Question

Questions AP4.38 and AP4.39 refer to the following situation. Do children’s fear levels change over time and, if so, in what ways? Little research has been done on the prevalence and persistence of fears in children. Several years ago, two researchers surveyed a randomly selected group of 94 third- and fourth-grade children, asking them to rate their level of fearfulness about a variety of situations. Two years later, the children again completed the same survey. The researchers computed the overall fear rating for each child in both years and were interested in the relationship between these ratings. They then assumed that the true regression line was μlater rating=β0+β1(initial rating) and that the assumptions for regression inference were satisfied. This model was fitted to the data using least-squares regression. The following results were obtained from statistical software.

Here is a scatterplot of the later ratings versus the initial ratings and a plot of the residuals versus the initial ratings:

AP4.39 George’s initial fear rating was 0.2 higher than Jonny’s. What does the model predict about their later fear ratings?

a. George’s will be about 0.96 higher than Jonny’s.

b. George’s will be about 0.40 higher than Jonny’s.

c. George’s will be about 0.20 higher than Jonny’s

d. George’s will be about 0.08 higher than Jonny’s.

e. George’s will be about the same as Jonny’s.

Short answer

The correct answer is option (d) George’s will be about $0.08$ higher than Jonny’s.

Step-by-step solution

Given information

To determine that the model predict about their later fear ratings.

Explanation

The researchers are interested in learning if and how the children's terror levels vary over time.
In the question, there is a scatterplot.
As a result, the slope of linear regression is $0.397911$.
The slope of linear regression multiplied by the initial rating difference as:
$0.2\times 0.397911$
$=0.0796$
$\approx 0.08$
As a result, option (d) is the correct answer.