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Consider the linear regression model Y₁ = i = α + ßx; + €i, i=1,2,... ..., n with unknown parameters & and ß, known constants x; and random variables Y₁, €₁. Least squares estimates of the parameters are = Sxy/Sxx and â = y - x, where (a) (b) (c) (d) n Sxx = (x₁ - x)² and Sxy =£(x₁ i=1 State the usual assumptions made about the errors €₁. Use least squares to derive the regression coefficient estimates & and B. Show that may be written as 1 C₁Y; for certain coefficients c₁, and hence show that E(B) = p. ii. Thirteen specimens of copper/nickel alloys, each with a specific iron content, were tested on a corrosion wheel. The wheel was rotated in salt water at 10 meters/sec for sixty days. Recorded corrosion, measured as weight loss in mg/sq. decimetre/day, was: Iron Content, x Corrosion Loss, y 127.6 Note: n = i. line. Iron Content, x 1.44 0.71 Corrosion Loss, y 92.3 113.1 = 0.01 0.48 0.71 124.0 110.8 n Σ(x₁ - x)(y₁ - y) i=1 0.95 1.19 0.01 0.48 103.9 101.5 130.1 122.0 1.96 0.01 1.44 1.96 83.7 128.0 91.4 86.2 B-B s/√Sxx 13, x=0.873, y=108.82, Sxx = 5.709, Sxy -137.13 Calculate & and 3, and give the equation of the fitted regression The residual standard error s = 3.058. Give a 90% confidence interval for B. Hint: You may state the form of your interval without proof. Also note: tn-2 iii. It has been suggested that for each unit increase in iron content, the weight loss in copper/nickel alloys due to corrosion decreases by 20 mg/sq. decimetre/day. Using standard notation, state hypotheses that are appropriate to test this claim. Carry out the test at a 5% significance level and give your conclusions.

Consider the linear regression model Y₁ = i = α + ßx; + €i, i=1,2,... ..., n with unknown parameters & and ß, known constants x; and random variables Y₁, €₁. Least squares estimates of the parameters are = Sxy/Sxx and â = y - x, where (a) (b) (c) (d) n Sxx = (x₁ - x)² and Sxy =£(x₁ i=1 State the usual assumptions made about the errors €₁. Use least squares to derive the regression coefficient estimates & and B. Show that may be written as 1 C₁Y; for certain coefficients c₁, and hence show that E(B) = p. ii. Thirteen specimens of copper/nickel alloys, each with a specific iron content, were tested on a corrosion wheel. The wheel was rotated in salt water at 10 meters/sec for sixty days. Recorded corrosion, measured as weight loss in mg/sq. decimetre/day, was: Iron Content, x Corrosion Loss, y 127.6 Note: n = i. line. Iron Content, x 1.44 0.71 Corrosion Loss, y 92.3 113.1 = 0.01 0.48 0.71 124.0 110.8 n Σ(x₁ - x)(y₁ - y) i=1 0.95 1.19 0.01 0.48 103.9 101.5 130.1 122.0 1.96 0.01 1.44 1.96 83.7 128.0 91.4 86.2 B-B s/√Sxx 13, x=0.873, y=108.82, Sxx = 5.709, Sxy -137.13 Calculate & and 3, and give the equation of the fitted regression The residual standard error s = 3.058. Give a 90% confidence interval for B. Hint: You may state the form of your interval without proof. Also note: tn-2 iii. It has been suggested that for each unit increase in iron content, the weight loss in copper/nickel alloys due to corrosion decreases by 20 mg/sq. decimetre/day. Using standard notation, state hypotheses that are appropriate to test this claim. Carry out the test at a 5% significance level and give your conclusions.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Consider the linear regression model Y₁ = = a + Bxi + €i, i=1,2,...,n with unknown parameters & and ß, known constants x; and random variables Y₁, €₁. Least squares estimates of the parameters are = Sxy/Sxx and â = y - x, where (a) (b) (c) (d) Sxx n n =Σ(x₁ - x)² and_ Sxy=[(x₁ - x)(yi — ÿ) i=1 i=1 State the usual assumptions made about the errors €₁. Use least squares to derive the regression coefficient estimates & and ß. Show that may be written as ₁ c₁Y; for certain coefficients c₁, and hence show that E() = p. =1 ii. = Thirteen specimens of copper/nickel alloys, each with a specific iron content, were tested on a corrosion wheel. The wheel was rotated in salt water at 10 meters/sec for sixty days. Recorded corrosion, measured as weight loss in mg/sq. decimetre/day, was: line. Iron Content, x Corrosion Loss, y 0.01 127.6 0.48 0.71 124.0 110.8 103.9 1.44 0.01 1.44 1.96 Iron Content, x 0.71 1.96 Corrosion Loss, y 92.3 113.1 83.7 128.0 91.4 86.2 Note: n 13, x=0.873, y = 108.82, Sxx = 5.709, Sxy = -137.13 i. 0.95 1.19 0.01 0.48 101.5 130.1 122.0 Calculate & and B, and give the equation of the fitted regression The residual standard error s = 3.058. Give a 90% confidence interval for ß. Hint: You may state the form of your interval without proof. Also note: B-B s/√Sxx ~tn-2 iii. It has been suggested that for each unit increase in iron content, the weight loss in copper/nickel alloys due to corrosion decreases by 20 mg/sq. decimetre/day. Using standard notation, state hypotheses that are appropriate to test this claim. Carry out the test at a 5% significance level and give your conclusions.
Transcribed Image Text:Consider the linear regression model Y₁ = = a + Bxi + €i, i=1,2,...,n with unknown parameters & and ß, known constants x; and random variables Y₁, €₁. Least squares estimates of the parameters are = Sxy/Sxx and â = y - x, where (a) (b) (c) (d) Sxx n n =Σ(x₁ - x)² and_ Sxy=[(x₁ - x)(yi — ÿ) i=1 i=1 State the usual assumptions made about the errors €₁. Use least squares to derive the regression coefficient estimates & and ß. Show that may be written as ₁ c₁Y; for certain coefficients c₁, and hence show that E() = p. =1 ii. = Thirteen specimens of copper/nickel alloys, each with a specific iron content, were tested on a corrosion wheel. The wheel was rotated in salt water at 10 meters/sec for sixty days. Recorded corrosion, measured as weight loss in mg/sq. decimetre/day, was: line. Iron Content, x Corrosion Loss, y 0.01 127.6 0.48 0.71 124.0 110.8 103.9 1.44 0.01 1.44 1.96 Iron Content, x 0.71 1.96 Corrosion Loss, y 92.3 113.1 83.7 128.0 91.4 86.2 Note: n 13, x=0.873, y = 108.82, Sxx = 5.709, Sxy = -137.13 i. 0.95 1.19 0.01 0.48 101.5 130.1 122.0 Calculate & and B, and give the equation of the fitted regression The residual standard error s = 3.058. Give a 90% confidence interval for ß. Hint: You may state the form of your interval without proof. Also note: B-B s/√Sxx ~tn-2 iii. It has been suggested that for each unit increase in iron content, the weight loss in copper/nickel alloys due to corrosion decreases by 20 mg/sq. decimetre/day. Using standard notation, state hypotheses that are appropriate to test this claim. Carry out the test at a 5% significance level and give your conclusions.
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