The question asks to determine if the linear correlation between $X$ and $Y$ is statistically significant given a sample size, the Pearson correlation coefficient ($R$), a significance level ($\alpha$), and a critical value.

Hypothesis Testing for Correlation

We perform a hypothesis test to check for significant linear correlation. The hypotheses are:

• Null Hypothesis ($H_0$): There is no linear correlation between $X$ and $Y$ (i.e., $\rho = 0$).
• Alternative Hypothesis ($H_a$): There is a linear correlation between $X$ and $Y$ (i.e., $\rho \neq 0$).

Calculating the Test Statistic

Given:

• Number of observations, $n = 62$
• Pearson correlation coefficient, $R = 0.27$
• Significance level, $\alpha = 0.05$
• Critical t-value, $t_{\alpha/2, n-2} = t_{0.025, 60} = 2.0$

The test statistic formula is provided:

$ t = \frac{R\sqrt{n - 2}}{\sqrt{1 - R^2}} $

Substitute the given values:

$ t = \frac{0.27\sqrt{62 - 2}}{\sqrt{1 - (0.27)^2}} $

$ t = \frac{0.27\sqrt{60}}{\sqrt{1 - 0.0729}} $

$ t = \frac{0.27 \times 7.745966...}{\sqrt{0.9271}} $

$ t = \frac{2.091411...}{0.962860...} $

$ t \approx 2.172 $

Comparing Test Statistic with Critical Value

We compare the absolute value of the calculated t-statistic with the critical t-value:

• Calculated t-statistic: $|t| \approx 2.172$
• Critical t-value: $t_{0.025, 60} = 2.0$

Since $|2.172| > 2.0$, the calculated test statistic falls in the rejection region.

Conclusion

We reject the null hypothesis ($H_0$). This indicates that there is sufficient statistical evidence to conclude that a linear correlation exists between $X$ and $Y$ at the $\alpha = 0.05$ significance level.

Therefore, $X$ and $Y$ are significantly linearly correlated.