Question

A study was conducted to determine if larger universities tend to have more property crime. Let x represent student enrollment in thousands and let y^ represent the number of burlaries in a year on campus. ^y=1.326x-4.12. Consider university with 13000 students. a) calculate what would be the predicted number of burglries for this univeristy? b) if in fact 4 burglaries happened in this univeristy, what is residual? Is it overestimated or not? c) What is the vlaue of slope? d) if R2= 58.5% what does this mean. Calculate the correlation r and explain.

Answer 1

$$\hat{y}=1.326x-4.12$$

a) Given that $x=13$

$$\hat{y}(13)=1.326(13)-4.12=13.118$$

b) The residual is

$$e_i=y_i-\hat{y_i}$$

$$e=4-13.118=-9.118$$

The residual is negative. The model overestimates the observation (the predicted value is too big).

c)

$$slope=1.326>0$$

If the student enrollment x increases by one thousand, then the number of burlaries for this university increases by 1.326

d) R-squared (coefficient of determination) is the percentage of the dependent variable variation that a linear model explains.

$$R^2=0.585$$

This means that the model explains 58.5 % of the variation of the number of burlaries.

The correlation coefficient is

$$R=\sqrt{0.585}\approx0.7649>0.7$$

Strong positive correlation (linear relationship).