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Chapter 12: Q149SE (page 808)

Question: Defects in nuclear missile housing parts. The technique of multivariable testing (MVT) was discussed in the Journal of the Reliability Analysis Center (First Quarter, 2004). MVT was shown to improve the quality of carbon-foam rings used in nuclear missile housings. The rings are produced via a casting process that involves mixing ingredients, oven curing, and carving the finished part. One type of defect analyzed was the number y of black streaks in the manufactured ring. Two variables found to impact the number of defects were turntable speed (revolutions per minute),x1, and cutting-blade position (inches from center), x2.

  1. The researchers discovered 鈥渁n interaction between blade position and turntable speed.鈥� Hypothesize a regression model for E(y) that incorporates this interaction.
  2. The researchers reported a positive linear relationship between number of defects (y) and turntable speed (x1)but found that the slope of the relationship was much steeper for lower values of cutting-blade position (x2). What does this imply about the interaction term in the model, part a? Explain.

Short Answer

Expert verified

Answer

  1. A regression model for E(y) with x1 and x2 as functions while there is an interaction amongst the variable can be written asEy=尾0+尾1x1+尾2x2+尾3x1x2
  2. Since for lower value of cutting blade position the relationship between y and x2 is much steeper and there is a positive relationship between y and this means that for lower values of x1 and x2 there is some relationship between x1 and x2 as the lines will interact at lower values of x1 and x2 .

Step by step solution

01

Definition

Multivariate testing is technique used for testing a hypothesis in which multiple variables are modified. This testing is used to determine which combination of variations performs well among all the possible combinations

02

Regression model for E(y)

b.

Since the researchers have observed an interaction amongst the blade position and the turntable speed, a new variable needs to be added in the model to represent this interaction in the model. A regression model for E(y) with x1 and x2 as functions while there is an interaction amongst the variable which can be written as Ey=尾0+尾1x1+尾2x2+尾3x1x2.

The hypothesis is that the rings are being produced via a casting process that involves mixing ingredients, oven curing, and carving the finished part, where one type of defect analyzed was the number y of black streaks in the manufactured ring and the two variables found to impact the number of defects were turntable speed (revolutions per minute),x1 and cutting-blade position (inches from center), x2.Therefore, the null and the alternative hypothesis are:H0:尾0=尾1=0

and At least one of the 尾should be zero respectively.

03

Interpretation of the interaction term

b.

Since for lower value of cutting blade position the relationship between y and x2 is much steeper and there is a positive relationship between y and this means that for lower values of x1 and x2 there is some relationship between x1 and x2 as the lines will interact at lower values of x1 and x2 .

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Most popular questions from this chapter

Question: Write a first-order model relating to

  1. Two quantitative independent variables.
  2. Four quantitative independent variables.
  3. Five quantitative independent variables.

To model the relationship between y, a dependent variable, and x, an independent variable, a researcher has taken one measurement on y at each of three different x-values. Drawing on his mathematical expertise, the researcher realizes that he can fit the second-order model Ey=尾0+尾1x+尾2x2 and it will pass exactly through all three points, yielding SSE = 0. The researcher, delighted with the excellent fit of the model, eagerly sets out to use it to make inferences. What problems will he encounter in attempting to make inferences?

Suppose you fit the second-order model y=尾0+尾1x+尾2x2+蔚to n = 25 data points. Your estimate of尾2is尾^2= 0.47, and the estimated standard error of the estimate is 0.15.

  1. TestH0:尾2=0againstHa:尾2鈮�0. Use伪=0.05.
  2. Suppose you want to determine only whether the quadratic curve opens upward; that is, as x increases, the slope of the curve increases. Give the test statistic and the rejection region for the test for伪=0.05. Do the data support the theory that the slope of the curve increases as x increases? Explain.

Question: Novelty of a vacation destination. Many tourists choose a vacation destination based on the newness or uniqueness (i.e., the novelty) of the itinerary. The relationship between novelty and vacationing golfers鈥� demographics was investigated in the Annals of Tourism Research (Vol. 29, 2002). Data were obtained from a mail survey of 393 golf vacationers to a large coastal resort in the south-eastern United States. Several measures of novelty level (on a numerical scale) were obtained for each vacationer, including 鈥渃hange from routine,鈥� 鈥渢hrill,鈥� 鈥渂oredom-alleviation,鈥� and 鈥渟urprise.鈥� The researcher employed four independent variables in a regression model to predict each of the novelty measures. The independent variables were x1 = number of rounds of golf per year, x2 = total number of golf vacations taken, x3 = number of years played golf, and x4 = average golf score.

  1. Give the hypothesized equation of a first-order model for y = change from routine.
  1. A test of H0: 尾3 = 0 versus Ha: 尾3< 0 yielded a p-value of .005. Interpret this result if 伪 = .01.
  1. The estimate of 尾3 was found to be negative. Based on this result (and the result of part b), the researcher concluded that 鈥渢hose who have played golf for more years are less apt to seek change from their normal routine in their golf vacations.鈥� Do you agree with this statement? Explain.
  1. The regression results for three dependent novelty measures, based on data collected for n = 393 golf vacationers, are summarized in the table below. Give the null hypothesis for testing the overall adequacy of the first-order regression model.
  1. Give the rejection region for the test, part d, for 伪 = .01.
  1. Use the test statistics reported in the table and the rejection region from part e to conduct the test for each of the dependent measures of novelty.
  1. Verify that the p-values reported in the table support your conclusions in part f.
  1. Interpret the values of R2 reported in the table.

Question: Write a regression model relating E(y) to a qualitative independent variable that can assume three levels. Interpret all the terms in the model.
See all solutions

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