Motor Trend Project
options(width = 100)
knitr::opts_chunk$set(fig.width=12, fig.height=8, fig.path='Figs/',
echo=FALSE, warning=FALSE, message=FALSE)Project Summary
Suppose I work for Motor Trend, a magazine about the automobile industry. Looking at a data set of a collection of cars, they are interested in exploring the relationship between a set of variables and miles per gallon (MPG) (outcome). They are particularly interested in the following two questions:
#1 “Is an automatic or manual transmission better for MPG”
#2 “Quantify the MPG difference between automatic and manual transmissions”
We conclude that manual transmission is better than automatic transmission for MPG from both a t-test and a linear regression model.
We will consider mtcars, a data frame with 32 observations on 11 (numeric) variables.
[, 1] mpg Miles/(US) gallon
[, 2] cyl Number of cylinders [, 3] disp Displacement (cu.in.)
[, 4] hp Gross horsepower
[, 5] drat Rear axle ratio
[, 6] wt Weight (1000 lbs)
[, 7] qsec 1/4 mile time
[, 8] vs Engine (0 = V-shaped, 1 = straight)
[, 9] am Transmission (0 = automatic, 1 = manual)
[,10] gear Number of forward gears
[,11] carb Number of carburetors
And we added:
[,12] trans automatic, manual
Install the appropriate packages:
Now, we will summarize and explore the data
Next, we add the 12th column to our dataframe
Now, we plot the different transmission types and their mpg.
Compare the two means
The mean mpg for a manual transmission is 7.245 higher than for an automatic transmission.
Now, we perform a t-test
Based on the results, the p-value = 0.001374 which is less than 0.05, we can reject the null hypothesis that there is no difference in MPG, and conclude that manual transmission is better than automatic transmission for MPG.
So now, we perform a linear regression.
We note the t-values are far away from 0, lending strength to our rejecting the null hypothesis
The R-squared value of .3598 tells us that only about 36% of the variance found in the response variable (mpg) can be explained by the predictor variable (transmission type).
So now, we consider multiple regression.
All five predictors account for about 86% of the variance in mpg. But, it seems that transmission and weight are the only significant predictors of mpg.
So now we consider a regression with only transmission and weight.
Using the 2 predictors account for about 75% of the variance in mpg. And transmission and weight are both significant from the p-values.
From fit 5, hp has a p-value very close to .05, so let’s try including that.
Now, using the 3 predictors account for about 84% of the variance in mpg. And transmission, weight and hp are all significant from the p-values. This seems like the best model.
Finally, we examine the residuals.
The Residuals vs Fitted Plot suggests the data does not have any obvious distinct pattern. While it is slightly curved, it has equally spread residuals around the horizontal line without a distinct pattern. This is an indication it may not be a non-linear relationship.
Residuals should be normally distributed and the Q-Q Plot shows this, since the residuals follow close to a straight line on this plot. It is a good indication they are normally distributed.
The spread location shows that we have homoscedasticity i.e., the residuals have equal variance along the regression line.
The Residuals vs. Leverage plots helps us identify influential data points on our model. Our plot doesn’t show any influential cases as all of the cases are within the the dashed Cook’s distance line.
Conclusions
Our results tell us that manual transmission is better than automatic transmission when we examine mpg’s. To quantify the MPG difference between automatic and manual, we see an increase by 7.245. However, transmission type is not the only factor accounting for MPG, horsepower, and weight are also important factors in affecting the MPG. And in our final model, manual transmission accounts for a 2.084 increase in MPG.
