Chapter 6.1, Problem 36E

(a)

To determine

Equation of the least squares regression line relating posttest and pretest scores.

Answer to Problem 36E

Equation of the least squares regression line,

  $\widehat{y}=17.897+0.78301x$

Explanation of Solution

Given information:

For the least − squares regression line,

General equation:

  $\widehat{y}=b_0+b_{1}x$

From the computer output,

The estimate of the slope b₁ is provided in the column “Coef” and in the row “Pretest”:

  $b_1=0.78301$

Also,

The estimate of the slope b₀ is provided in the column “Coef” and in the row “Constant”:

  $b_0=17.897$

Thus,

In the general equation of the least − squares regression line,

Replace b₁ by 0.78301 and b₀ by 17.897:

  $\widehat{y}=b_0+b_{1}x=17.897+0.78301x$

Where,

x represents the Pretest score.

y represents the Posttest score.

(b)

To determine

Whether a linear model is appropriate for describing this relationship.

Answer to Problem 36E

Yes, a linear model is appropriate for describing this relationship.

Explanation of Solution

Given information:

In this case,

There is no strong curvature present in the scatterplot.

Also,

No strong curvature present in the residual plot.

Moreover,

The points in the residual plot appear to be randomly scattered.

Also,

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)

To determine

Typical predictions while using the least − square regression line to predict students’ posttest scores from their pretest scores.

Answer to Problem 36E

The predicted posttest score deviates on an average by 12.55 points from the actual posttest score.

Explanation of Solution

Given information:

In the computer output,

The standard error of the estimate s is given after “S=”.

Such that

  $s=12.55$

The standard deviation of the residual or the standard error of the estimate ( s ) represents the average error of predictions.

Thus,

This shows average deviation between actual y − values and the predicted y − values.

Thereby,

Using the equation of the least − squares regression line,

The predicted posttest score deviates on an average by 12.55 points from the actual posttest score.