Answer: 4.169, β is significantly different from zero

## Explanation We are testing the hypothesis: - H₀: β = 0 (null hypothesis) - H₁: β ≠ 0 (alternative hypothesis) **Step 1: Calculate the slope coefficient β** β = S_xy/S_xx = 8.0/12.0 = 0.667 **Step 2: Calculate the test statistic** The test statistic follows a t-distribution with n-2 degrees of freedom: t = (β - β₀)/se(β) = (0.667 - 0)/0.16 = 4.169 **Step 3: Determine the critical value** For a two-tailed test at 1% significance level with n-2 = 16 degrees of freedom: - α = 0.01 - α/2 = 0.005 for each tail - t_0.005,16 = 2.921 (from t-distribution table) **Step 4: Make the decision** Since |4.169| > 2.921, we reject the null hypothesis H₀. **Conclusion**: β is significantly different from zero at the 1% significance level. **Why the other options are incorrect**: - **A (0.667, β is not significantly different from zero)**: 0.667 is the slope coefficient, not the test statistic, and the conclusion is wrong. - **B (0.4169, β is not significantly different from zero)**: 0.4169 is incorrect calculation of the test statistic, and the conclusion is wrong. - **C (0.667, β is significantly different from zero)**: While the conclusion is correct, 0.667 is the slope coefficient, not the test statistic. Only option D provides both the correct test statistic (4.169) and the correct conclusion.

Author: Nikitesh Somanthe