By NEAS - 8/5/2018 10:03:29 PM
MS Module 17 Confidence interval and prediction interval practice exam questions
(The attached PDF file has better formatting.)
A linear regression analysis relates Y to X.
● The X values are {1, 2, …, 7}
● The least squares estimator for β1 is a linear function of the Y values = ciYi
● The error sum of squares (SSE) is 25
Question 17.2:
What is , the average X value?
Answer 17.2: (1 + 7) / 2 = 4
Question 17.3: Sxx
What is Sxx, the sum of squared residuals for the X values?
Answer 17.3: (1 – 4)2 + (2 – 4)2 + (3 – 4)2 + (4 – 4)2 + (5 – 4)2 + (6 – 4)2 + (7 – 4)2 = 28
Question 17.4: Least squares estimate of σ2
What is s2, the estimate of σ2?
Answer 17.4: 25 / (7 – 2) = 5
(least squares estimate of σ2 = SSE / (number of observations – 2) )
Question 17.5: Least squares estimate of σ
What is s, the estimate of σ?
Answer 17.5: 50.5 = 2.2361
(standard deviation = square root of variance)
Question 17.6: Standard deviation of the least squares estimate of β1
What is the standard deviation of the least squares estimate of β1?
Answer 17.6: 2.2361 / 280.5 = 0.4226
(standard deviation = square root of variance)
Question 17.7: t value for 90% two-sided confidence interval
What is the t value for a 90% two-sided confidence interval?
Answer 17.7: 2.015
(Table look-up)
Question 17.8: Width of confidence interval
What is the width of the 90% confidence interval at x = 2?
Answer 17.8: 2 × 2.2361 × 2.015 × (1 / 7 + (2 – 4)2 / 28)0.5 = 4.8168
(width of the confidence interval is 2 × tα/2,n-2 × s × (1/n + (x* – )2 / Sxx )½ )
Question 17.9: Width of prediction interval
What is the width of the 90% prediction interval at x = 2?
Answer 17.9: 2 × 2.2361 × 2.015 × (1 + 1 / 7 + (2 – 4)2 / 28)0.5 = 10.2181
(width of the confidence interval is 2 × tα/2,n-2 × s × (1 + 1/n + (x* – )2 / Sxx )½ )
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