
## cylinders:horsepower:weight:year 1.863e-04
## displacement:horsepower:weight:year 1.258e-05
## cylinders:displacement:acceleration:year 1.668e-02
## cylinders:horsepower:acceleration:year 2.951e-02
## displacement:horsepower:acceleration:year 2.111e-03
## cylinders:weight:acceleration:year 1.042e-03
## displacement:weight:acceleration:year 6.427e-05
## horsepower:weight:acceleration:year 1.252e-04
## cylinders:displacement:horsepower:origin 1.380e+00
## cylinders:displacement:weight:origin 3.286e-02
## cylinders:horsepower:weight:origin 6.533e-02
## displacement:horsepower:weight:origin 2.022e-03
## cylinders:displacement:acceleration:origin 1.622e+00
## cylinders:horsepower:acceleration:origin 2.162e-01
## displacement:horsepower:acceleration:origin 8.997e-02
## cylinders:weight:acceleration:origin 9.461e-03
## displacement:weight:acceleration:origin 3.037e-03
## horsepower:weight:acceleration:origin 4.625e-03
## cylinders:displacement:year:origin 1.247e+00
## cylinders:horsepower:year:origin 7.737e-02
## displacement:horsepower:year:origin 1.938e-02
## cylinders:weight:year:origin 1.927e-01
## displacement:weight:year:origin 6.625e-04
## horsepower:weight:year:origin 9.857e-04
## cylinders:acceleration:year:origin 1.533e+01
## displacement:acceleration:year:origin 9.168e-02
## horsepower:acceleration:year:origin 1.337e-01
## weight:acceleration:year:origin 4.536e-03
## cylinders:displacement:horsepower:weight:acceleration 4.466e-06
## cylinders:displacement:horsepower:weight:year 9.705e-07
## cylinders:displacement:horsepower:acceleration:year 1.748e-04
## cylinders:displacement:weight:acceleration:year 5.405e-06
## cylinders:horsepower:weight:acceleration:year 1.111e-05
## displacement:horsepower:weight:acceleration:year 7.632e-07
## cylinders:displacement:horsepower:weight:origin 4.152e-04
## cylinders:displacement:horsepower:acceleration:origin NA
## cylinders:displacement:weight:acceleration:origin NA
## cylinders:horsepower:weight:acceleration:origin NA
## displacement:horsepower:weight:acceleration:origin 3.791e-05
## cylinders:displacement:horsepower:year:origin NA
## cylinders:displacement:weight:year:origin NA
## cylinders:horsepower:weight:year:origin NA
## displacement:horsepower:weight:year:origin 8.158e-06
## cylinders:displacement:acceleration:year:origin NA
## cylinders:horsepower:acceleration:year:origin NA
## displacement:horsepower:acceleration:year:origin 1.169e-03
## cylinders:weight:acceleration:year:origin NA
## displacement:weight:acceleration:year:origin 3.907e-05
## horsepower:weight:acceleration:year:origin 5.950e-05
## cylinders:displacement:horsepower:weight:acceleration:year 5.819e-08
## cylinders:displacement:horsepower:weight:acceleration:origin NA
## cylinders:displacement:horsepower:weight:year:origin NA
## cylinders:displacement:horsepower:acceleration:year:origin NA
## cylinders:displacement:weight:acceleration:year:origin NA
## cylinders:horsepower:weight:acceleration:year:origin NA
## displacement:horsepower:weight:acceleration:year:origin 4.889e-07
## cylinders:displacement:horsepower:weight:acceleration:year:origin NA
## t value
## (Intercept) 1.342
## cylinders -1.616
## displacement -1.434
## horsepower -1.360
## weight 1.094
## acceleration -1.413
## year 0.860
## origin -1.323
## cylinders:displacement 1.464
## cylinders:horsepower 1.461
## displacement:horsepower 1.402
## cylinders:weight -1.161
## displacement:weight 1.365
## horsepower:weight 1.267
## cylinders:acceleration 1.456
## displacement:acceleration 1.056
## horsepower:acceleration 0.465
## weight:acceleration 