Question

A researcher took a sample of 25 observations and calculated the correlation coefficient for X and Y and found r = 0.68. Test at the 5% level the null hypothesis that there is no correlation between X and Y.

Answer 1

Solution:

The null and alternative hypotheses are as follows:

i.e. The population correlation coefficient between X and Y is 0.

  i.e. The population correlation coefficient between X and Y is not 0.

The obtained value of sample correlation coefficient is r_{​​​​​} = 0.68 and sample size n = 25.

We shall compare the obtained value of r_{​​​​​} with the two-tailed critical value of correlation coefficient r at 5% significance level and (n-2) degrees of freedom. If the obtained value of r_{​​​​​} is greater than the critical value of r then, we reject the null hypothesis. If the obtained value of r_{​​​​​} is less than the critical value of r then, we fail to reject the null hypothesis.

The two-tailed critical value of r at 5% = 0.05 significance level and (n-2) = (25-2) = 23 degrees of freedom is 0.396.

(0.68 > 0.396)

Since, the obtained value of the r_{​​​​​} is greater than the critical value of r at 5% significance level and 23 degrees of freedom, therefore we shall reject the null hypothesis at 5% significance level.

Conclusion: At 5% significance level, there is not enough evidence to conclude that there is no correlation between X and Y.