Consider the following data:Σ Χ = 20, Σ Υ = 40, Σ (Χ - X̅)2= 40, Σ (Υ - Y̅)2 = 124Σ (Χ - X̅) (Υ - Y̅)= 70, n = 5I. Estimate the parameters α and β of the model:Yi= α+ βXi +UiII. Estimate the standard errors of these estimates.III. How would test ẞ?(t0.025,3 = 3.182)
Question
Consider the following data:Σ Χ = 20, Σ Υ = 40, Σ (Χ - X̅)2= 40, Σ (Υ - Y̅)2 = 124Σ (Χ - X̅) (Υ - Y̅)= 70, n = 5I. Estimate the parameters α and β of the model:Yi= α+ βXi +UiII. Estimate the standard errors of these estimates.III. How would test ẞ?(t0.025,3 = 3.182)
Solution
I. To estimate the parameters α and β of the model, we can use the following formulas:
β = Σ (Χ - X̅) (Υ - Y̅) / Σ (Χ - X̅)2 α = Y̅ - βX̅
Using the given data, we can calculate the values:
X̅ = Σ Χ / n = 20 / 5 = 4 Y̅ = Σ Υ / n = 40 / 5 = 8
Σ (Χ - X̅)2 = 40 Σ (Υ - Y̅)2 = 124 Σ (Χ - X̅) (Υ - Y̅) = 70
Plugging these values into the formulas, we get:
β = 70 / 40 = 1.75 α = 8 - (1.75 * 4) = 1
Therefore, the estimated parameters are α = 1 and β = 1.75.
II. To estimate the standard errors of these estimates, we can use the following formulas:
SE(α) = sqrt(Σ (Υ - Y̅)2 / (n - 2) * (1/n + X̅^2 / Σ (Χ - X̅)2)) SE(β) = sqrt(Σ (Υ - Y̅)2 / (n - 2) / Σ (Χ - X̅)2)
Using the given data, we can calculate the values:
SE(α) = sqrt(124 / (5 - 2) * (1/5 + 4^2 / 40)) = sqrt(124 / 3 * (0.2 + 0.4)) = sqrt(124 / 3 * 0.6) = sqrt(24.8) ≈ 4.98 SE(β) = sqrt(124 / (5 - 2) / 40) = sqrt(124 / 3 / 40) = sqrt(1.033) ≈ 1.02
Therefore, the estimated standard errors of α and β are approximately 4.98 and 1.02, respectively.
III. To test β, we can calculate the t-statistic using the formula:
t = β / SE(β)
Using the given value t0.025,3 = 3.182, we can compare it with the calculated t-value. If the calculated t-value is greater than t0.025,3, we can reject the null hypothesis that β = 0.
In this case, the calculated t-value is 1.75 / 1.02 ≈ 1.72, which is less than t0.025,3. Therefore, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant relationship between X and Y.
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