6414-hw3-f23-v3

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Georgia Institute Of Technology *

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6414

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Mechanical Engineering

Date

Oct 30, 2023

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pdf

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Uploaded by BaronLlamaMaster789

1 ISyE 6414 — Fall 2023 Homework #3 To study the performance of a cooling method that uses high pressure inlet fogging, 67 gas turbines were tested and the heat rate (kilojoules per kilowatt per hour) of each measured. Among the other variables measured was speed (revolutions per minute) of the engine. The attached data set, 6414 HW3 Turbine , contains the results for 67 gas turbines where: y : Heat rate (kj/kwhr) x : Speed (rpm). By using the data set, answer Questions 1–9. Submit the relevant outputs and R codes. 1. Construct and submit a scatter plot of y versus x . Is a simple linear regression model approriate to relate Heat rate to Speed? 2. Construct and run a simple linear regression model using x as the predictor variable and y as the response variable. By using the output, identify and report ˆ β 0 , ˆ β 1 , ˆ σ , se ( ˆ β 0 ), and se ( ˆ β 1 ). Note that se stands for the standard error. 3. Calculate a 95% confidence interval for the slope β 1 and give a statement in plain English interpreting this interval. 4. Conduct a hypothesis test using α = 0.05 to see whether or not the predictor x is statistically significant. State the null and alternative hypotheses, the test statistic, critical t -value, reject or fail to reject, and your conclusion. Give a statement in plain English interpreting this result. 5. Identify the p -value for testing the intercept β 0 . By using the p -value, state whether or not the intercept is statistically significant. 6. Identify the p -value for testing β 1 . Using the p -value, determine whether we can reject H 0 by setting α to 0.10, 0.05, and 0.005. What do you conclude about the strength of the relationship between y and x ? 7. By using R, find a point estimate of and a 95% confidence interval for the mean “heat rate” when the speed is 8327 rpm (= average speed). 8. By using R, find a point prediction of and a 95% prediction interval for the heat rate when the speed is 8327 rpm. Is this interval wider than the confidence interval for the mean heat rate found in Question 7? Answer by comparing the half-lengths numerically.

2 9. Find a point estimate of and a 95% confidence interval for the mean heat rate when the speed is is 3600 rpm. Is the half-length of this confidence interval identical to the one found in Question 7? If not, why? Answer by comparing the half-lengths numerically. Derivation Questions : Answer Questions 10–12 by considering the following model: Suppose you know that the true β 0 (intercept) is zero in a simple linear regression model, and you do not want to include the intercept in the model, i.e., the no-intercept model is y = β 1 x + ε . 10. Show that the least squares estimate of β 1 for such a model that minimizes SSE = ∑ n i =1 ( y i − ˆ y i ) 2 is ˆ β 1 = ∑ n i =1 x i y i ∑ n i =1 x 2 i . 11. Also show that Var( ˆ β 1 ) = σ 2 / ∑ n i =1 x 2 i for the no-intercept model. 12. For the no-intercept model, prove or disprove that the estimators ˆ y i = ˆ β 1 x i satisfy the property for all samples: the sum of the residuals should be zero, i.e., ∑ n i =1 ( y i − ˆ y i ) = 0.

6414-hw3-f23-v3

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