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Chapter 12: Q29E (page 733)

Cooling method for gas turbines. Refer to the Journal of Engineering for Gas Turbines and Power (January 2005) study of a high-pressure inlet fogging method for a gas turbine engine, Exercise 12.19 (p. 726). Recall that you fit a first-order model for heat rate (y) as a function of speed (x1) , inlet temperature (x2) , exhaust temperature (x3) , cycle pressure ratio (x4) , and airflow rate (x5) . A Minitab printout with both a 95% confidence interval for E(y) and prediction interval for y for selected values of the x’s is shown below.

a. Interpret the 95% prediction interval for y in the words of the problem.

b. Interpret the 95% confidence interval forE(y)in the words of the problem.

c. Will the confidence interval for E(y) always be narrower than the prediction interval for y? Explain.

Short Answer

Expert verified

(a) Given the variable setting, it can be concluded with 95% accuracy that the value of y will lie within the interval (12157.9, 13107.1).

(b) Given the variable setting, it can be concluded with 95% accuracy that the value of E(y) will lie within the interval (13599.6, 13665.5).

(c) Prediction interval must also include the uncertainty in estimating the mean plus the variation in estimating an individual value, the prediction interval is always wider than the confidence interval.

Step by step solution

01

Confidence interval for y

95% confidence interval for y here is (12157.9, 13107.1) where the variable setting set at speed = 7500, inlet temperature = 1000, exhaust temperature = 525, cycle pressure ratio = 13.5, and airflow = 10. Given the variable setting, it can be concluded with 95% accuracy that the value of y will lie within the interval (12157.9, 13107.1).

02

Certainty of interval for E(y)

95% confidence interval for E(y) here is (13599.6, 13665.5) where the variable setting set at speed = 7500, inlet temperature = 1000, exhaust temperature = 525, cycle pressure ratio = 13.5, and airflow = 10. Given the variable setting, it can be concluded with 95% accuracy that the value of E(y) will lie within the interval (13599.6, 13665.5).

03

Fortitude interval for E(y) and prediction interval for y

A prediction interval for y predicts the range in which the individual value of y will lie. While the confidence interval shows the range of values for E(y). Since the prediction interval must also include the uncertainty in estimating the mean plus the variation in estimating an individual value, the prediction interval is always wider than the confidence interval.

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Most popular questions from this chapter

Question: The Excel printout below resulted from fitting the following model to n = 15 data points: y=β0+β1x1+β2x2+ε

Where,

x1=(1iflevel20ifnot)x2=(1iflevel30ifnot)

Question: Shared leadership in airplane crews. Refer to the Human Factors (March 2014) study of shared leadership by the cockpit and cabin crews of a commercial airplane, Exercise 8.14 (p. 466). Recall that simulated flights were taken by 84 six-person crews, where each crew consisted of a 2-person cockpit (captain and first officer) and a 4-person cabin team (three flight attendants and a purser.) During the simulation, smoke appeared in the cabin and the reactions of the crew were monitored for teamwork. One key variable in the study was the team goal attainment score, measured on a 0 to 60-point scale. Multiple regression analysis was used to model team goal attainment (y) as a function of the independent variables job experience of purser (x1), job experience of head flight attendant (x2), gender of purser (x3), gender of head flight attendant (x4), leadership score of purser (x5), and leadership score of head flight attendant (x6).

a. Write a complete, first-order model for E(y) as a function of the six independent variables.

b. Consider a test of whether the leadership score of either the purser or the head flight attendant (or both) is statistically useful for predicting team goal attainment. Give the null and alternative hypotheses as well as the reduced model for this test.

c. The two models were fit to the data for the n = 60 successful cabin crews with the following results: R2 = .02 for reduced model, R2 = .25 for complete model. On the basis of this information only, give your opinion regarding the null hypothesis for successful cabin crews.

d. The p-value of the subset F-test for comparing the two models for successful cabin crews was reported in the article as p 6 .05. Formally test the null hypothesis using α = .05. What do you conclude?

e. The two models were also fit to the data for the n = 24 unsuccessful cabin crews with the following results: R2 = .14 for reduced model, R2 = .15 for complete model. On the basis of this information only, give your opinion regarding the null hypothesis for unsuccessful cabin crews.

f. The p-value of the subset F-test for comparing the two models for unsuccessful cabin crews was reported in the article as p < .10. Formally test the null hypothesis using α = .05. What do you conclude?

Question: Cooling method for gas turbines. Refer to the Journal of Engineering for Gas Turbines and Power (January 2005) study of a high-pressure inlet fogging method for a gas turbine engine, Exercise 12.19 (p. 726). Consider a model for heat rate (kilojoules per kilowatt per hour) of a gas turbine as a function of cycle speed (revolutions per minute) and cycle pressure ratio. The data are saved in the file.

a. Write a complete second-order model for heat rate (y).

b. Give the null and alternative hypotheses for determining whether the curvature terms in the complete second-order model are statistically useful for predicting heat rate (y).

c. For the test in part b, identify the complete and reduced model.

d. The complete and reduced models were fit and compared using SPSS. A summary of the results are shown in the accompanying SPSS printout. Locate the value of the test statistic on the printout.

e. Find the rejection region for α = .10 and locate the p-value of the test on the printout.

f. State the conclusion in the words of the problem.

Forecasting movie revenues with Twitter. Refer to the IEEE International Conference on Web Intelligence and Intelligent Agent Technology (2010) study on using the volume of chatter on Twitter.com to forecast movie box office revenue, Exercise 12.10 (p. 723). The researchers modelled a movie’s opening weekend box office revenue (y) as a function of tweet rate (x1 ) and ratio of positive to negative tweets (x2) using a first-order model.

a) Write the equation of an interaction model for E(y) as a function of x1 and x2 .

b) In terms of theβ in the model, part a, what is the change in revenue (y) for every 1-tweet increase in the tweet rate (x1 ) , holding PN-ratio (x2)constant at a value of 2.5?

c) In terms of the in the model, part a, what is the change in revenue (y) for every 1-tweet increase in the tweet rate (x1 ) , holding PN-ratio (x2)constant at a value of 5.0?

d) In terms of theβ in the model, part a, what is the change in revenue (y) for every 1-unit increase in the PN-ratio (x2) , holding tweet rate (x1 )constant at a value of 100?

e) Give the null hypothesis for testing whether tweet rate (x1 ) and PN-ratio (x2) interact to affect revenue (y).

Question: Shopping on Black Friday. Refer to the International Journal of Retail and Distribution Management (Vol. 39, 2011) study of shopping on Black Friday (the day after Thanksgiving), Exercise 6.16 (p. 340). Recall that researchers conducted interviews with a sample of 38 women shopping on Black Friday to gauge their shopping habits. Two of the variables measured for each shopper were age (x) and number of years shopping on Black Friday (y). Data on these two variables for the 38 shoppers are listed in the accompanying table.

  1. Fit the quadratic model, E(y)=β0+β1x+β2x2, to the data using statistical software. Give the prediction equation.
  2. Conduct a test of the overall adequacy of the model. Use α=0.01.
  3. Conduct a test to determine if the relationship between age (x) and number of years shopping on Black Friday (y) is best represented by a linear or quadratic function. Use α=0.01.
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