Graduation Rates. Refer to Problem 11. At the 2.5 % significance level, do the data provide sufficient evidence to conclude that the variables student-to-faculty ratio and graduation rate are positively linearly correlated?

The test hypotheses are as follows:

Null hypothesis:

$H_0:\rho =0$

That is, the data does not provide sufficient evidence to conclude that the variables student-tofaculty ratio and graduation rate are positively linearly correlated.

Alternative hypothesis:

$H_a:\rho >0$

That is, the data provide sufficient evidence to conclude that the variables student-to-faculty ratio and graduation rate are positively linearly correlated.

Obtain the correlation and $p$-value between x and y by using MINITAB.

MINITAB procedure:

Step 1: Select Stat >Basic Statistics > Correlation.

Step 2: In Variables, select GRADRATE and SF RATIOfrom the box on the left.

Step 3: Click OK.

MINITAB output:

From the above MINITAB output, the value of correlation between BIRDIES and SCORE is $0.511$and the $p$-value for left-tail test is $0.0655\left(=\frac{0.131}{2}\right)$

Use the significance level, $\alpha =0.025$

Here, $p$-value is greater than the level of significance.

That is, $p$-value $(=0.0655)>\alpha (=0.025)$.

Therefore, by the rejection rule, it can be concluded that there is no evidence to reject the null hypothesis $\left(H_0\right)$at $\alpha =0.025$

That is, the data does not provide sufficient evidence to conclude that the variables student-tofaculty ratio and graduation rate are positively linearly correlated.