Question

Exercises 35 and 36 refer to the following setting. Many chess masters and chess advocates

believe that chess play develops general intelligence, analytical skill, and the ability to

concentrate. According to such beliefs, improved reading skills should result from study to

improve chess-playing skills. To investigate this belief, researchers conducted a study. All the

subjects in the study participated in a comprehensive chess program, and their reading

performances were measured before and after the program. The graphs and numerical

summaries that follow provide information on the subjects’ pretest scores, posttest scores, and

the difference (Post − Pre) between these two scores.

Predicting posttest scores (3.2)

a. What is the equation of the least-squares regression line relating posttest and pretest

scores? Define any variables used.

b. Is a linear model appropriate for describing this relationship? Justify your answer.

c. If we use the least-squares regression line to predict students’ posttest scores from their

pretest scores, how far off will our predictions typically be?

Short answer

a) $y=17.897+0.78301x$where x represents the pretest score and y represents the posttest score.

b) Yes, a linear model appropriate for describing this relationship.

c) The predicted posttest score deviates on average by$12.55$ points from the actual posttest score.

Step-by-step solution

Step 1. Given information  

We have given data.

a) Step 1. To find the equation of the least-squares regression line.

General equation of the least squares regression line is

$y=b_0+b_{1}x$

Eliminate,role="math" localid="1654232338347" $b_0=17.897and{b}_1=0.78301$

Thus,

$y=17.897+0.78301x$

b) Step 1  Justify a linear model is appropriate for describing this relationship.

A linear model is appropriate for describing the relationship in this case, because there is no strong curvature present in the scatterplot and there is no strong curvature present in the residual plot. Moreover, the points in the residual plot appear to be randomly scattered and there appear to be no strong outliers (points that deviate strongly from the pattern in the other points), thus the model does indeed appear to be appropriate.

c) Step 1. To find how far off posttest scores from their pretest scores,

The standard error of the estimate s is given after$"S="$in the computer output:$S=12.55$

The standard error of the estimate s represents the average error of predictions, thus the average deviation between actual y-values and the predicted y-values.