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Chapter 13: Problem 1

Suppose that \(E(y)\) is related to two predictor variables, \(x_{1}\) and \(x_{2}\), by the equation $$ E(y)=3+x_{1}-2 x_{2} $$ a. Graph the relationship between \(E(y)\) and \(x_{1}\) when \(x_{2}=2 .\) Repeat for \(x_{2}=1\) and for \(x_{2}=0\). b. What relationship do the lines in part a have to one another?

Short Answer

Expert verified
Based on the given relationship between E(y), x鈧�, and x鈧�, and the graphs created for different values of x鈧�, describe the relationship between E(y) and x鈧� for different values of x鈧�.

Step by step solution

01

Find the equation for \(E(y)\) when \(x_{2} = 2, 1\) and \(0\)

To find the equation for the different values of \(x_{2}\), we just need to plug in the given value of \(x_{2}\) in the given equation and simplify. For \(x_{2} = 2\), our equation will be: $$ E(y) = 3 + x_{1} - 2(2) $$ For \(x_{2} = 1\), our equation will be: $$ E(y) = 3 + x_{1} - 2(1) $$ For \(x_{2} = 0\), our equation will be: $$ E(y) = 3 + x_{1} - 2(0) $$ Now, let's simplify these equations.
02

Simplify the equations for \(E(y)\)

Now we will simplify the equations we found in Step 1. For \(x_{2} = 2\): $$ E(y) = 3 + x_{1} - 4 = x_{1} - 1 $$ For \(x_{2} = 1\): $$ E(y) = 3 + x_{1} - 2 = x_{1} + 1 $$ For \(x_{2} = 0\): $$ E(y) = 3 + x_{1} - 0 = x_{1} + 3 $$ The equations for \(E(y)\) are: - For \(x_{2} = 2\), \(E(y) = x_{1} - 1\) - For \(x_{2} = 1\), \(E(y) = x_{1} + 1\) - For \(x_{2} = 0\), \(E(y) = x_{1} + 3\)
03

Graph the relationships

Now, we will graph the three equations on the same coordinate axes. These are the lines that represent the relationship between \(E(y)\) and \(x_{1}\) for different values of \(x_{2}\). To do that, you can use graphing software or simply sketch the lines on a graph paper. The lines will look like: - Line 1 (for \(x_{2} = 2\)): A line passing through \((0, -1)\) and with a slope of 1. - Line 2 (for \(x_{2} = 1\)): A line passing through \((0, 1)\) and with a slope of 1. - Line 3 (for \(x_{2} = 0\)): A line passing through \((0, 3)\) and with a slope of 1. After plotting the lines, you will see that all three lines are parallel and have the same slope (1).
04

Analyze the relationship between the lines

Upon observing the graph, we can see that the lines representing the relationship between \(E(y)\) and \(x_{1}\) for different values of \(x_{2}\) are parallel. They have the same slope (1) but different y-intercepts. This means that as \(x_{2}\) changes, the relationship between \(E(y)\) and \(x_{1}\) remains the same, but they are shifted vertically by a constant amount determined by the change in \(x_{2}\). In conclusion, the relationship between the lines in part a is that they are parallel and have the same slope (1). As \(x_{2}\) increases, the y-intercept decreases, shifting the line downward.

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Key Concepts

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

Predictor Variables
Predictor variables, often referred to as independent variables, are the input factors that we believe have an impact on the dependent variable鈥攐ur outcome of interest. In the context of multiple linear regression, we deal with more than one predictor variable influencing the expected value of the dependent variable.

For example, in the exercise presented, the equation \( E(y)=3+x_{1}-2x_{2} \) includes two predictor variables, \( x_{1} \) and \( x_{2} \). These variables contribute information that helps us predict or estimate the value of \( E(y) \), the expected value of the outcome variable. The coefficients, 1 and -2, tell us how much influence each predictor variable has on the expected value respectively.
Expected Value
The expected value in regression analysis is the mean value of the dependent variable given the predictor variables. It is the focal point of the regression equation, computed from the known values of the predictors and the associated coefficients. In our exercise, \( E(y) \) represents the expected value of the outcome variable y, and it is computed based on the values of \( x_{1} \) and \( x_{2} \).

The concept of expected value is central to regression as it provides the predicted outcome for specific combinations of predictor variables under the linear model assumption. The expected value changes with alterations in the predictor variables, and this change is graphically represented by shifts in the position of the regression line.
Graphing Relationships
Graphing the relationships in regression involves plotting the expected values against the predictor variables to visualize how changes in the predictors affect the outcome. It's a powerful way to understand the data and the model.