0.473
## cylinders:year -1.049
## displacement:year 1.371
## horsepower:year 0.324
## weight:year -1.187
## acceleration:year 1.415
## cylinders:origin 1.395
## displacement:origin 1.367
## horsepower:origin 1.340
## weight:origin -1.236
## acceleration:origin 1.267
## year:origin -1.341
## cylinders:displacement:horsepower -1.515
## cylinders:displacement:weight -1.515
## cylinders:horsepower:weight -1.448
## displacement:horsepower:weight -1.339
## cylinders:displacement:acceleration -1.790
## cylinders:horsepower:acceleration -0.719
## displacement:horsepower:acceleration -0.421
## cylinders:weight:acceleration -0.485
## displacement:weight:acceleration -0.641
## horsepower:weight:acceleration -0.381
## cylinders:displacement:year -1.510
## cylinders:horsepower:year -0.349
## displacement:horsepower:year -0.462
## cylinders:weight:year 1.227
## displacement:weight:year -0.660
## horsepower:weight:year -0.327
## cylinders:acceleration:year -1.461
## displacement:acceleration:year -0.589
## horsepower:acceleration:year -0.555
## weight:acceleration:year -0.625
## cylinders:displacement:origin -1.394
## cylinders:horsepower:origin -1.373
## displacement:horsepower:origin -1.361
## cylinders:weight:origin 1.252
## displacement:weight:origin -1.298
## horsepower:weight:origin -1.340
## cylinders:acceleration:origin -1.278
## displacement:acceleration:origin -1.042
## horsepower:acceleration:origin -0.356
## weight:acceleration:origin -0.437
## cylinders:year:origin 1.390
## displacement:year:origin -1.279
## horsepower:year:origin -0.264
## weight:year:origin 1.281
## acceleration:year:origin -1.236
## cylinders:displacement:horsepower:weight 1.492
## cylinders:displacement:horsepower:acceleration 0.605
## cylinders:displacement:weight:acceleration 0.737
## cylinders:horsepower:weight:acceleration 0.420
## displacement:horsepower:weight:acceleration 0.504
## cylinders:displacement:horsepower:year 0.502
## cylinders:displacement:weight:year 0.630
## cylinders:horsepower:weight:year 0.276
## displacement:horsepower:weight:year 0.545
## cylinders:displacement:acceleration:year 0.910
## cylinders:horsepower:acceleration:year 0.819
## displacement:horsepower:acceleration:year 0.477
## cylinders:weight:acceleration:year 0.724
## displacement:weight:acceleration:year 0.711
## horsepower:weight:acceleration:year 0.467
## cylinders:displacement:horsepower:origin 1.430
## cylinders:displacement:weight:origin 1.392
## cylinders:horsepower:weight:origin 1.400
## displacement:horsepower:weight:origin 1.309
## cylinders:displacement:acceleration:origin 1.451
## cylinders:horsepower:acceleration:origin -0.252
## displacement:horsepower:acceleration:origin 0.325
## cylinders:weight:acceleration:origin -1.254
## displacement:weight:acceleration:origin 0.507
## horsepower:weight:acceleration:origin 0.472
## cylinders:displacement:year:origin 1.344
## cylinders:horsepower:year:origin -1.126
## displacement:horsepower:year:origin 0.430
## cylinders:weight:year:origin -1.291
## displacement:weight:year:origin 0.590
## horsepower:weight:year:origin 0.517
## cylinders:acceleration:year:origin 1.239
## displacement:acceleration:year:origin 0.384
## horsepower:acceleration:year:origin 0.451
## weight:acceleration:year:origin 0.643
## cylinders:displacement:horsepower:weight:acceleration -0.514
## cylinders:displacement:horsepower:weight:year -0.430
## cylinders:displacement:horsepower:acceleration:year -0.643
## cylinders:displacement:weight:acceleration:year -0.795
## cylinders:horsepower:weight:acceleration:year -0.491
