An analyst is investigating the linear association between two continuous variables X and Y using n = 7 paired observations. The following summary statistics are available:
Sample standard deviation of X: s_X = 4
Sample standard deviation of Y: s_Y = 3
Sum of cross-products of deviations: Σ(x_i − x̄)(y_i − ȳ) = 36
Using Pearson's correlation coefficient and testing at the α = 0.05 significance level (two-tailed) for H₀: ρ = 0, which statement correctly states both the value of the sample correlation r and the appropriate decision on the null hypothesis?
r ≈ 0.83 and fail to reject the null hypothesis (no statistically significant linear correlation)
r = 0.50 and reject the null hypothesis (statistically significant linear correlation)
r ≈ 0.83 and reject the null hypothesis (statistically significant linear correlation)
r = 0.50 and fail to reject the null hypothesis (no statistically significant linear correlation)
The Pearson correlation for a sample is r = Σ(x_i − x̄)(y_i − ȳ) / [(n − 1)s_X s_Y]. Substituting the given values gives r = 36 / (6 × 4 × 3) = 0.50. To test significance, use the t statistic t = r √(n − 2) / √(1 − r²). With n = 7, t = 0.50 × √5 / √(1 − 0.25) ≈ 1.29. The critical value for a two-tailed test with df = 5 at α = 0.05 is about ±2.57, so |1.29| does not exceed the critical value. Therefore the analyst fails to reject H₀; the correlation is not statistically significant at the 0.05 level.
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