Fuente:DataSource.ai
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Executive Summary
Motor Trend is a magazine about the automobile industry. It is interested in exploring the relationship between a set of variables and miles per gallon (MPG) (outcome), particularly:
“Is an automatic or manual transmission better for MPG” “Quantify the MPG difference between automatic and manual transmissions”
Using a data set from Motor Trend Magazine along with linear regression and hypothesis testing, it can be concluded that there is a significant difference between the MPG of automatic and manual transmission cars.
To quantify the MPG difference between automatic and manual transmission cars, a linear regression model that took into account the weight, the type of transmission and the acceleration (qsec) was used. Controlling for these factors, manual transmission cars have a better fuel efficiency of 2.94 MPG more than automatic transmission cars.
You 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:
“Is an automatic or manual transmission better for MPG” “Quantify the MPG difference between automatic and manual transmissions”
Data Processing
The data was extracted from the 1974 Motor Trend US magazine, and comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973–74 models).
Coefficents
From this simple linear regression model of mpg against am, manual transmission cars have 7.24 MPG more than automatic transmission cars. The R^2 value of this model is 0.3598, meaning that it only explains 35.98% of the variance.
[1] 32 11summary(mtcars)
mpg cyl disp hp
Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0
1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5
Median :19.20 Median :6.000 Median :196.3 Median :123.0
Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7
3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0
Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0
drat wt qsec vs
Min. :2.760 Min. :1.513 Min. :14.50 Min. :0.0000
1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 1st Qu.:0.0000
Median :3.695 Median :3.325 Median :17.71 Median :0.0000
Mean :3.597 Mean :3.217 Mean :17.85 Mean :0.4375
3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90 3rd Qu.:1.0000
Max. :4.930 Max. :5.424 Max. :22.90 Max. :1.0000
am gear carb
Min. :0.0000 Min. :3.000 Min. :1.000
1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000
Median :0.0000 Median :4.000 Median :2.000
Mean :0.4062 Mean :3.688 Mean :2.812
3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000
Max. :1.0000 Max. :5.000 Max. :8.000 Is an automatic or manual transmission better for MPG? For automatic:
summary(mtcars[mtcars$am==0,])
mpg cyl disp hp
Min. :10.40 Min. :4.000 Min. :120.1 Min. : 62.0
1st Qu.:14.95 1st Qu.:6.000 1st Qu.:196.3 1st Qu.:116.5
Median :17.30 Median :8.000 Median :275.8 Median :175.0
Mean :17.15 Mean :6.947 Mean :290.4 Mean :160.3
3rd Qu.:19.20 3rd Qu.:8.000 3rd Qu.:360.0 3rd Qu.:192.5
Max. :24.40 Max. :8.000 Max. :472.0 Max. :245.0
drat wt qsec vs
Min. :2.760 Min. :2.465 Min. :15.41 Min. :0.0000
1st Qu.:3.070 1st Qu.:3.438 1st Qu.:17.18 1st Qu.:0.0000
Median :3.150 Median :3.520 Median :17.82 Median :0.0000
Mean :3.286 Mean :3.769 Mean :18.18 Mean :0.3684
3rd Qu.:3.695 3rd Qu.:3.842 3rd Qu.:19.17 3rd Qu.:1.0000
Max. :3.920 Max. :5.424 Max. :22.90 Max. :1.0000
am gear carb
Min. :0 Min. :3.000 Min. :1.000
1st Qu.:0 1st Qu.:3.000 1st Qu.:2.000
Median :0 Median :3.000 Median :3.000
Mean :0 Mean :3.211 Mean :2.737
3rd Qu.:0 3rd Qu.:3.000 3rd Qu.:4.000
Max. :0 Max. :4.000 Max. :4.000 summary(mtcars[mtcars$am==1,])
mpg cyl disp hp
Min. :15.00 Min. :4.000 Min. : 71.1 Min. : 52.0
1st Qu.:21.00 1st Qu.:4.000 1st Qu.: 79.0 1st Qu.: 66.0
Median :22.80 Median :4.000 Median :120.3 Median :109.0
Mean :24.39 Mean :5.077 Mean :143.5 Mean :126.8
