Question

1. the table and scatterplot below show the relationship between student enrollment (in thousands) and total number of campus property crimes (burglary and theft) in one year for eight colleges in a certain u.s. state. the equation of the least - squares regression line is \\( \hat { y } = - 112.58 + 21.83 x \\), where \\( y = \\) number of property crimes and \\( x = \\) student enrollment in thousands. a) the standard deviation of the residuals for this model is \\( s = 89.18 \\). interpret this value. b) the \\( r ^ { 2 } \\) value for this model is 0.80. interpret \\( r ^ { 2 } \\) value.

Explanation:

a) The standard deviation of the residuals ($s = 89.18$) represents the typical distance that the observed number of property crimes falls from the number of property crimes predicted by the least - squares regression line. In other words, on average, the actual number of campus property crimes (burglary and theft) for these colleges differs from the number predicted by the regression model ($\hat{y}=-112.58 + 21.83x$) by about 89.18.
b) The coefficient of determination ($r^{2}=0.80$) means that 80% of the variation in the number of campus property crimes (burglary and theft) can be explained by the linear relationship with student enrollment (in thousands) as described by the least - squares regression line $\hat{y}=-112.58 + 21.83x$. The remaining 20% of the variation is due to other factors not accounted for in this simple linear regression model.

Answer:

a) On average, the actual number of campus property crimes differs from the predicted number by about 89.18.
b) 80% of the variation in the number of campus property crimes is explained by the linear relationship with student enrollment.