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Chapter 12: Q57E. (page 747)

Commercial refrigeration systems. The role of maintenance in energy saving in commercial refrigeration was the topic of an article in the Journal of Quality in Maintenance Engineering (Vol. 18, 2012). The authors provided the following illustration of data relating the efficiency (relative performance) of a refrigeration system to the fraction of total charges for cooling the system required for optimal performance. Based on the data shown in the graph (next page), hypothesize an appropriate model for relative performance (y) as a function of fraction of charge (x). What is the hypothesized sign (positive or negative) of the β2parameter in the model?

Short Answer

Expert verified

The appropriate model for the scatterplot above is a quadratic model of y on x.

The sign of β2will be positive as it can be seen in the graph that the parabola is an upward-sloping curve.

Step by step solution

01

model for the fitted data

The second-order model equation for the fitted data isy=β0+β1x+β2x2

02

Sign of β2

The sign of β2will be positive as it can be seen in the graph that the parabola is an upward-sloping curve.

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Most popular questions from this chapter

Can money spent on gifts buy love? Refer to the Journal of Experimental Social Psychology (Vol. 45, 2009) study of whether buying gifts truly buys love, Exercise 9.9 (p. 529). Recall those study participants were randomly assigned to play the role of gift-giver or gift-receiver. Gift-receivers were asked to provide the level of appreciation (measured on a 7-point scale where 1 = “not at all” and 7 = “to a great extent”) they had for the last birthday gift they received from a loved one. Gift-givers were asked to recall the last birthday gift they gave to a loved one and to provide the level of appreciation the loved one had for the gift.

  1. Write a dummy variable regression model that will allow the researchers to compare the average level of appreciation for birthday gift-giverswith the average for birthday gift-receivers.
  2. Express each of the model’s β parameters in terms ofand.
  3. The researchers hypothesize that the average level of appreciation is higher for birthday gift-givers than for birthday gift-receivers. Explain how to test this hypothesis using the regression model.

Question: Chemical plant contamination. Refer to Exercise 12.18 (p. 725) and the U.S. Army Corps of Engineers study. You fit the first-order model,E(Y)=β0+β1x1+β2x2+β3x3 , to the data, where y = DDT level (parts per million),X1= number of miles upstream,X2= length (centimeters), andX3= weight (grams). Use the Excel/XLSTAT printout below to predict, with 90% confidence, the DDT level of a fish caught 300 miles upstream with a length of 40 centimeters and a weight of 1,000 grams. Interpret the result.

Question: Glass as a waste encapsulant. Because glass is not subject to radiation damage, encapsulation of waste in glass is considered to be one of the most promising solutions to the problem of low-level nuclear waste in the environment. However, chemical reactions may weaken the glass. This concern led to a study undertaken jointly by the Department of Materials Science and Engineering at the University of Florida and the U.S. Department of Energy to assess the utility of glass as a waste encapsulant. Corrosive chemical solutions (called corrosion baths) were prepared and applied directly to glass samples containing one of three types of waste (TDS-3A, FE, and AL); the chemical reactions were observed over time. A few of the key variables measured were

y = Amount of silicon (in parts per million) found in solution at end of experiment. (This is both a measure of the degree of breakdown in the glass and a proxy for the amount of radioactive species released into the environment.)

x1 = Temperature (°C) of the corrosion bath

x2 = 1 if waste type TDS-3A, 0 if not

x3 = 1 if waste type FE, 0 if not

(Waste type AL is the base level.) Suppose we want to model amount y of silicon as a function of temperature (x1) and type of waste (x2, x3).

a. Write a model that proposes parallel straight-line relationships between amount of silicon and temperature, one line for each of the three waste types.

b. Add terms for the interaction between temperature and waste type to the model of part a.

c. Refer to the model of part b. For each waste type, give the slope of the line relating amount of silicon to temperature.

e. Explain how you could test for the presence of temperature–waste type interaction.

Suppose you fit the quadratic model E(y)=β0+β1x+β2x2to a set of n = 20 data points and found R2=0.91, SSyy=29.94, and SSE = 2.63.

a. Is there sufficient evidence to indicate that the model contributes information for predicting y? Test using a = .05.

b. What null and alternative hypotheses would you test to determine whether upward curvature exists?

c. What null and alternative hypotheses would you test to determine whether downward curvature exists?

Write a model relating E(y) to one qualitative independent variable that is at four levels. Define all the terms in your model.
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