In the provided exercise, the relationship between \( E(y) \) and \( x_{1} \) is graphed for different values of \( x_{2} \). When \( x_{2} \) is set to 2, 1, and 0, we obtain different linear equations that could be sketched on the same graph. The line representing each equation is characterized by its slope and y-intercept. Since the slope remains constant in our example, and only the intercepts change, this indicates the lines are parallel, showcasing the consistent impact of \( x_{1} \) on \( E(y) \), across different levels of \( x_{2} \).
Slope of a Line
The slope of a line in the context of linear regression denotes the rate at which the expected value of the dependent variable changes concerning the change in a predictor variable. Mathematically, it's the rise over run or the change in y divided by the change in x.

In our exercise, all three lines obtained by graphing the relationship between \( E(y) \) and \( x_{1} \) for various \( x_{2} \) values had the same slope of 1. This means that for every one-unit increase in \( x_{1} \), the expected value \( E(y) \) increases by one unit, regardless of the value of \( x_{2} \). The slope being constant signifies a stable and direct relationship between \( x_{1} \) and \( E(y) \), a foundational understanding for interpreting linear regression results.

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Most popular questions from this chapter

Suppose you wish to predict production yield \(y\) as a function of several independent predictor variables. Indicate whether each of the following independent variables is qualitative or quantitative. If qualitative, define the appropriate dummy variable(s). a. The prevailing interest rate in the area b. The price per pound of one item used in the production process c. The plant (A, B, or C) at which the production yield is measured d. The length of time that the production machine has been in operation e. The shift (night or day) in which the yield is measured

A quality control engineer is interested in predicting the strength of particle board \(y\) as a function of the size of the particles \(x_{1}\) and two types of bonding compounds. If the basic response is expected to be a quadratic function of particle size, write a linear model that incorporates the qualitative variable "bonding compound" into the predictor equation.

Refer to Exercise \(13.26 .\) Use a computer software package to perform the multiple regression analysis and obtain diagnostic plots if possible. a. Comment on the fit of the model, using the analysis of variance \(F\) -test, \(R^{2},\) and the diagnostic plots to check the regression assumptions. b. Find the prediction equation, and graph the three department sales lines. c. Examine the graphs in part b. Do the slopes of the lines corresponding to the children's wear \(\mathrm{B}\) and men's wear A departments appear to differ? Test the null hypothesis that the slopes do not differ \(\left(H_{0}: \beta_{4}=0\right)\) versus the alternative hypothesis that the slopes are different. d. Are the interaction terms in the model significant? Use the methods described in Section 13.5 to test \(H_{0}: \beta_{4}=\beta_{5}=0 .\) Do the results of this test suggest that the fitted model should be modified? e. Write a short explanation of the practical implications of this regression analysis.

Refer to Exercise 13.12 . A command in the MINITAB regression menu provides output in which \(R^{2}\) and \(R^{2}\) (adj) are calculated for all possible subsets of the four independent variables. The printout is provided here. a. If you had to compare these models and choose the best one, which model would you choose? Explain. b. Comment on the usefulness of the model you chose in part a. Is your model valuable in predicting a taste score based on the chosen predictor variables?

Groups of 10 -day-old chicks were randomly assigned to seven treatment groups in which a basal diet was supplemented with \(0,50,100,150,200,250,\) or 300 micrograms/kilogram \((\mu \mathrm{g} / \mathrm{kg})\) of biotin. The table gives the average biotin intake \((x)\) in micrograms per day and the average weight gain (y) in grams per day. \(^{6}\) $$ \begin{array}{rcc} \text { Added Biotin } & \text { Biotin Intake, } x & \text { Weight Gain, } y \\\ \hline 0 & .14 & 8.0 \\ 50 & 2.01 & 17.1 \\ 100 & 6.06 & 22.3 \\ 150 & 6.34 & 24.4 \\ 200 & 7.15 & 26.5 \\ 250 & 9.65 & 23.4 \\ 300 & 12.50 & 23.3 \end{array} $$ In the MINITAB printout, the second-order polynomial model $$ E(y)=\beta_{0}+\beta_{1} x+\beta_{2} x^{2} $$ is fitted to the data. Use the printout to answer the questions. a. What is the fitted least-squares line? b. Find \(R^{2}\) and interpret its value. c. Do the data provide sufficient evidence to conclude that the model contributes significant information for predicting \(y ?\) d. Find the results of the test of \(H_{0}: \beta_{2}=0 .\) Is there sufficient evidence to indicate that the quadratic model provides a better fit to the data than a simple linear model does? e. Do the residual plots indicate that any of the regression assumptions have been violated? Explain.
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