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)

Which of the following change if the two variables of enrollment and burglaries are switched: the value of r= 0.499, the P-value of 0.393, the critical values of$\pm$0.878?

Short answer

No value will change if the variables are switched.

Step-by-step solution

Given information

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

The value of$r\mathrm{i}\mathrm{s}0.499$.

The P-value is 0.393.

The critical values are $\pm$ 0.878.

Discuss each of the measures

Theformula of t-statistic for the correlation coefficient is stated as follows:

$t=\frac{r}{\sqrt{\frac{1--r^2}{n--2}}}$

Where

$r=\frac{n\left(\sum xy\right)--\left(\sum x\right)\left(\sum y\right)}{\sqrt{\left(\left(n\sum x^2\right)--{\left(\sum x\right)}^2\right)\left(\left(n\sum y^2\right)--{\left(\sum y\right)}^2\right)}}$

As per the formula of the correlation coefficient, the measure will be unaffected if the role of variables is changed.

Resultant of this, the test statistic measure and hence the P-value will be unaltered.

The critical values are obtained from the t-distribution table at n–2 degrees of freedom, which are unaffected by switching the variables.

Thus, if the variable enrolment and burglaries are switched, there will be no change in any of the given values.

Therefore, r, P-values, and critical values won’t change if the variables are switched.