3rd Qu.:30.40 3rd Qu.:6.000 3rd Qu.:160.0 3rd Qu.:113.0
Max. :33.90 Max. :8.000 Max. :351.0 Max. :335.0
drat wt qsec vs
Min. :3.54 Min. :1.513 Min. :14.50 Min. :0.0000
1st Qu.:3.85 1st Qu.:1.935 1st Qu.:16.46 1st Qu.:0.0000
Median :4.08 Median :2.320 Median :17.02 Median :1.0000
Mean :4.05 Mean :2.411 Mean :17.36 Mean :0.5385
3rd Qu.:4.22 3rd Qu.:2.780 3rd Qu.:18.61 3rd Qu.:1.0000
Max. :4.93 Max. :3.570 Max. :19.90 Max. :1.0000
am gear carb
Min. :1 Min. :4.000 Min. :1.000
1st Qu.:1 1st Qu.:4.000 1st Qu.:1.000
Median :1 Median :4.000 Median :2.000
Mean :1 Mean :4.385 Mean :2.923
3rd Qu.:1 3rd Qu.:5.000 3rd Qu.:4.000
Max. :1 Max. :5.000 Max. :8.000 Hence, the mean of mpg is greater for manual at 24.4 than automatic at 17.1
Quantify the MPG difference between automatic and manual transmissions.boxplot(mpg ~ am, data = mtcars, xlab = "Transmission", ylab = "Miles per gallon", main="Miles per gallon by Transmission Type")
Manual (represented by 1) has a higher mean for mpg than automatic (represented by 0).
Hypothesis Testing
aggregate(mpg~am, data = mtcars, mean)
am mpg
1 0 17.14737
2 1 24.39231auto <- mtcars[mtcars$am == 0,]
manual <- mtcars[mtcars$am == 1,]
t.test(auto$mpg, manual$mpg)
Welch Two Sample t-test
data: auto$mpg and manual$mpg
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 of x mean of y
17.14737 24.39231 Since p-value = 0.001374, we reject the null hypothesis. There is a major difference between mpg of manual and automatic transmissions.
In other words, the p-value of the t-test is 0.001374, which falls within the 95% confidence interval. Hence, controlling for all other variables, there is a significant difference between the mean MPG of automatic and manual cars.
Call:
lm(formula = mpg ~ am, 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 ***
am 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: 0.000285From the above, we may conclude that automatic run at 17.15mpg, while manual have 7.24 more mpg.
Also, R^2 is 0.36, hence the model only accounts for 36% variance.
In other words, from this simple linear regression model of mpg against am, manual transmission cars have 7.24 MPG more than automatic transmission cars. The R^2 value of this model is 0.3598, meaning that it only explains 35.98% of the variance.
Performing multivariate linear regression:
Analysis of Variance Table
Model 1: mpg ~ am
Model 2: mpg ~ am + wt + hp + cyl
Res.Df RSS Df Sum of Sq F Pr(>F)
1 30 720.9
2 27 170.0 3 550.9 29.166 1.274e-08 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(model)
Call:
lm(formula = mpg ~ am + wt + hp + cyl, data = mtcars)
Residuals:
Min 1Q Median 3Q Max
-3.4765 -1.8471 -0.5544 1.2758 5.6608
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 36.14654 3.10478 11.642 4.94e-12 ***
am 1.47805 1.44115 1.026 0.3142
wt -2.60648 0.91984 -2.834 0.0086 **
hp -0.02495 0.01365 -1.828 0.0786 .
cyl -0.74516 0.58279 -1.279 0.2119
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.509 on 27 degrees of freedom
Multiple R-squared: 0.849, Adjusted R-squared: 0.8267
F-statistic: 37.96 on 4 and 27 DF, p-value: 1.025e-10The only significant coefficient of the model is wt with a p value of 0.0086 ** and an impact of -2.60648 of the coefficient wt on the dependent variable, although the intercept is significant. This is not interpretable for this model. So the model can be reformulated with a simple regression with mpg = y and wt = x lm(formula = mpg ~ wt -1, data = mtcars)
Conclusion This model explains 84.9% of the variance. It may be concluded that on average, manual transmissions have 1.478 more mpg than automatic.
In conclusion, holding the weight and acceleration (qsec) of the car constant, manual transmission cars offer 2.94 MPG better fuel efficiency.
APPENDIX Model Residualsplot(model)
