3.4.3.2 - Example: Interpreting Output

This example uses the "CAOSExam" dataset available from http://www.lock5stat.com/datapage.html.

CAOS stands for Comprehensive Assessment of Outcomes in a First Statistics course. It is a measure of students' statistical reasoning skills. Here we have data from 10 students who took the CAOS at the beginning (pre-test) and end (post-test) of a statistics course. 

Research question: How can we use students' pre-test scores to predict their post-test scores?

Minitab Express was used to construct a simple linear regression model. The two pieces of output that we are going to interpret here are the regression equation and the scatterplot containing the regression line.

Regression output from Minitab Express: Posttest = 21.43 + 0.8394 Pretest

Minitab Express output. Fitted line plot for linear model. This is a scatterplot with the regression line drawn on it.

Let's work through a few common questions.

 

 

What is the regression model?

The "regression model" refers to the regression equation. This is \(\widehat {posttest}=21.43 + 0.8394(Pretest)\)

 

Identify and interpret the slope.

The slope is 0.8694. For every one point increase in a student's pre-test score, their predicted post-test score increases by 0.8694 points. 

 

Identify and interpret the y-intercept. 

The y-intercept is 21.43. A student with a pre-test score of 0 would have a predicted post-test score of 21.43.  However, in this scenario, we should not actually use this model to predict the post-test score of someone who scored 0 on the pre-test because that would be extrapolation. This model should only be used to predict the post-test score of students from a comparable population whose pre-test scores were between approximately 35 and 65.

 

One student scored 60 on the pre-test and 65 on the post-test. Calculate and interpret that student's residual. 

This student's observed x value was 60 and their observed y value was 65. 

\(e=y- \widehat y\)

We have y.  We can compute \(\widehat y\) using the x value and regression equation that we have.

\(\widehat y = 21.43 + 0.8394(60) = 71.794\)

\(e=65-71.794=-6.794\)

This student's residual is -6.794. They scored 6.794 points lower on the post-test than we predicted given their pre-test score.