The sample correlation coefficient, r, between two random...

The sample correlation coefficient, $r$, between two random variables is 0.625 .
(a) If $n=21$, test the null hypothesis $H_0: \rho=0$ against the alternative hypothesis $H_1: \rho \neq 0$ at the $5 \%$ level of significance.
(b) What is the $p$-value of the test?

Key Concepts

p-value

A p-value represents the probability of obtaining a test statistic at least as extreme as the one computed from the sample data, assuming that the null hypothesis is true. In hypothesis testing, a smaller p-value indicates stronger evidence against the null hypothesis, and it is compared against a predetermined significance level to decide whether to reject the null.

Significance Level

The significance level, often denoted as alpha (?), is the threshold used to determine whether the observed data are sufficiently unlikely under the null hypothesis. A common significance level is 5% (0.05), meaning that there is a 5% risk of rejecting the null hypothesis when it is actually true.

Degrees of Freedom

Degrees of freedom reflect the number of independent values that can vary in an analysis without violating any constraints. In the context of testing the significance of a correlation coefficient, the degrees of freedom are given by the sample size minus two (n - 2), accounting for the estimation of the means for both variables.

Student's t-Distribution

The Student's t-distribution is used in scenarios where the sample size is small and the population standard deviation is unknown, which is common in testing a correlation coefficient. By transforming the correlation coefficient into a t-statistic, one can leverage the t-distribution to evaluate the probability of observing a statistic as extreme or more extreme than the computed value under the null hypothesis.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to make inferences about a population parameter based on sample data. In testing the significance of a correlation coefficient, the null hypothesis typically assumes that there is no linear relationship between the variables (i.e., the population correlation is zero) against an alternative hypothesis that assumes a nonzero relationship.

Pearson Correlation Coefficient

The Pearson correlation coefficient quantifies the strength and direction of the linear relationship between two continuous variables. It ranges between -1 and 1, where values close to 1 or -1 indicate strong positive or negative linear relationships, respectively, and values near 0 imply a weak or no linear relationship.

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