/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 You obtain the multiple regressi... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 6

You obtain the multiple regression equation \(\hat{y}=-5-9 x_{1}+4 x_{2}\) from a set of sample data. (a) Interpret the slope coefficients for \(x_{1}\) and \(x_{2}\), (b) Determine the regression equation with \(x_{1}=10\). Graph the regression equation with \(x_{1}=10 .\) (c) Determine the regression equation with \(x_{1}=15\). Graph the regression equation with \(x_{1}=15\) (d) Determine the regression equation with \(x_{1}=20 .\) Graph the regression equation with \(x_{1}=20\). (e) What is the effect of changing the value \(x_{1}\) on the graph of the regression equation?

Short Answer

Expert verified
The slope for \(x_1\) is -9 and for \(x_2\) is 4. For \(x_1 = 10\), the equation is \(\hat{y} = -95 + 4 x_{2}\); for \(x_1 = 15\), \(\hat{y} = -140 + 4 x_{2}\); for \(x_1 = 20\), \(\hat{y} = -185 + 4 x_{2}\). Increasing \(x_1\) shifts the intercept downward.

Step by step solution

01

Interpret the slope coefficients

The multiple regression equation is given by \(\hat{y}=-5-9 x_{1}+4 x_{2}\). The slope coefficient for \(x_1\) is -9, which means for every one unit increase in \(x_1\), \(y\) is expected to decrease by 9 units, holding \(x_2\) constant. The slope coefficient for \(x_2\) is 4, which means for every one unit increase in \(x_2\), \(y\) is expected to increase by 4 units, holding \(x_1\) constant.
02

Determine and graph the regression equation with \(x_1 = 10\)

Substitute \(x_1 = 10\) into the regression equation: \(\hat{y}=-5-9(10)+4 x_{2}\). Simplifying this gives \(\hat{y}=-95 + 4 x_{2}\). This is a linear equation in the form \(\hat{y} = -95 + 4 x_{2}\). To graph this equation, plot a line with a y-intercept at -95 and a slope of 4.
03

Determine and graph the regression equation with \(x_1 = 15\)

Substitute \(x_1 = 15\) into the regression equation: \(\hat{y}=-5-9(15)+4 x_{2}\). Simplifying this gives \(\hat{y}=-140 + 4 x_{2}\). This is a linear equation in the form \(\hat{y} = -140 + 4 x_{2}\). To graph this equation, plot a line with a y-intercept at -140 and a slope of 4.
04

Determine and graph the regression equation with \(x_1 = 20\)

Substitute \(x_1 = 20\) into the regression equation: \(\hat{y}=-5-9(20)+4 x_{2}\). Simplifying this gives \(\hat{y}=-185 + 4 x_{2}\). This is a linear equation in the form \(\hat{y} = -185 + 4 x_{2}\). To graph this equation, plot a line with a y-intercept at -185 and a slope of 4.
05

Effect of changing \(x_1\) on the regression equation graph

Changing the value of \(x_1\) shifts the y-intercept of the regression equation but does not change the slope, which remains at 4. As \(x_1\) increases, the y-intercept moves downward.

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Key Concepts

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

slope interpretation
Understanding the slope coefficients of a regression equation is crucial. The equation provided is \(\hat{y}=-5-9 x_{1}+4 x_{2}\). Here, the slope coefficient for \(x_1\) is -9. This means if \(x_1\) increases by 1 unit, \(y\) decreases by 9 units, assuming \(x_2\) remains constant. On the other hand, the slope coefficient for \(x_2\) is 4. This implies that if \(x_2\) increases by 1 unit, \(y\) increases by 4 units, provided \(x_1\) stays the same. These coefficients help in understanding the relationship between the dependent variable \(y\) and the independent variables \(x_1\) and \(x_2\).
regression equation
A regression equation is a mathematical representation that shows the relationship between a dependent variable and one or more independent variables. For this case, the provided multiple regression equation is \(\hat{y}=-5-9 x_{1}+4 x_{2}\). By substituting specific values of \(x_1\), we can derive new regression equations:

  • For \(x_1 = 10\), the equation becomes \(\hat{y} = -95 + 4x_2\).
  • For \(x_1 = 15\), the equation is \(\hat{y} = -140 + 4x_2\).
  • For \(x_1 = 20\), the regression equation is \(\hat{y} = -185 + 4x_2\).
These equations reflect how \(x_2\) impacts \(y\) after accounting for specific values of \(x_1\).
graphing linear equations
Graphing linear equations helps visualize relationships. To graph equations from our regression, use the y-intercept and slope:

  • For \(x_1 = 10\), \(\hat{y} = -95 + 4x_2\): y-intercept is -95, slope is 4.
  • For \(x_1 = 15\), \(\hat{y} = -140 + 4x_2\): y-intercept is -140, slope is 4.
  • For \(x_1 = 20\), \(\hat{y} = -185 + 4x_2\): y-intercept is -185, slope is 4.
Here's how to plot: locate the y-intercept on the graph, then add the slope. For instance, from the y-intercept, rise 4 units for each unit you move right. This yields a straight line showing how \(y\) changes with \(x_2\).
effect of variable change
Changing variables in a regression equation affects its graph and interpretation. When \(x_1\) changes:

- \(x_1\) increases: the y-intercept drops. For example, from \(\hat{y} = -95 + 4x_2\) to \(\hat{y} = -140 + 4x_2\).
- \(x_1\) shifts lower, resulting in downward movement. Despite this, the slope remains consistent for \(x_2\) (4).