## displacement:horsepower:weight:acceleration:year -0.578
## cylinders:displacement:horsepower:weight:origin -1.414
## cylinders:displacement:horsepower:acceleration:origin NA
## cylinders:displacement:weight:acceleration:origin NA
## cylinders:horsepower:weight:acceleration:origin NA
## displacement:horsepower:weight:acceleration:origin -0.444
## cylinders:displacement:horsepower:year:origin NA
## cylinders:displacement:weight:year:origin NA
## cylinders:horsepower:weight:year:origin NA
## displacement:horsepower:weight:year:origin -0.541
## cylinders:displacement:acceleration:year:origin NA
## cylinders:horsepower:acceleration:year:origin NA
## displacement:horsepower:acceleration:year:origin -0.388
## cylinders:weight:acceleration:year:origin NA
## displacement:weight:acceleration:year:origin -0.569
## horsepower:weight:acceleration:year:origin -0.556
## cylinders:displacement:horsepower:weight:acceleration:year 0.566
## cylinders:displacement:horsepower:weight:acceleration:origin NA
## cylinders:displacement:horsepower:weight:year:origin NA
## cylinders:displacement:horsepower:acceleration:year:origin NA
## cylinders:displacement:weight:acceleration:year:origin NA
## cylinders:horsepower:weight:acceleration:year:origin NA
## displacement:horsepower:weight:acceleration:year:origin 0.516
## cylinders:displacement:horsepower:weight:acceleration:year:origin NA
## Pr(>|t|)
## (Intercept) 0.1808
## cylinders 0.1073
## displacement 0.1527
## horsepower 0.1751
## weight 0.2750
## acceleration 0.1589
## year 0.3906
## origin 0.1869
## cylinders:displacement 0.1444
## cylinders:horsepower 0.1450
## displacement:horsepower 0.1622
## cylinders:weight 0.2468
## displacement:weight 0.1735
## horsepower:weight 0.2063
## cylinders:acceleration 0.1467
## displacement:acceleration 0.2920
## horsepower:acceleration 0.6421
## weight:acceleration 0.6367
## cylinders:year 0.2949
## displacement:year 0.1715
## horsepower:year 0.7464
## weight:year 0.2361
## acceleration:year 0.1582
## cylinders:origin 0.1641
## displacement:origin 0.1726
## horsepower:origin 0.1813
## weight:origin 0.2175
## acceleration:origin 0.2064
## year:origin 0.1809
## cylinders:displacement:horsepower 0.1310
## cylinders:displacement:weight 0.1308
## cylinders:horsepower:weight 0.1488
## displacement:horsepower:weight 0.1816
## cylinders:displacement:acceleration 0.0745 .
## cylinders:horsepower:acceleration 0.4726
## displacement:horsepower:acceleration 0.6744
## cylinders:weight:acceleration 0.6279
## displacement:weight:acceleration 0.5220
## horsepower:weight:acceleration 0.7035
## cylinders:displacement:year 0.1323
## cylinders:horsepower:year 0.7270
## displacement:horsepower:year 0.6447
## cylinders:weight:year 0.2208
## displacement:weight:year 0.5098
## horsepower:weight:year 0.7439
## cylinders:acceleration:year 0.1452
## displacement:acceleration:year 0.5565
## horsepower:acceleration:year 0.5794
## weight:acceleration:year 0.5326
## cylinders:displacement:origin 0.1644
## cylinders:horsepower:origin 0.1708
## displacement:horsepower:origin 0.1747
## cylinders:weight:origin 0.2118
## displacement:weight:origin 0.1953
## horsepower:weight:origin 0.1813
## cylinders:acceleration:origin 0.2025
## displacement:acceleration:origin 0.2985
## horsepower:acceleration:origin 0.7223
## weight:acceleration:origin 0.6624
## cylinders:year:origin 0.1657
## displacement:year:origin 0.2019
## horsepower:year:origin 0.7919
## weight:year:origin 0.2014
## acceleration:year:origin 0.2175
## cylinders:displacement:horsepower:weight 0.1369
## cylinders:displacement:horsepower:acceleration 