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Chapter 12: Q13E (page 723)

Question: Predicting elements in aluminum alloys. Aluminum scraps that are recycled into alloys are classified into three categories: soft-drink cans, pots and pans, and automobile crank chambers. A study of how these three materials affect the metal elements present in aluminum alloys was published in Advances in Applied Physics (Vol. 1, 2013). Data on 126 production runs at an aluminum plant were used to model the percentage (y) of various elements (e.g., silver, boron, iron) that make up the aluminum alloy. Three independent variables were used in the model: x1 = proportion of aluminum scraps from cans, x2 = proportion of aluminum scraps from pots/pans, and x3 = proportion of aluminum scraps from crank chambers. The first-order model, , was fit to the data for several elements. The estimates of the model parameters (p-values in parentheses) for silver and iron are shown in the accompanying table.

(A) Is the overall model statistically useful (at α = .05) for predicting the percentage of silver in the alloy? If so, give a practical interpretation of R2.

(b)Is the overall model statistically useful (at a = .05) for predicting the percentage of iron in the alloy? If so, give a practical interpretation of R2.

(c)Based on the parameter estimates, sketch the relationship between percentage of silver (y) and proportion of aluminum scraps from cans (x1). Conduct a test to determine if this relationship is statistically significant at α = .05.

(d)Based on the parameter estimates, sketch the relationship between percentage of iron (y) and proportion of aluminum scraps from cans (x1). Conduct a test to determine if this relationship is statistically significant at α = .05.

Short Answer

Expert verified

(a) At α = .05 we can conclude that the model is not statistically useful for predicting the percentage of silver in the alloy. R2 value for silver is 0.075 meaning that around 7% of the variation in the regression is explained by the model. Here, 7% is very low and thus indicates that the model is not a good fit for the data.

(b) At α = .05 we can conclude that the model is not statistically useful for predicting the percentage of silver in the alloy. R2 value for iron is 0.783 meaning that around 78% of the variation in the regression is explained by the model. Here, 78% is very high and thus indicates that the model is a good fit for the data.

(c) At 95% confidence interval, al,valueofβ1≠0

(d) At 95% confidence interval,valueofβ1≠0

Step by step solution

01

Step-by-Step Solution Step 1: Significance of model

For predicting the percentage of silver in the alloy, the F p-value is 0.049 from the table. The model is said to be statistically useful if p-value > α. Here, p-value is 0.049 so at α = .05 we can conclude that the model is not statistically useful for predicting the percentage of silver in the alloy.

R2 value for silver is 0.075 meaning that around 7% of the variation in the regression is explained by the model. Higher value of R2 denotes that the model is a good fit for the data. Here, 7% is very low and thus indicates that the model is not a good fit for the data.

02

Significance of model

For predicting the percentage of iron in the alloy, the F p-value < 0.001 from the table. The model is said to be statistically useful if p-value > α. Here, p-value is less than 0.001 so at α = .05 we can conclude that the model is not statistically useful for predicting the percentage of silver in the alloy.

R2 value for silver is 0.783 meaning that around 78% of the variation in the regression is explained by the model. Higher value of R2 denotes that the model is a good fit for the data. Here, 78% is very high and thus indicates that the model is a good fit for the data.

03

Graph and hypothesis testing 

04

Graph and hypothesis testing 

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Most popular questions from this chapter

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: Orange juice demand study. A chilled orange juice warehousing operation in New York City was experiencing too many out-of-stock situations with its 96-ounce containers. To better understand current and future demand for this product, the company examined the last 40 days of sales, which are shown in the table below. One of the company’s objectives is to model demand, y, as a function of sale day, x (where x = 1, 2, 3, c, 40).

  1. Construct a scatterplot for these data.
  2. Does it appear that a second-order model might better explain the variation in demand than a first-order model? Explain.
  3. Fit a first-order model to these data.
  4. Fit a second-order model to these data.
  5. Compare the results in parts c and d and decide which model better explains variation in demand. Justify your choice.

Suppose the mean value E(y) of a response y is related to the quantitative independent variables x1and x2

E(y)=2+x1-3x2-x1x2

a) Identify and interpret the slope forx2

b) Plot the linear relationship between E(y) andx2for role="math" localid="1649796003444" x1=0,1,2, whererole="math" localid="1649796025582" 1≤x2≤3

c) How would you interpret the estimated slopes?

d) Use the lines you plotted in part b to determine the changes in E(y) for eachrole="math" localid="1649796051071" x1=0,1,2.

e) Use your graph from part b to determine how much E(y) changes whenrole="math" localid="1649796075921" 3≤x1≤5androle="math" localid="1649796084395" 1≤x2≤3.

Question: Risk management performance. An article in the International Journal of Production Economics (Vol. 171, 2016) investigated the factors associated with a firm’s supply chain risk management performance (y). Five potential independent variables (all measured quantitatively) were considered: (1) firm size, (2) supplier orientation, (3) supplier dependency, (4) customer orientation, and (5) systemic purchasing. Consider running a stepwise regression to find the best subset of predictors for risk management performance.

a. How many 1-variable models are fit in step 1 of the stepwise regression?

b. Assume supplier orientation is selected in step 1. How many 2-variable models are fit in step 2 of the stepwise regression?

c. Assume systemic purchasing is selected in step 2. How many 3-variable models are fit in step 3 of the stepwise regression?

d. Assume customer orientation is selected in step 3. How many 4-variable models are fit in step 4 of the stepwise regression?

e. Through the first 4 steps of the stepwise regression, determine the total number of t-tests performed. Assuming each test uses an a = .05 level of significance, give an estimate of the probability of at least one Type I error in the stepwise regression.

Question: Do blondes raise more funds? Refer to the Economic Letters (Vol. 100, 2008) study of whether the color of a female solicitor’s hair impacts the level of capital raised, Exercise 12.75 (p. 756). Recall that 955 households were contacted by a female solicitor to raise funds for hazard mitigation research. In addition to the household’s level of contribution (in dollars) and the hair color of the solicitor (blond Caucasian, brunette Caucasian, or minority female), the researcher also recorded the beauty rating of the solicitor (measured quantitatively, on a 10-point scale).

  1. Write a first-order model (with no interaction) for mean contribution level, E(y), as a function of a solicitor’s hair color and her beauty rating.
  2. Refer to the model, part a. For each hair color, express the change in contribution level for each 1-point increase in a solicitor’s beauty rating in terms of the model parameters.
  3. Write an interaction model for mean contribution level, E(y), as a function of a solicitor’s hair color and her beauty rating.
  4. Refer to the model, part c. For each hair color, express the change in contribution level for each 1-point increase in a solicitor’s beauty rating in terms of the model parameters.
  5. Refer to the model; part c. Illustrate the interaction with a graph.
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