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Chapter 12: Q.66E (page 755)

Question: Write a regression model relating the mean value of y to a qualitative independent variable that can assume two levels. Interpret all the terms in the model.

Short Answer

Expert verified

Answer:

Regression model for one qualitative independent variable with two level isx1=(1IFyisobservedatleveli+10otherwise) .Here尾0denotes渭xithe mean for base level尾1and尾2denotes difference between the mean levels for different dummy variables. While,x1andx2denotes different levels of dummy variables used in the model which can take value of either 0 or 1.

Step by step solution

01

Regression model for one qualitative independent variable

A regression model relating the mean value of y to a qualitative independent variable that can assume two levels can be written as

E(y)=尾0+尾1x+尾2x2

Where, x1 is the dummy variable for i+1level

Meaning x1=(1IFyisobservedatleveli+10otherwise)

02

Interpretation of the terms in the model

Here尾0denotesrole="math" localid="1649848061325" 渭xithe mean for base level

尾1and尾2denotes difference between the mean levels for different dummy variables.

While,role="math" localid="1649848167521" x1and role="math" localid="1649848182825" x2denotes different levels of dummy variables used in the model

which can take value of either 0 or 1.

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Most popular questions from this chapter

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 尾2parameter in the model?

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鈮�x2鈮�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鈮�x1鈮�5androle="math" localid="1649796084395" 1鈮�x2鈮�3.

Question:How is the number of degrees of freedom available for estimating 蟽2(the variance of蔚 ) related to the number of independent variables in a regression model?

Question: Failure times of silicon wafer microchips. Refer to the National Semiconductor study of manufactured silicon wafer integrated circuit chips, Exercise 12.63 (p. 749). Recall that the failure times of the microchips (in hours) was determined at different solder temperatures (degrees Celsius). The data are repeated in the table below.

  1. Fit the straight-line modelEy=尾0+尾1xto the data, where y = failure time and x = solder temperature.

  2. Compute the residual for a microchip manufactured at a temperature of 149掳C.

  3. Plot the residuals against solder temperature (x). Do you detect a trend?

  4. In Exercise 12.63c, you determined that failure time (y) and solder temperature (x) were curvilinearly related. Does the residual plot, part c, support this conclusion?

Suppose you fit the model y =尾0+尾1x1+尾1x22+尾3x2+尾4x1x2+蔚to n = 25 data points with the following results:

尾^0=1.26,尾^1= -2.43,尾^2=0.05,尾^3=0.62,尾^4=1.81s尾^1=1.21,s尾^2=0.16,s尾^3=0.26, s尾^4=1.49SSE=0.41 and R2=0.83

  1. Is there sufficient evidence to conclude that at least one of the parameters b1, b2, b3, or b4 is nonzero? Test using a = .05.

  2. Test H0: 尾1 = 0 against Ha: 尾1 < 0. Use 伪 = .05.

  3. Test H0: 尾2 = 0 against Ha: 尾2 > 0. Use 伪 = .05.

  4. Test H0: 尾3 = 0 against Ha: 尾3 鈮� 0. Use 伪 = .05.
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