Elementary Statistics – Regression Examples

 

Example 1

Data have been collected on 93 cars of various makes and models from the year 1993. You are interested in predicting the highway miles per gallon of a car using the weight (in pounds) of a car. After looking at the scatterplot, it seems reasonable to fit a least-squares regression line. The equation of the least-squares line (via computer) is:  Summary statistics for the two variables are also included below.

 

 

Variable          N      Mean     StDev 

Highway MPG      93    29.086     5.332  

Weight (in lbs)  93   3072.90    589.90  

 

Correlation Coefficient of Highway MPG and Weight = -0.811

 

  1. Using the given summary statistics, verify the equation of the regression line.

 

 

  1. A regression model can be used to explain the response variable.
    1. In words, carefully interpret the value of the slope of the regression line.

 

 

 

    1. In words, carefully interpret the value of the y-intercept of the regression line.

 

 

 

  1. A regression model can also be used to predict the response variable.
    1. What is the predicted MPG for a car that weighs 3,000 pounds?

 

 

 

    1. What MPG would you predict for a 5,000-pound car?

 

 

 

  1. It’s also important to check whether your regression model is any good (using regression diagnostics).
    1. Determine the value of  for this regression, and, in words, carefully interpret the value.

 

 

    1. One of the cars has a weight of 2,530 pounds and gets 30 MPG. What is the residual associated with this car?

      2.
    2. The residual plot from this regression is shown below. What does this plot tell you about the adequacy of the regression model?

 

Example 2

For a hypothetical class, the exams scores and study time (in hours) are shown in the scatterplot below. Because the relationship seems linear, a regression line is fit (predicted exam score = 56.03 + 4.58studytime). The residual plot from the regression is also shown below. Does the residual plot show a random scatter of points? A pattern? What does this tell you about the regression model?

 

 

 

Example 3

For a sample of girls, the age and average height are recorded. These variables are shown in the scatterplot below. The linear relationship is very strong, with a correlation coefficient of 0.994. A regression line is fit to the data (predicted height = 27.62 + 2.58age). The residual plot from the regression is also shown below. What does the residual plot tell you about the regression model?