This consistency highlights how \(x_2\) influences \(y\) independently of \(x_1\)'s value. Understanding these shifts helps predict and interpret data relationships efficiently.

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Most popular questions from this chapter

Explain the difference between the coefficient of determination, \(R^{2}\), and the adjusted coefficient of determination, \(R_{\text {adj. }}^{2}\) Which is better for determining whether an additional explanatory variable should be added to the regression model?

(a) Draw a scatter diagram of the data. What type of relation appears to exist between \(x\) and \(y ?\) (b) Find the quadratic regression equation \(\hat{y}=b_{0}+b_{1} x+b_{2} x^{2}\) (c) Draw a residual plot against the fitted values, \(x,\) and \(x^{2}\). Also. draw a boxplot of the residuals. Are there any problems with the model? (d) Interpret the coefficient of determination. (e) Does the \(F\) -test indicate that we should reject \(H_{0}: \beta_{1}=\beta_{2}=0 ?\) Is either coefficient not significantly different from zero? (f) Construct and interpret \(95 \%\) confidence and prediction intervals for \(x=4\) $$ \begin{array}{cc} x & y \\ \hline 2.3 & 19.3 \\ \hline 2.7 & 14.8 \\ \hline 3.2 & 10.2 \\ \hline 4.1 & 4.8 \\ \hline 4.9 & 2.9 \\ \hline 5.6 & 3.9 \\ \hline 6.4 & 7.9 \\ \hline \end{array} $$

An economist was interested in modeling the relation among annual income, level of education, and work experience. The level of education is the number of years of education beyond eighth grade, so 1 represents completing I year of high school, 8 means completing 4 years of college, and so on. Work experience is the number of years employed in the current profession. From a random sample of 12 individuals, he obtained the following data: $$ \begin{array}{ccc} \begin{array}{l} \text { Work Experience } \\ \text { (years) } \end{array} & \begin{array}{l} \text { Level of } \\ \text { Education } \end{array} & \begin{array}{l} \text { Annual Income } \\ \text { (\$ thousands) } \end{array} \\ \hline 21 & 6 & 34.7 \\ \hline 14 & 3 & 17.9 \\ \hline 4 & 8 & 22.7 \\\ \hline 16 & 8 & 63.1 \\ \hline 12 & 4 & 33.0 \\ \hline 20 & 4 & 41.4 \\\ \hline 25 & 1 & 20.7 \\ \hline 8 & 3 & 14.6 \\ \hline 24 & 12 & 97.3 \\\ \hline 28 & 9 & 72.1 \\ \hline 4 & 11 & 49.1 \\ \hline 15 & 4 & 52.0 \\\ \hline \end{array} $$ (a) Construct a correlation matrix between work experience, level of education, and annual income. Is there any reason to be concerned with multicollinearity based on the correlation matrix? (b) Find the least-squares regression equation \(\hat{y}=b_{0}+\) \(b_{1} x_{1}+b_{2} x_{2},\) where \(x_{1}\) is work experience, \(x_{2}\) is level of education, and \(y\) is the response variable, annual income. (c) Draw residual plots and a boxplot of the residuals to assess the adequacy of the model. (d) Interpret the regression coefficients for the least-squares regression equation. (e) Determine and interpret \(R^{2}\) and the adjusted \(R^{2}\) (f) Test \(H_{0}: \beta_{1}=\beta_{2}=0\) versus \(H_{1} ;\) at least one of the \(\beta_{i} \neq 0\) at the \(\alpha=0.05\) level of significance. (g) Test the hypotheses \(H_{0}: \beta_{1}=0\) versus \(H_{1}: \beta_{1} \neq 0\) and \(H_{0}: \beta_{2}=0\) versus \(H_{1}: \beta_{2} \neq 0\) at the \(\alpha=0.05\) level of significance. (h) Predict the mean income of all individuals whose experience is 12 years and level of education is 4 (i) Predict the income of a single individual whose experience is 12 years and level of education is 4 (j) Construct \(95 \%\) confidence and prediction intervals for income when experience is 12 years and level of education is 4

Divorce Rates The given data represent the percentage, \(y,\) of the population that is divorced for various ages, \(x\), in the United States in 2010 based on sample data obtained from the United States Statistical Abstract in \(2012 .\) $$ \begin{array}{cc} \text { Age, } x & \text { Percentage Divorced, } y \\ \hline 22 & 0.9 \\ \hline 27 & 3.6 \\ \hline 32 & 7.4 \\ \hline 37 & 10.4 \\ \hline 42 & 12.7 \\ \hline 50 & 15.7 \\ \hline 60 & 16.2 \\ \hline 70 & 13.1 \\ \hline 80 & 6.5 \end{array} $$ (a) Draw a scatter diagram of the data. What type of relation appears to exist between \(x\) and \(y ?\) (b) Find the quadratic regression equation \(\hat{y}=b_{0}+b_{1} x+b_{2} x^{2}\) (c) Draw a residual plot against the fitted values, \(x,\) and \(x^{2}\). Also, draw a boxplot of the residuals. Are there any problems with the model? (d) Interpret the coefficient of determination. (e) Does the \(F\) -test indicate that we should reject \(H_{0}: \beta_{1}=\beta_{2}=0 ?\) Is either coefficient not significantly different from zero? (f) Construct and interpret a \(95 \%\) confidence interval for percent divorced among all 30 years olds.

Suppose we wish to develop a regression equation that models the selling price of a home. The researcher wishes to include the variable garage in the model. She has identified three possibilities for a garage: (1) attached, (2) detached, (3) no garage. Define the indicator variables necessary to incorporate the variable "garage" into the model.
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