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?

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

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

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.

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.