MS Module 16 Regression summary statistics practice exam questions
(The attached PDF file has better formatting.)
A regression analysis on 11 data points has summary statistics
● xi = 8
● yi = 15
● xi2 = 41
● yi2 = 55
● xiyi = 41
Question 16.1:
What is , the average X value?
Answer 16.1: 8 / 11 = 0.727273
(average = total / number of observations)
Question 16.2:
What is , the average Y value?
Answer 16.2: 15 / 11 = 1.363636
(average = total / number of observations)
Question 16.3: Sxx
What is Sxx, the sum of squares of the X values?
Answer 16.3: 41 – 0.7272732 × 11 = 35.182
(Sxx, the sum of squared deviations of the X values, is xi2 – N × 2)
Question 16.4: Syy
What is Syy, the sum of squares of the Y values?
Answer 16.4: 55 – 1.3636362 × 11 = 34.545
(Syy, the sum of squares of the Y values, is yi2 – N × 2)
Question 16.5: Sxy
What is Sxy, the cross sum of squares of the X and Y values?
Answer 16.5: 41 – 8 × 15 / 11 = 30.091
(Sxy, the cross sum of squares of the X and Y values, is xiyi – N × × = xiyi – xi × yi / N)
Question 16.6: Least squares estimate for β1
What is the least squares estimate for β1?
Answer 16.6: 30.091 / 35.182 = 0.855
(least squares estimate for β1 = Sxy / Sxx)
Question 16.7: Least squares estimate for β0
What is the least squares estimate for β0?
Answer 16.7: 1.364 – 0.727 × 0.855 = 0.742
(least squares estimate for β0 = – × β1)
Question 16.8: Error sum of squares
What is the error sum of squares?
Answer 16.8: 55 – 0.742 × 15 – 0.855 × 41 = 8.815; with more significant digits for β0 and β1, ESS = 8.809
(error sum of squares SSE is yi2 – β0 × yi – β1 × xiyi)
Question 16.9: Least squares estimate for σ2
What is s2, the least squares estimate for σ2?
Answer 16.9: 8.809 / (11 – 2) = 0.979
(least squares estimate for σ2 = error sum of squares / (number of observations – 2) )
Question 16.10: Least squares estimate for σ
What is s, the least squares estimate for σ?
Answer 16.10: 0.9790.5 = 0.989
(standard deviation = square root of variance)
Question 16.11: Standard deviation of least squares estimate for β1
What is the standard deviation of the least squares estimate for β1?
Answer 16.11: 0.989 / 35.1820.5 = 0.167
(the standard deviation of the least squares estimate for β1 = σ / Sxx0.5)
Question 16.12: R2
What is the least squares estimate for R2?
Answer 16.12: 1 – 8.809 / 34.545 = 0.745
(the least squares estimate for R2 = 1 – error sum of squares / Syy)
Question 16.13: Correlation
What is the estimated correlation ρ between X and Y?
Answer 16.13: 30.091 / (35.182 × 34.545)0.5 = 0.863
(the estimated correlation ρ between X and Y = Sxy / (Sxx × Syy)0.5
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