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).

Conclusion The linear correlation coefficient r is found to be 0.499, the P-value is 0.393, and the critical values for a 0.05 significance level are\( \pm 0.878\). What should you conclude?

Short answer

There is no evidence to conclude that there is a significant linear relationship between the number of enrolment and burglaries at 0.05 significance level.

Step-by-step solution

Given information

For the two variables, enrolment, and burglaries,

Correlation coefficient (r) is 0.499 with associated P-value as 0.393.

Critical values are -0.878 and 0.878 at 0.05 level of significance.

Identify the hypothesis

Here\(\rho \)represents the population correlation coefficient.

\(\begin{array}{l}{H_0}:\rho = 0\\{H_1}:\rho \ne 0\end{array}\)

State the decision rule

Comparing the P-value and significance level,

If\(P - value < \alpha \), reject the null hypothesis.

Thus, it can be concluded that there is a significant linear relationship between the two variables.

If\(P - value > \alpha \), fail to reject the null hypothesis.

Thus, it can be concluded that there is nosignificant linear relationship between the two variables.

Asimilar result can be concluded using critical values.

• If the correlation coefficient value lies between critical values, the result for correlation coefficient is insignificant.
• If the correlation coefficient value does lie between critical values, the result for correlation coefficient is significant.

Step 4:State the conclusion for the correlation coefficient

Here\(P - value = 0.393\)and\(\alpha = 0.05\).

As\(P - value > 0.05\), wefail to reject the null hypothesis.

In a similar way, 0.499 lies between -0.878 and 0.878, which leads to the rejection of null hypothesis.

Thus, it can be concluded that there is not enough evidence fora significant linear correlation between enrolment and burglaries.