Question

The following exercises are based on the following sample data consisting of numbers of enrolled students (in thousands) and numbers of burglaries for randomly selected large colleges in a recent year (based on data from the New York Times).

The sample data result in a linear correlation coefficient of r= 0.499 and the regression equation\(\hat y = 3.83 + 2.39x\). What is the best predicted number of burglaries, given an enrollment of 50 (thousand), and how was it found?

Short answer

Thebest-predicted number of burglaries when the enrollment is 50 (thousand) is 92.6.

The predicted number of burglaries is computed as the mean sampled observation for all burglaries.

Step-by-step solution

Given information

The table represents the number of enrolled students (in thousands) and the number of burglaries for randomly selected large colleges in recent years.

Linear correlation coefficient,\(r = 0.499\)

The regression equation is

\(\hat y = 3.83 + 2.39x\).

Discuss the type of model

A model is categorized as good or bad based on the following criteria:

• The fit of the observations is approximately linear.
• The measure of the correlation coefficient is significant.
• The observation at which the response is predicted is not extreme.

A bad model determines the prediction as the average of sample observations of response measures.

Compute the best-predicted number of burglaries

Here, the correlation coefficient is small which implies weak linear association; Thus, it is not significant.

Therefore,it is a bad model and the regression equation must not be used topredict the number of burglaries, given an enrollment of 50 (thousand).

Thus, the best-predicted number of burglaries, with an enrollment of 50 (thousand) is obtained by computing the mean of sampled burglaries.

Let y represent the number of burglaries.

The mean value is obtained as the average of five measures.

\(\begin{array}{c}\bar y = \frac{{\sum\limits_{i = 1}^n {{y_i}} }}{n}\\ = \frac{{86 + 57 + 32 + 131 + 157}}{5}\\ = 92.6\end{array}\)

Therefore, the best-predicted number of burglaries, with an enrollment of 50 (thousand), is 92.6.