Tables and Plots
## 'data.frame': 32 obs. of 11 variables:
## $ mpg : num 21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
## $ cyl : num 6 6 4 6 8 6 8 4 4 6 ...
## $ disp: num 160 160 108 258 360 ...
## $ hp : num 110 110 93 110 175 105 245 62 95 123 ...
## $ drat: num 3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
## $ wt : num 2.62 2.88 2.32 3.21 3.44 ...
## $ qsec: num 16.5 17 18.6 19.4 17 ...
## $ vs : num 0 0 1 1 0 1 0 1 1 1 ...
## $ am : num 1 1 1 0 0 0 0 0 0 0 ...
## $ gear: num 4 4 4 3 3 3 3 4 4 4 ...
## $ carb: num 4 4 1 1 2 1 4 2 2 4 ...## mpg cyl disp hp drat wt
## Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0 Min. :2.760 Min. :1.513
## 1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5 1st Qu.:3.080 1st Qu.:2.581
## Median :19.20 Median :6.000 Median :196.3 Median :123.0 Median :3.695 Median :3.325
## Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7 Mean :3.597 Mean :3.217
## 3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0 3rd Qu.:3.920 3rd Qu.:3.610
## Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0 Max. :4.930 Max. :5.424
## qsec vs am gear carb
## Min. :14.50 Min. :0.0000 Min. :0.0000 Min. :3.000 Min. :1.000
## 1st Qu.:16.89 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000
## Median :17.71 Median :0.0000 Median :0.0000 Median :4.000 Median :2.000
## Mean :17.85 Mean :0.4375 Mean :0.4062 Mean :3.688 Mean :2.812
## 3rd Qu.:18.90 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :22.90 Max. :1.0000 Max. :1.0000 Max. :5.000 Max. :8.000
##
## Welch Two Sample t-test
##
## data: mtcars$mpg by mtcars$trans
## t = -3.7671, df = 18.332, p-value = 0.001374
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -11.280194 -3.209684
## sample estimates:
## mean in group Automatic mean in group Manual
## 17.14737 24.39231##
## Call:
## lm(formula = mpg ~ trans, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.3923 -3.0923 -0.2974 3.2439 9.5077
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.147 1.125 15.247 1.13e-15 ***
## transManual 7.245 1.764 4.106 0.000285 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.902 on 30 degrees of freedom
## Multiple R-squared: 0.3598, Adjusted R-squared: 0.3385
## F-statistic: 16.86 on 1 and 30 DF, p-value: 0.000285##
## Call:
## lm(formula = mpg ~ trans + cyl + disp + hp + wt, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.5952 -1.5864 -0.7157 1.2821 5.5725
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 38.20280 3.66910 10.412 9.08e-11 ***
## transManual 1.55649 1.44054 1.080 0.28984
## cyl -1.10638 0.67636 -1.636 0.11393
## disp 0.01226 0.01171 1.047 0.30472
## hp -0.02796 0.01392 -2.008 0.05510 .
## wt -3.30262 1.13364 -2.913 0.00726 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.505 on 26 degrees of freedom
## Multiple R-squared: 0.8551, Adjusted R-squared: 0.8273
## F-statistic: 30.7 on 5 and 26 DF, p-value: 4.029e-10##
## Call:
## lm(formula = mpg ~ trans + wt, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.5295 -2.3619 -0.1317 1.4025 6.8782
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 37.32155 3.05464 12.218 5.84e-13 ***
## transManual -0.02362 1.54565 -0.015 0.988
## wt -5.35281 0.78824 -6.791 1.87e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.098 on 29 degrees of freedom
## Multiple R-squared: 0.7528, Adjusted R-squared: 0.7358
## F-statistic: 44.17 on 2 and 29 DF, p-value: 1.579e-09##
## Call:
## lm(formula = mpg ~ trans + wt + hp, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4221 -1.7924 -0.3788 1.2249 5.5317
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 34.002875 2.642659 12.867 2.82e-13 ***
## transManual 2.083710 1.376420 1.514 0.141268
## wt -2.878575 0.904971 -3.181 0.003574 **
## hp -0.037479 0.009605 -3.902 0.000546 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.538 on 28 degrees of freedom
## Multiple R-squared: 0.8399, Adjusted R-squared: 0.8227
## F-statistic: 48.96 on 3 and 28 DF, p-value: 2.908e-11