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.

You can see there is a positive relationship between weight and price. Smaller cars are cheaper while the heavier cars are more expensive.

## '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 (the coefficient’s absolute value > 0.6) positive (its sign) association between weight and height.

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

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

Yes, our statistics show that the coefficient of weight is significant at 5%. Changes in the weight is statisitically significant to the price of the vehicle.

Q3. What price does the model predict for a car that weighs 4000 pounds?

For a car that weighs in at 4,000 pounds, I predict that the car will be priced around $43,000 dollars. This was found by using this equation:

W = 2171 + (43*P); 4,000 = 2171 + 43P; 4,000/2171 = 43P; 1829/43 = 43P; 42.53 = P

Hint: Check the units of the variables in the openintro manual.

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

The reported residual standard error dound in our stats is 433 pounds on 52 degress of freedom. This means that it is the typical difference between the actual weight and the weight being predicted.

Q5. What is the reported adjusted R squared? What does it mean?

The reported Adjusted R-squared is 0.7209 in our statistics. The R^2 of 0.5136 means that 51.36% of the variability in weight can be explained by height.

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.

Model 2 better fits the data in our stats. This is because model 2 has a smaller residual standard error at 347 while the first model is at 433.

Build regression model

## 
## Call:
## lm(formula = weight ~ price, data = cars)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1328.29  -228.09    10.92   258.19   924.27 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 2171.113    118.956  18.251  < 2e-16 ***
## price         43.331      5.169   8.383 3.17e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 433 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 = weight ~ price + passengers, data = cars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -976.81 -201.56    6.13  151.33  799.88 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   294.25     356.98   0.824    0.414    
## price          35.99       4.36   8.256 5.80e-11 ***
## passengers    395.91      72.56   5.456 1.44e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 347.4 on 51 degrees of freedom
## Multiple R-squared:  0.7315, Adjusted R-squared:  0.7209 
## F-statistic: 69.46 on 2 and 51 DF,  p-value: 2.748e-15

Interpretation

• 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.