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Chapter 13: Problem 32

Consider a regression study involving a dependent variable \(y,\) a quantitative independent variable \(x_{1},\) and a qualitative variable with two levels (level 1 and level 2 ). a. Write a multiple regression equation relating \(x_{1}\) and the qualitative variable to \(y\) b. What is the expected value of \(y\) corresponding to level 1 of the qualitative variable? c. What is the expected value of \(y\) corresponding to level 2 of the qualitative variable? d. Interpret the parameters in your regression equation.

Short Answer

Expert verified
a. y = 尾0 + 尾1x1 + 尾2x2 b. E(y|x2=0) = 尾0 + 尾1x1 c. E(y|x2=1) = 尾0 + 尾1x1 + 尾2 d. 尾0 is the intercept; 尾1 is the effect of x1; 尾2 is the effect of level 2 over level 1.

Step by step solution

01

Formulate the Regression Equation

We consider a multiple regression model that includes both the quantitative variable \(x_1\) and the qualitative variable with two levels. For this purpose, we use a dummy variable \(x_2\) to represent the qualitative variable, where \(x_2 = 0\) for level 1 and \(x_2 = 1\) for level 2. The multiple regression equation can be written as:\[ y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon \] where \(\epsilon\) is the error term.
02

Expected Value of y for Level 1

For level 1 of the qualitative variable, we have \(x_2 = 0\). Substitute \(x_2 = 0\) into the regression equation to determine the expected value of \(y\):\[ E(y|x_2 = 0) = \beta_0 + \beta_1x_1 + \beta_2(0) = \beta_0 + \beta_1x_1 \] This simplifies to \(E(y|x_2 = 0) = \beta_0 + \beta_1x_1\).
03

Expected Value of y for Level 2

For level 2 of the qualitative variable, we have \(x_2 = 1\). Substitute \(x_2 = 1\) into the regression equation to determine the expected value of \(y\):\[ E(y|x_2 = 1) = \beta_0 + \beta_1x_1 + \beta_2(1) = \beta_0 + \beta_1x_1 + \beta_2 \] This simplifies to \(E(y|x_2 = 1) = \beta_0 + \beta_1x_1 + \beta_2\).
04

Interpretation of Parameters

Each parameter in the multiple regression equation plays a specific role:- \(\beta_0\): This is the intercept, which represents the expected value of \(y\) when \(x_1 = 0\) and the qualitative variable is at level 1 (i.e., \(x_2 = 0\)).- \(\beta_1\): This is the coefficient of the quantitative variable \(x_1\), representing the expected change in \(y\) for a one-unit change in \(x_1\), assuming other variables remain constant.- \(\beta_2\): This is the coefficient of the dummy variable and indicates the expected difference in \(y\) between level 2 and level 1 of the qualitative variable, for a given \(x_1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dependent Variable
In the realm of multiple regression, the dependent variable is the primary focus of our analysis. It is the outcome we are trying to predict or explain. Imagine you're trying to understand the factors that affect your test scores; the test score here would be the dependent variable.
  • Often denoted as \(y\), it is the variable that changes in response to other variables in your equation.
  • In the context of the regression model given, \(y\) is influenced by both a quantitative variable \(x_1\) and a qualitative factor encapsulated by a dummy variable \(x_2\).
  • Understanding the dependent variable helps us frame the predictions or insights the analysis is aiming to provide.
Always remember, the dependent variable is what you are most interested in explaining, based on other variables in your study.
Qualitative Variable
Qualitative variables are those that describe categories or classes. They are not numerical but rather descriptive or categorical.
  • In regression analysis, handling qualitative variables requires a bit of extra work, as they can't be directly used in calculations like quantitative data.
  • In our exercise, the qualitative variable has two levels, which might represent anything from gender categories to satisfaction levels.
  • To include such a variable in a regression model, we turn it into a numerical format using a technique called dummy coding, which transforms categories into binary values.
In summary, qualitative variables enrich your analysis by introducing categorical distinctions, allowing for a more comprehensive view of the factors affecting your dependent variable.
Dummy Variable
A dummy variable is a kinda magical tool in regression analysis when dealing with qualitative data. Since regression models require numerical input, dummy variables make it possible to include qualitative data.
  • A dummy variable usually takes on values of 0 and 1.
  • In our current model, \(x_2\) acts as a dummy variable, where value 0 corresponds to level 1 of the qualitative variable, and value 1 corresponds to level 2.
  • This setup allows us to seamlessly integrate qualitative factors into a quantitative regression framework.
The essential goal of using dummy variables is to ensure that categorical data can be accurately interpreted and used in regression models. This technique allows us to maintain the integrity and inclusiveness of the data in a numerical environment.
Expected Value
The concept of expected value helps us understand what the average outcome would be if an experiment was repeated many times. In the context of regression, it's about estimating the average value of \(y\) given particular levels of your variables.
  • For example, the expected value \(E(y|x_2 = 0)\) gives us the average \(y\) when the qualitative variable is at level 1. It's calculated based on the regression equation with \(x_2 = 0\).
  • Similarly, \(E(y|x_2 = 1)\) provides the expected \(y\) for level 2, obtained by inputting \(x_2 = 1\).
  • This helps you understand and compare outcomes across different levels of qualitative variables.
Expected value acts as a foundational statistical concept that enables sound and insightful predictions in regression analysis, thereby informing decisions or further research.

