Question

Consider 62 pair of observations \((x_i, y_i)\), \(i = 1, \ldots, 62\), obtained from the distribution of \((X, Y)\). The Pearson correlation coefficient (\(R\)) of these observations is 0.27.

Which one of the following statements is CORRECT for a significance level \(\alpha = 0.05\) and \(t_{0.025,60} = 2.0\)?

Hint: Test statistic \( t = \dfrac{R\sqrt{n-2}}{\sqrt{1-R^2}} \)

• A. \(X\) and \(Y\) are significantly linearly correlated
• B. \(X\) and \(Y\) are statistically independent
• C. \(X\) and \(Y\) are not significantly linearly correlated
• D. \(X\) and \(Y\) are physically related

Hint:

Compute the t statistic from R and n using the given formula and compare its magnitude with the critical value \(t_{0.025,60}=2.0\) to decide if the correlation is significant.

Answer 1

The Correct Option is A

Step 1: Write down the given data and the hypotheses.
We are given \(n = 62\) paired observations with sample correlation coefficient \(R = 0.27\). We test \(H_0: \rho = 0\), no linear correlation in the population, against \(H_1: \rho \neq 0\), at significance level \(\alpha = 0.05\), using the given critical value \(t_{0.025,60} = 2.0\).

Step 2: Compute the test statistic using the given formula.
\[ t = \frac{R\sqrt{n-2}}{\sqrt{1-R^2}} \] With \(n - 2 = 60\), \(\sqrt{60} = 7.746\), \(R^2 = 0.27^2 = 0.0729\), and \(1 - R^2 = 0.9271\), \(\sqrt{0.9271} = 0.9629\): \[ t = \frac{0.27 \times 7.746}{0.9629} = \frac{2.0914}{0.9629} = 2.172 \]

Step 3: Compare the computed \(t\) to the critical value.
The calculated \(|t| = 2.172\) exceeds the two tailed critical value \(t_{0.025,60} = 2.0\). Since the test statistic falls in the rejection region, we reject \(H_0: \rho = 0\) at the 5 percent significance level.

Step 4: Interpret the result and eliminate wrong options.
Rejecting \(H_0\) means the sample correlation of 0.27 is unlikely to have arisen from an uncorrelated population purely by chance, so \(X\) and \(Y\) are significantly linearly correlated, which is option (A). Option (B), statistical independence, is a much stronger and unrelated claim that cannot be concluded from a correlation test. Option (C) contradicts the result of the hypothesis test just performed. Option (D), physically related, is not a statistical conclusion this test can support.

\[ \boxed{\text{Option (A): } X \text{ and } Y \text{ are significantly linearly correlated}} \]