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).

Repeat the preceding exercise, assuming that the linear correlation coefficient is r= 0.997.

Short answer

Thebest predicted number of burglaries, with an enrollment of 50 (thousand), is 123.3.

The predictednumber of burglaries is obtained using the regression equation by substituting the measure of x as 50.

Step-by-step solution

Given information

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

The linear correlation coefficient is\(r = 0.997\).

From Exercise 5, the regression equation is\(\hat y = 3.83 + 2.39x\).

Discuss the type of model

A regression model is good if it follows the criteria stated below:

• The scatterplot shows linear pattern.
• The correlation coefficient measure is significant.
• Extrapolation is not done for predicting the values.

A good model predicts the measure using the regression equation, while the predicted value of a bad model is the mean of the sampled response variables.

Check the type of model

The scatterplot for the samples is described below:

• Mark enrolment on the x-axis and burglaries on the y-axis.
• Scale the axes as per the observations.
• Mark the paired observations on the plot.

The resultant graph is as follows.

The correlation measure is significant and the value 50 is close to the sampled enrollment data. Thus,the model is good.

Determine the predicted value

Using the regression equation, thebest predicted number of burglaries with an enrollment of 50 (thousand) is

\(\begin{array}{c}\hat y = 3.83 + \left( {2.39 \times 50} \right)\\ = 123.3.\end{array}\)

Therefore,thebest predicted number of burglaries with an enrollment of 50 (thousand) is 123.3.

The predicted number of burglaries for the enrollment of 50 (thousand) is obtained by substituting 50 for x in the provided regression equation.