0.5458
## cylinders:displacement:weight:acceleration 0.4620
## cylinders:horsepower:weight:acceleration 0.6751
## displacement:horsepower:weight:acceleration 0.6147
## cylinders:displacement:horsepower:year 0.6160
## cylinders:displacement:weight:year 0.5292
## cylinders:horsepower:weight:year 0.7831
## displacement:horsepower:weight:year 0.5860
## cylinders:displacement:acceleration:year 0.3639
## cylinders:horsepower:acceleration:year 0.4134
## displacement:horsepower:acceleration:year 0.6339
## cylinders:weight:acceleration:year 0.4696
## displacement:weight:acceleration:year 0.4776
## horsepower:weight:acceleration:year 0.6411
## cylinders:displacement:horsepower:origin 0.1538
## cylinders:displacement:weight:origin 0.1649
## cylinders:horsepower:weight:origin 0.1626
## displacement:horsepower:weight:origin 0.1917
## cylinders:displacement:acceleration:origin 0.1479
## cylinders:horsepower:acceleration:origin 0.8012
## displacement:horsepower:acceleration:origin 0.7456
## cylinders:weight:acceleration:origin 0.2109
## displacement:weight:acceleration:origin 0.6127
## horsepower:weight:acceleration:origin 0.6372
## cylinders:displacement:year:origin 0.1800
## cylinders:horsepower:year:origin 0.2611
## displacement:horsepower:year:origin 0.6672
## cylinders:weight:year:origin 0.1977
## displacement:weight:year:origin 0.5558
## horsepower:weight:year:origin 0.6055
## cylinders:acceleration:year:origin 0.2162
## displacement:acceleration:year:origin 0.7011
## horsepower:acceleration:year:origin 0.6525
## weight:acceleration:year:origin 0.5208
## cylinders:displacement:horsepower:weight:acceleration 0.6078
## cylinders:displacement:horsepower:weight:year 0.6672
## cylinders:displacement:horsepower:acceleration:year 0.5206
## cylinders:displacement:weight:acceleration:year 0.4270
## cylinders:horsepower:weight:acceleration:year 0.6240
## displacement:horsepower:weight:acceleration:year 0.5636
## cylinders:displacement:horsepower:weight:origin 0.1584
## cylinders:displacement:horsepower:acceleration:origin NA
## cylinders:displacement:weight:acceleration:origin NA
## cylinders:horsepower:weight:acceleration:origin NA
## displacement:horsepower:weight:acceleration:origin 0.6573
## cylinders:displacement:horsepower:year:origin NA
## cylinders:displacement:weight:year:origin NA
## cylinders:horsepower:weight:year:origin NA
## displacement:horsepower:weight:year:origin 0.5893
## cylinders:displacement:acceleration:year:origin NA
## cylinders:horsepower:acceleration:year:origin NA
## displacement:horsepower:acceleration:year:origin 0.6983
## cylinders:weight:acceleration:year:origin NA
## displacement:weight:acceleration:year:origin 0.5700
## horsepower:weight:acceleration:year:origin 0.5789
## cylinders:displacement:horsepower:weight:acceleration:year 0.5719
## cylinders:displacement:horsepower:weight:acceleration:origin NA
## cylinders:displacement:horsepower:weight:year:origin NA
## cylinders:displacement:horsepower:acceleration:year:origin NA
## cylinders:displacement:weight:acceleration:year:origin NA
## cylinders:horsepower:weight:acceleration:year:origin NA
## displacement:horsepower:weight:acceleration:year:origin 0.6064
## cylinders:displacement:horsepower:weight:acceleration:year:origin NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.418 on 279 degrees of freedom
## Multiple R-squared: 0.9315, Adjusted R-squared: 0.904
## F-statistic: 33.88 on 112 and 279 DF, p-value: < 2.2e-16**(9f)
###10
head(Carseats)## Sales CompPrice Income Advertising Population Price ShelveLoc Age Education
