Question

a.​Present (here) the plot of the residuals of this simple linear regression model against its fitted values.

​b. Describe (here) the appearance of this residual plot.

c.​State (here) the RMSE of this regression.

MPG 43.1 19.9 19.2 Horsepower 48 110 105 165 139 103 115 155 142 150 71 76 65 100 84 58 88 92 139 110 90 17.7 18.1 20.3 21.5 16.9 ISS 185 27.2 41.5 46.6 23.7 27.2 39.1 28.0 24.0 20.2 20.5 28.0 34.7 36.1 35.7 20.2 23.9 29.9 30.4 36.0 22.6 36.4 27.5 33.7 44.6 32.9 38.0 24.2 38.1 39.4 25.4 31.3 34.1 34.0 31.0 27.4 223 28.0 17.6 34.4 20.6 Weight 16RS 3365 3535 3445 3205 2830 3345 4360 4054 3940 2190 2044 2110 2420 2490 1455 2305 2465 3570 3155 2678 2015 1800 2015 2965 2920 2380 2850 1980 2800 2950 2560 1910 66 80 85 90 65 67 74 110 87 95 75 Transmission manual manual manual manual manual manual automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic manual manual manual manual manual manual manual manual automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic automatic manual manual manual manual manual manual 1850 100 67 120 60 70 116 75 2615 1963 3230 1968 2070 2985 1842 1985 2395 2720 2470 2890 2025 3465 3465 3380 82 80 79 85 105

Answer 1

For the given data, we fit the regression model,

$y_i = \beta_0+\beta_1x_1+\beta_2x_2+\epsilon_i,\,i\in\{1,\ldots,50\}$,

where $\epsilon_i$ are independent Gaussian variables.

We rewrite this in matrix form as

$Y=X\beta+\epsilon$.

Performing linear regression, we get

.

2.5771 @ = (X'X)-'X'y = 1000 -0.0356 0.0120

Then we have the fitted values given by

$\hat Y = X\hat\beta$.

The residues are given by

$E = Y-\hat Y$.

a. The plot of $E\text{ vs }\hat Y$ is given in the following image.

1400 1200 1000 800 600 400 200 0 -200 -400 -600 1500 2000 2500 3000 3500 4000

b. The residue plot appears Gaussian centred at around (2600,-20).

c. The RSME of the regression is given by

$RSME = \sqrt{\sum_{i=1}^{50}\frac{(y_i-\hat y_i)^2}{50}} = \sqrt{\frac{(Y-\hat Y)'(Y-\hat Y)}{50}}=345.4715$.