Question

A random sample of 30 pairs of observations from a normal population gives a correlation coefficient 0.45. is it likely that the variables in the population are uncorrelated?

Answer 1

$n=30 \\ r=0.45$

Test-statistics to check that it is likely that the variables in the population are uncorrelated.

$t= \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \\ = \frac{0.45 \sqrt{30-2}}{\sqrt{1-0.45^2}} \\ = \frac{2.381}{0.893} \\ = 2.666$

The value of the test statistic is 2.666.

Degrees of freedom = (n - 2) = (30 - 2) = 28

Since, our test is two-tailed test, therefore we shall obtain two-tailed p-value for the test statistic.

The EXCEL formula to find the p-value for a two-tailed t-test and df=28 is

=tdist(2.666, 28, 2)

p-value = 0.0126

p<0.05

At significance level of 5%, we have sufficient evidence to conclude that, it is not likely that variables in the population are uncorrelated.