Question

Answer the following question by showing the codes in R

2. Consider the dataset mtcars and suppose we are interested in modeling the mpg of a vehicle based on a single variable presented in the dataset. a) Use the cor ) function in R, apply it to only numerical variables in the dataset. Identify the numerical variable that shows the most significant correlation, and generate a scatterplot between this variable and mpg. b) Use the 1m() function in R to build a simple regression model using the formula mpg x, where x is the variable you identified from a). Interpret the coefficient generated by the model by calling summary on the model object. Plot the model residuals on a histogram, what do you notice?

Answer 1

using the cor(t(mtcars)) the following is the correlation table among the variables and found that

since there are N=32 observations( number of cars)

so the significant correlation coefficient=0.349 at two tailed alpha=0.05 with n-2=32-2=30. therefore all the variables significantly correlated to mpg, as the all the absolute correlation coefficient is more than 0.349 ( please look critical r on standard book or internet). But the wt is most significant correlated(negative).

the scatter plot mpg(x1) and wt( x6) is given as

	mpg	cyl	disp	hp	drat	wt	qsec	vs	am	gear	carb
mpg	1.0000	-0.8522	-0.8476	-0.7762	0.6812	-0.8677	0.4187	0.6640	0.5998	0.4803	-0.5509
cyl	-0.8522	1.0000	0.9020	0.8324	-0.6999	0.7825	-0.5912	-0.8108	-0.5226	-0.4927	0.5270
disp	-0.8476	0.9020	1.0000	0.7909	-0.7102	0.8880	-0.4337	-0.7104	-0.5912	-0.5556	0.3950
hp	-0.7762	0.8324	0.7909	1.0000	-0.4488	0.6587	-0.7082	-0.7231	-0.2432	-0.1257	0.7498
drat	0.6812	-0.6999	-0.7102	-0.4488	1.0000	-0.7124	0.0912	0.4403	0.7127	0.6996	-0.0908
wt	-0.8677	0.7825	0.8880	0.6587	-0.7124	1.0000	-0.1747	-0.5549	-0.6925	-0.5833	0.4276
qsec	0.4187	-0.5912	-0.4337	-0.7082	0.0912	-0.1747	1.0000	0.7445	-0.2299	-0.2127	-0.6562
vs	0.6640	-0.8108	-0.7104	-0.7231	0.4403	-0.5549	0.7445	1.0000	0.1683	0.2060	-0.5696
am	0.5998	-0.5226	-0.5912	-0.2432	0.7127	-0.6925	-0.2299	0.1683	1.0000	0.7941	0.0575
gear	0.4803	-0.4927	-0.5556	-0.1257	0.6996	-0.5833	-0.2127	0.2060	0.7941	1.0000	0.2741
carb	-0.5509	0.5270	0.3950	0.7498	-0.0908	0.4276	-0.6562	-0.5696	0.0575	0.2741	1.0000

(b) the regression analysis shows the independent variable wt is significantly explaining the dependent variable mpg as the p-value is less than the typical value of alpha=0.05

following are the R-ouput

> lm(mpg~wt)

Call:
lm(formula = mpg ~ wt)

Coefficients:
(Intercept) wt  
37.285 -5.344  

> summary(lm(mpg~wt))

Call:
lm(formula = mpg ~ wt)

Residuals:
Min 1Q Median 3Q Max
-4.5432 -2.3647 -0.1252 1.4096 6.8727

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 37.2851 1.8776 19.858 < 2e-16 ***
wt -5.3445 0.5591 -9.559 1.29e-10 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.046 on 30 degrees of freedom
Multiple R-squared: 0.7528, Adjusted R-squared: 0.7446
F-statistic: 91.38 on 1 and 30 DF, p-value: 1.294e-10