B obtain a point estimate of the expected number of

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b. Obtain a point estimate of the expected number of broken ampules when X = 1 transfer is made. c. Estimate the increase in the expected number of ampules broken when there are 2 transfers as compared to 1 transfer. d. Verify that your fitted regression line goes through the point (? ̅ , ? ̅ ). Q1.22) Plastic hardness. Refer to Problems 1.3 and 1.14. Sixteen batches of the plastic were made, and from each batch one test item was molded. Each test item was randomly assigned to one of the four predetermined time levels, and the hardness was measured after the assigned elapsed time. The results are shown below; X is the elapsed time in hours? and Y is hardness in Brinell units. Assume that first-order regression model (1.1) is appropria'te. a. Obtain the estimated regression function. Plot the estimated regression function and the data. Does a linear regression function appear to give a good fit here? b. Obtain a point estimate of the mean hardness when X = 40 hours. c. Obtain a point estimate of the change in mean hardness when X increases by 1 hour. Solution: ? ̅ = 28, ? ̅ = 225.5625 ∑ (? 𝑖 − ? ̅ ) 𝑛=120 𝑖=1 (? 𝑖 − ? ̅ ) = 2604 ∑ (? 𝑖 − ? ̅ ) 2 𝑛=120 𝑖=1 = 1280
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5 ∑ (? 𝑖 − ? ̅ ) 2 𝑛=120 𝑖=1 = 5443.938 ? 1 = ? 1 ̂ = (? 𝑖 − ? ̅ )(? 𝑖 − ? ̅ ) (? 𝑖 − ? ̅ ) 2 𝑛=120 𝑖=1 𝑛=120 𝑖=1 = 2.034375 ? 0 = ? 0 ̂ = ? ̅ − ? 1 ? ̅ = 168.6 ? ̂ = 168.6 + 2.034375 ? At X=40 ? ̂ = 168.6 + 2.034375 (40) = 249.975 Q1.24) Refer to Copier maintenance Problem 1.20. a Obtain the residuals 𝒆 𝒊 and the sum of the squared residuals ∑ 𝒆 𝒊 ? . What is the relation between the sum of the squared residuals here and the quantity Q in (1.8)? b. Obtain point estimates of 𝝈 ? and . In what units is 𝝈 expressed? ∑ ? 𝑖 2 = 3416.377 ∑ ? 𝑖 2 = ? 𝜎 2 ̂ = ∑ ? 𝑖 2 ? − 2 = 3416.377 43 = 79.45063 = ??? 𝜎 = √??? = √79.45063
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6 Q1.25) (H.W) Refer to Airfreight breakage Problem 1.21. a. Obtain the residual for the first case. What is its relation to 𝒆 ? ? b. Compute ∑ 𝒆 𝒊 ? and MSE. What is estimated by MSE? Q1.26) (H.W) Refer to Plastic hardness Problem 1.22. a. Obtain the residuals ej. Do they sum to zero in accord with (1.17)? b. Estimate 𝝈 ? and . In what units is 𝝈 expressed? Q1.21) Solution (H.W) Airfreight breakage. A substance used in biological and medical research is shipped by airfreight to users in cartons of 1,000 ampules. The data below, involving 10 shipments, were collected on the number of times the carton was transferred from one aircraft to another over the shipment route (X) and the number of ampules found to be broken upon arrival (Y). Assume that first-order regression model (1.1) is appropriate. a. Obtain the estimated regression function. Plot the estimated regression function and the data. Does a linear regression function appear to give a good fit here? ? ̂ = 10.2 + 4.0? b. Obtain a point estimate of the expected number of broken ampules when X = 1 transfer is made. If X=1 Then ? ̂ = 10.2 + 4.0(1) = 14.20 c. Estimate the increase in the expected number of ampules broken when there are 2 transfers as compared to 1 transfer.
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