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Chapter 12: 69E (page 755)

The following model was used to relate E(y) to a single qualitative variable with four levels:E(y)=尾0+尾1x1+尾2x2+尾3x3where

x1=1iflevel20ifnotx2=1iflevel30ifnotx3=1iflevel40ifnot

This model was fit to n = 30 data points, and the following result was obtained:

y^=10.2-4x1+12x2+2x3

  1. Use the least-squares prediction equation to find the estimate of E(y) for each level of the qualitative independent variable.
  2. Specify the null and alternative hypotheses you would use to test whether E(y) is the same for all four levels of the independent variable.

Short Answer

Expert verified

a.For x1=1,x2=0and x3=0;

y^=10.2-4(1)+12(0)+2(0)=6.2


For x2=1,x1=0and x3=0

y^=10.2-4(0)+12(1)+2(0)=1.8


. Forx3=1,x1=0andx2=0

y^=10.2-4(0)+12(0)+2(1)=8.2

b. The hypothesis to be tested can be written asH0:尾1=尾2=尾3=0

Ha:At least one of the parameters 尾1,尾2and尾3differfrom0

Step by step solution

01

Mean values for each level of the qualitative independent variable

The value E(Y) for a different level of the qualitative independent variable

Can be found by substituting in place of other two variables

Forx1=1,x2=0andx3=0;

y^=10.2-4(1)+12(0)+2(0)=6.2

Forx2=1,x1=0,andx3=0;

y^=10.2-4(0)+12(1)+2(0)=1.8

Forx3=1,x1=0,andx2=0;

y^=10.2-4(0)+12(0)+2(1)=8.2

02

Hypothesis testing

Here, the null hypothesis becomes that the means for the three groups are equal meaning 渭1=渭2=渭3while the alternate hypothesis implies that at least two of the three means (渭1,渭2,and 渭3) differ.

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Most popular questions from this chapter

Question: Adverse effects of hot-water runoff. The Environmental Protection Agency (EPA) wants to determine whether the hot-water runoff from a particular power plant located near a large gulf is having an adverse effect on the marine life in the area. The goal is to acquire a prediction equation for the number of marine animals located at certain designated areas, or stations, in the gulf. Based on past experience, the EPA considered the following environmental factors as predictors for the number of animals at a particular station:

X1 = Temperature of water (TEMP)

X2 = Salinity of water (SAL)

X3 = Dissolved oxygen content of water (DO)

X4 = Turbidity index, a measure of the turbidity of the water (TI)

x5 = Depth of the water at the station (ST_DEPTH)

x6 = Total weight of sea grasses in sampled area (TGRSWT)

As a preliminary step in the construction of this model, the EPA used a stepwise regression procedure to identify the most important of these six variables. A total of 716 samples were taken at different stations in the gulf, producing the SPSS printout shown below. (The response measured was y, the logarithm of the number of marine animals found in the sampled area.)

a. According to the SPSS printout, which of the six independent variables should be used in the model? (Use 伪 = .10.)

b. Are we able to assume that the EPA has identified all the important independent variables for the prediction of y? Why?

c. Using the variables identified in part a, write the first-order model with interaction that may be used to predict y.

d. How would the EPA determine whether the model specified in part c is better than the first-order model?

e.Note the small value of R2. What action might the EPA take to improve the model?

Question:Suppose you fit the first-order model y=尾0+尾1x1+尾2x2+尾3x3+尾4x4+尾5x5+蔚to n=30 data points and obtain SSE = 0.33 and R2=0.92

(A) Do the values of SSE and R2suggest that the model provides a good fit to the data? Explain.

(B) Is the model of any use in predicting Y ? Test the null hypothesis H0:尾1=尾2=尾3=尾4=尾5=0 against the alternative hypothesis {H}at least one of the parameters 尾1,尾2,...,尾5 is non zero.Use伪=0.05 .

Question: Revenues of popular movies. The Internet Movie Database (www.imdb.com) monitors the gross revenues for all major motion pictures. The table on the next page gives both the domestic (United States and Canada) and international gross revenues for a sample of 25 popular movies.

  1. Write a first-order model for foreign gross revenues (y) as a function of domestic gross revenues (x).
  2. Write a second-order model for international gross revenues y as a function of domestic gross revenues x.
  3. Construct a scatterplot for these data. Which of the models from parts a and b appears to be the better choice for explaining the variation in foreign gross revenues?
  4. Fit the model of part b to the data and investigate its usefulness. Is there evidence of a curvilinear relationship between international and domestic gross revenues? Try using伪=0.05.
  5. Based on your analysis in part d, which of the models from parts a and b better explains the variation in international gross revenues? Compare your answer with your preliminary conclusion from part c.

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鈥檚 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.

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鈥檚 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 the尾 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 the尾 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).
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