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Most popular questions from this chapter

Consider a regression study involving a dependent variable \(y,\) a quantitative independent variable \(x_{1},\) and a qualitative independent variable with three possible levels (level \(1,\) level 2 and level 3). a. How many dummy variables are required to represent the qualitative variable? b. Write a multiple regression equation relating \(x_{1}\) and the qualitative variable to \(y\). c. Interpret the parameters in your regression equation.

In a regression analysis involving 30 observations, the following estimated regression equation was obtained. \\[\hat{y}=17.6+3.8 x_{1}-2.3 x_{2}+7.6 x_{3}+2.7 x_{4}\\] a. Interpret \(b_{1}, b_{2}, b_{3},\) and \(b_{4}\) in this estimated regression equation. b. Estimate \(y\) when \(x_{1}=10, x_{2}=5, x_{3}=1,\) and \(x_{4}=2\)

In exercise \(1,\) the following estimated regression equation based on 10 observations was presented. \\[\hat{y}=29.1270+.5906 x_{1}+.4980 x_{2}\\] a. Develop a point estimate of the mean value of \(y\) when \(x_{1}=180\) and \(x_{2}=310\) b. Develop a point estimate for an individual value of \(y\) when \(x_{1}=180\) and \(x_{2}=310\)

The owner of Showtime Movie Theaters, Inc., would like to estimate weekly gross revenue as a function of advertising expenditures. Historical data for a sample of eight weeks follow. $$\begin{array}{ccc} \text { Weekly } & \text { Television } & \text { Newspaper } \\ \text { Gross Revenue } & \text { Advertising } & \text { Advertising } \\ \text { (\$1000s) } & \text { (\$1000s) } & \text { (\$1000s) } \\ 96 & 5.0 & 1.5 \\ 90 & 2.0 & 2.0 \\ 95 & 4.0 & 1.5 \\ 92 & 2.5 & 2.5 \\ 95 & 3.0 & 3.3 \\ 94 & 3.5 & 2.3 \\ 94 & 2.5 & 4.2 \\ 94 & 3.0 & 2.5 \end{array}$$ a. Develop an estimated regression equation with the amount of television advertising as the independent variable. b. Develop an estimated regression equation with both television advertising and newspaper advertising as the independent variables. c. Is the estimated regression equation coefficient for television advertising expenditures the same in part (a) and in part (b)? Interpret the coefficient in each case. d. What is the estimate of the weekly gross revenue for a week when \(\$ 3500\) is spent on television advertising and \(\$ 1800\) is spent on newspaper advertising?

In exercise 2,10 observations were provided for a dependent variable \(y\) and two independent variables \(x_{1}\) and \(x_{2} ;\) for these data \(\mathrm{SST}=15,182.9,\) and \(\mathrm{SSR}=14,052.2\) a. Compute \(R^{2}\) b. Compute \(R_{\mathrm{a}}^{2}\) c. Does the estimated regression equation explain a large amount of the variability in the data? Explain.
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