A researcher computes sample correlation coefficients 11, 12, 13 and 14 from four different samples and obtains their p-values 0.99, 0.999, 0.05 and 0.005, respectively. Correlation coefficient significant at 1% level is

The correct answer is r₁₄ (the fourth correlation coefficient, with a p-value of 0.005).

To understand why, we first need to clearly understand what a p-value means in the context of hypothesis testing for a correlation coefficient.

When a researcher tests whether a sample correlation coefficient is statistically significant, they are essentially testing the null hypothesis:

H₀: ρ = 0 (there is no true correlation in the population)

The p-value tells us the probability of obtaining the observed result (or something more extreme) purely by chance, assuming H₀ is true. A smaller p-value means stronger evidence against H₀, i.e., stronger evidence that the correlation is real and not due to random chance.

The significance level (α) is the threshold we set before the test. For a 1% significance level, we set:

α = 0.01

The decision rule is straightforward:

If p-value ≤ α, we reject H₀ → the correlation is statistically significant at that level.
If p-value > α, we fail to reject H₀ → the correlation is NOT statistically significant at that level.

Now let us examine each of the four correlation coefficients and their respective p-values:

r₁₁ → p-value = 0.99
Compare: 0.99 > 0.01 → Not significant at the 1% level.

r₁₂ → p-value = 0.999
Compare: 0.999 > 0.01 → Not significant at the 1% level.

r₁₃ → p-value = 0.05
Compare: 0.05 > 0.01 → Not significant at the 1% level. (It would only be significant at the 5% level, not at 1%.)

r₁₄ → p-value = 0.005
Compare: 0.005 ≤ 0.01 → Significant at the 1% level. ✓

A p-value of 0.005 means there is only a 0.5% probability of observing the computed correlation coefficient by chance alone (under H₀). Since 0.5% is well below our 1% threshold, we have strong evidence to reject H₀ and conclude that the correlation is statistically real and meaningful.

In contrast, p-values of 0.99 and 0.999 indicate that the observed correlations (r₁₁ and r₁₂) are almost entirely attributable to random chance — there is a 99% and 99.9% probability respectively of such values arising under no true correlation. A p-value of 0.05 (for r₁₃) only clears the 5% significance bar, not the stricter 1% bar.

Therefore, the only correlation coefficient that is significant at the 1% level of significance is r₁₄, with a p-value of 0.005.