## 1 9.50 138 73 11 276 120 Bad 42 17
## 2 11.22 111 48 16 260 83 Good 65 10
## 3 10.06 113 35 10 269 80 Medium 59 12
## 4 7.40 117 100 4 466 97 Medium 55 14
## 5 4.15 141 64 3 340 128 Bad 38 13
## 6 10.81 124 113 13 501 72 Bad 78 16
## Urban US
## 1 Yes Yes
## 2 Yes Yes
## 3 Yes Yes
## 4 Yes Yes
## 5 Yes No
## 6 No Yesstr(Carseats)## 'data.frame': 400 obs. of 11 variables:
## $ Sales : num 9.5 11.22 10.06 7.4 4.15 ...
## $ CompPrice : num 138 111 113 117 141 124 115 136 132 132 ...
## $ Income : num 73 48 35 100 64 113 105 81 110 113 ...
## $ Advertising: num 11 16 10 4 3 13 0 15 0 0 ...
## $ Population : num 276 260 269 466 340 501 45 425 108 131 ...
## $ Price : num 120 83 80 97 128 72 108 120 124 124 ...
## $ ShelveLoc : Factor w/ 3 levels "Bad","Good","Medium": 1 2 3 3 1 1 3 2 3 3 ...
## $ Age : num 42 65 59 55 38 78 71 67 76 76 ...
## $ Education : num 17 10 12 14 13 16 15 10 10 17 ...
## $ Urban : Factor w/ 2 levels "No","Yes": 2 2 2 2 2 1 2 2 1 1 ...
## $ US : Factor w/ 2 levels "No","Yes": 2 2 2 2 1 2 1 2 1 2 ...**(a)
lm.fit=lm(Sales~Price+Urban+US ,data=Carseats)
summary(lm.fit)##
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9206 -1.6220 -0.0564 1.5786 7.0581
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.043469 0.651012 20.036 < 2e-16 ***
## Price -0.054459 0.005242 -10.389 < 2e-16 ***
## UrbanYes -0.021916 0.271650 -0.081 0.936
## USYes 1.200573 0.259042 4.635 4.86e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2335
## F-statistic: 41.52 on 3 and 396 DF, p-value: < 2.2e-16**(b)
The “Urban” area coefficient in this model, does not effect the stores sale total. Additionaly, The “US” coefficient is indicitive of selling around 1200 more car seats than in other countries. Lastly, when the overall unit price jumped to $1000, the number of carseats decreased by 54,459 units.
**(c)
**(d)
You can rejec the null hypothesis from the “Urban” predictor
**(e)
lm.fit2=lm(Sales~Price+US ,data=Carseats)
summary(lm.fit2)##
## Call:
## lm(formula = Sales ~ Price + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9269 -1.6286 -0.0574 1.5766 7.0515
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.03079 0.63098 20.652 < 2e-16 ***
## Price -0.05448 0.00523 -10.416 < 2e-16 ***
## USYes 1.19964 0.25846 4.641 4.71e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.469 on 397 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2354
## F-statistic: 62.43 on 2 and 397 DF, p-value: < 2.2e-16**(f)
When examing the R-squared values given in the summary table, there is only about a 24% change in response between these two models.
**(g)
confint(lm.fit2)## 2.5 % 97.5 %
## (Intercept) 11.79032020 14.27126531
## Price -0.06475984 -0.04419543
## USYes 0.69151957 1.70776632par(mfrow=c(2,2))
plot(lm.fit2)
There is no evidence of outliers or high leverage based on the Normal.q-q plot and Residuals vs Leverage plot.
12
**(a)
The coefficient estimate for the regression of X onto Y is the same as the coefficient estimate for the regression of Y onto X, when ∑jx2j=∑jy2j. .
**(b)
set.seed(1)
x<-1:100
sum(x^2)## [1] 338350y<-2*x+rnorm(100,sd=0.1)
sum(y^2)## [1] 1353606fit.Y<-lm(y~x)
fit.X<-lm(x~y)
summary(fit.Y)##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.234005 -0.060584 0.001551 0.058514 0.229747
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0131666 0.0181897 0.724 0.471
## x 1.9999549 0.0003127 6395.532 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.09027 on 98 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 4.09e+07 on 1 and 98 DF, p-value: < 2.2e-16summary(fit.X)##
## Call:
## lm(formula = x ~ y)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.114850 -0.029264 -0.000719 0.030278 0.116917
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.462e-03 9.096e-03 -0.71 0.479
## y 5.000e-01 7.818e-05 6395.53 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.04513 on 98 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 4.09e+07 on 1 and 98 DF, p-value: < 2.2e-16**(c)
x <- 1:100
sum(x^2)## [1] 338350y <- 100:1
sum(y^2)## [1] 338350fit.Y<-lm(y~x)
fit.X<-lm(x~y)
summary(fit.Y)## Warning in summary.lm(fit.Y): essentially perfect fit: summary may be unreliable##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.575e-14 -5.302e-15 -2.850e-15 4.300e-16 2.680e-13
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.010e+02 5.598e-15 1.804e+16 <2e-16 ***
## x -1.000e+00 9.624e-17 -1.039e+16 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.778e-14 on 98 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 1.08e+32 on 1 and 98 DF, p-value: < 2.2e-16summary(fit.X)## Warning in summary.lm(fit.X): essentially perfect fit: summary may be unreliable##
## Call:
## lm(formula = x ~ y)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.680e-13 -4.300e-16 2.850e-15 5.302e-15 3.575e-14
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.010e+02 5.598e-15 1.804e+16 <2e-16 ***
## y -1.000e+00 9.624e-17 -1.039e+16 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.778e-14 on 98 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 1.08e+32 on 1 and 98 DF, p-value: < 2.2e-16