The sample correlation for 68 pairs of annual returns on...

The sample correlation for 68 pairs of annual returns on common stocks in country $\mathrm{A}$ and country $\mathrm{B}$ was found to be 0.51. Test the null hypothesis that the population correlation is 0 against the alternative that it is positive.

Key Concepts

Pearson Correlation Coefficient

The Pearson correlation coefficient measures the linear relationship between two continuous variables. It quantifies the degree to which changes in one variable can be associated with changes in another, taking values between -1 and 1, where values close to the extremes indicate strong relationships and values near zero indicate weak or no linear relationship.

Hypothesis Testing

Hypothesis testing in this context involves formulating a null hypothesis stating that there is no linear relationship (the population correlation is zero) and an alternative hypothesis suggesting a positive linear relationship (the population correlation is greater than zero). This framework allows us to determine if the observed correlation could have occurred by chance under the assumption of no association.

t-Test for Correlation

The t-test for correlation is used to assess the significance of the observed Pearson correlation coefficient, converting the correlation into a t statistic using the formula t = r * sqrt((n-2) / (1-r^2)). This test statistic follows a t-distribution with n-2 degrees of freedom, and it enables us to compute a p-value to evaluate the evidence against the null hypothesis.

Degrees of Freedom

Degrees of freedom refer to the number of independent values in a calculation after certain restrictions are imposed. For the t-test associated with correlation, the degrees of freedom are computed as n-2, reflecting the estimation of two parameters (the slope and intercept) in the underlying regression model. This adjustment is crucial for determining the correct critical values from the t-distribution.

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