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:

AP4.38 Which of the following statements is supported by these plots?
a. The abundance of outliers and influential observations in the plots means that the assumptions for regression are clearly violated.
b. These plots contain dramatic evidence that the standard deviation of the response about the true regression line is not approximately the same for each x -value.
c. These plots call into question the validity of the assumption that the later ratings vary Normally about the least-squares line for each value of the initial ratings.
d. A linear model isn’t appropriate here because the residual plot shows no association.
e. There is no striking evidence that the assumptions for regression inference are violated

Short answer

The correct answer is option (a) The abundance of outliers and influential observations in the plots means that the assumptions for regression are clearly violated.

Step-by-step solution

Given information

To determine which of the statements is supported by these plots.

Explanation

The distribution between the actual and fitted response values is depicted in a residual plot. A random dispersion of points forms an approximately constant width band around the identity line in the ideal residual plot, also known as the null residual plot.
The researchers are interested in learning if and how the children's terror levels vary over time. In the question, there is a scatterplot.
There is no evident pattern in the residual plot, and neither the scatterplot nor the residual plot appear to have any outliers.
As a result, answer (a) is accurate, because the regression inference assumptions are not violated.