Quiz on Correlation and Regression

Suppose that you want to build a regression model that predicts the price of cars using a data set named cars.

Q1. Per the scatter plot and the computed correlation coefficient, describe relationships between the two variables - price and weight.

Make sure to interpret the direction and the magnitude of the relationship. In addition, keep in mind that correlation (or regression) coefficients do not show causation but only association.

Create scatterplots

## 'data.frame':    54 obs. of  6 variables:
##  $ type      : Factor w/ 3 levels "large","midsize",..: 3 2 2 2 2 1 1 2 1 2 ...
##  $ price     : num  15.9 33.9 37.7 30 15.7 20.8 23.7 26.3 34.7 40.1 ...
##  $ mpgCity   : int  25 18 19 22 22 19 16 19 16 16 ...
##  $ driveTrain: Factor w/ 3 levels "4WD","front",..: 2 2 2 3 2 2 3 2 2 2 ...
##  $ passengers: int  5 5 6 4 6 6 6 5 6 5 ...
##  $ weight    : int  2705 3560 3405 3640 2880 3470 4105 3495 3620 3935 ...

## [1] 0.758112

Interpretation

There is a strong and positive correlation between the price and the weight. The more the car weighs the more expensive the car is.

Run a regression model for price with one explanatory variable, weight, and answer Q2 through Q5.

## 
## Call:
## lm(formula = price ~ weight, data = cars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -12.767  -3.766  -1.155   2.568  35.440 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -20.295205   4.915159  -4.129 0.000132 ***
## weight        0.013264   0.001582   8.383 3.17e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.575 on 52 degrees of freedom
## Multiple R-squared:  0.5747, Adjusted R-squared:  0.5666 
## F-statistic: 70.28 on 1 and 52 DF,  p-value: 3.173e-11

Q2. Is the coefficient of weight statistically significant at 5%? Interpret the coefficient.

The coefficient is not signignificant at 5% and is not meaningful in explaining changes in weight. ## Q3. What price does the model predict for a car that weighs 4000 pounds? Hint: Check the units of the variables in the openintro manual. If a car weighs 4,000 pounds the price model predicts that it should cost about $48,000.

Q4. What is the reported residual standard error? What does it mean?

The reported residual standard error is 7.575lbs which is the difference between the actual weight and the weight predicted. ## Q5. What is the reported adjusted R squared? What does it mean? The adjusted R squared is 0.5666, meainging that 56% of the variability in the price of the car can be explained by its weight.

Run a second regression model for price with two explanatory variables: weight and passengers, and answer Q6.

Q6. Which of the two models better fits the data? Discuss your answer by comparing the residual standard error and the adjusted R squared between the two models.

Build regression model

## 
## Call:
## lm(formula = price ~ weight, data = cars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -12.767  -3.766  -1.155   2.568  35.440 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -20.295205   4.915159  -4.129 0.000132 ***
## weight        0.013264   0.001582   8.383 3.17e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.575 on 52 degrees of freedom
## Multiple R-squared:  0.5747, Adjusted R-squared:  0.5666 
## F-statistic: 70.28 on 1 and 52 DF,  p-value: 3.173e-11
## 
## Call:
## lm(formula = price ~ passengers, data = cars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -11.422  -7.522  -4.189   6.244  42.478 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   -6.244     11.322  -0.552   0.5836  
## passengers     5.133      2.195   2.338   0.0233 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 11.05 on 52 degrees of freedom
## Multiple R-squared:  0.09513,    Adjusted R-squared:  0.07773 
## F-statistic: 5.467 on 1 and 52 DF,  p-value: 0.02326

Interpretation

Linear model 1 fits the data better becuase when comparing the residual standard error and the adjusted R squared are both lower in this model.

• significance of coefficients *** at the end of the coefficient of height indicates that the coefficient is significant at 0.1% signficance level (low p-values). In other words, changes in the height are highly likely meaningful in explaining changes in the weight. The same can be said for the y-intercept.
• coefficient of height An one-centimeter increase in the height of a person is associated with an increase of 1.018 kg in the weight.
• intercept When a person is 0 centimeter tall, his/her weight is -105.011 kg. Obviously, the intercept is meaningless in this case.
• residual standard error The typical difference between the actual weight and the weight predicted by the model is about 9.3 kg. In other words, the model estimated weight misses the actual weight by about 9.3 kg.
• Adjusted R-squared The R^2 of 0.5136 means that 51.36% of the variability in weight can be explained by height.
• Making predictions Ben is predicted to be 81 Kg based on his his height while his actual weight is 74.8 kg. The model overestimated Ben’s weight 6.2 kg.