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Chapter 13: Problem 2

Problem \(12.4\) presented data on ten precincts. The information is reproduced here. $$ \begin{array}{lccc} \hline \hline \text { Precinct } & \begin{array}{c} \text { Percentage } \\ \text { Democrat } \end{array} & \begin{array}{c} \text { Percentage } \\ \text { Minority } \end{array} & \begin{array}{c} \text { Percentage } \\ \text { Voter Turnout } \end{array} \\ \hline \text { A } & 33 & 9 & 36 \\ \text { B } & 78 & 15 & 60 \\ \text { C } & 62 & 18 & 64 \\ \text { D } & 50 & 10 & 56 \\ \text { E } & 45 & 12 & 55 \\ \text { F } & 56 & 8 & 52 \\ \text { G } & 85 & 20 & 25 \\ \text { H } & 25 & 0 & 42 \\ \text { I } & 13 & 5 & 89 \\ \text { J } & 33 & 9 & 88 \\ \hline \end{array} $$ Take voter turnout as the dependent variable and do the following: a. Find the multiple regression equations (unstandardized). b. What turnout would you expect for a precinct in which \(0 \%\) of the voters were Democrats and \(5 \%\) were minorities? c. Compute beta-weights for each independent variable and compare their relative effect on turnout. Which was the more important factor? d. Compute \(R^{2}\). e. Write a paragraph summarizing your findings.

Short Answer

Expert verified
Use the regression equation: \[Y = b_0 + b_2(5)\]. Calculate beta weights to compare impacts, and use \(R^2\) to evaluate goodness of fit.

Step by step solution

01

Data Preparation

First, organize the data for Percentage Democrat, Percentage Minority, and Percentage Voter Turnout into matrices to prepare for regression analysis.
02

Formulate the Multiple Regression Equation

The formula for multiple regression is given by: \[Y = b_0 + b_1X_1 + b_2X_2\] where Y is the dependent variable (Percentage Voter Turnout), and \(X_1\) and \(X_2\) are the independent variables (Percentage Democrat and Percentage Minority, respectively). Calculate the coefficients \(b_0\), \(b_1\), and \(b_2\) using statistical software or by solving the normal equations.
03

Calculate Turnout for Given Precinct

To find the expected voter turnout for a precinct with 0% Democrats and 5% minorities, substitute these values into the regression equation obtained in Step 2: \[Y = b_0 + b_1(0) + b_2(5)\]
04

Compute Beta-Weights

Standardize the coefficients \(b_1\) and \(b_2\) to obtain the beta weights. The beta weights are given by: \[ \beta_i = b_i \times \frac{S_{X_i}}{S_Y} \] where \(S_{X_i}\) and \(S_Y\) are the standard deviations of the independent variables and the dependent variable, respectively.
05

Determine the Relative Effect

Compare the computed beta weights to determine the variable with the more significant impact on voter turnout.
06

Compute R-Squared (\(R^2\))

Calculate the determination coefficient \(R^2\) using the formula: \[R^2 = 1 - \frac{SS_{res}}{SS_{tot}} \] where \(SS_{res}\) is the sum of squares of residuals and \(SS_{tot}\) is the total sum of squares.
07

Summarize the Findings

Explain the relationship between the independent variables and the dependent variable. Discuss the significance of the coefficients, the relative impact of each independent variable, and the goodness of fit of the regression model as indicated by \(R^2\).

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Key Concepts

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

Dependent Variable
In multiple regression analysis, the **dependent variable** is the outcome we seek to understand or predict. Here, the dependent variable is the Percentage Voter Turnout. This means we want to explain and predict voter turnout based on other factors.
In our case, voter turnout is influenced by many factors. By choosing it as the dependent variable, we aim to identify and quantify how various precinct characteristics impact voter turnout.
Independent Variables
Independent variables are the factors we believe influence our dependent variable. In this scenario, we have two independent variables: Percentage Democrat and Percentage Minority.
The independent variables serve as predictors. They help us understand how different scenarios can affect the dependent variable, voter turnout. By including these variables in our regression model, we can examine their individual effects on voter turnout.
  • Percentage Democrat: Represents the proportion of voters in a precinct who are Democrats.
  • Percentage Minority: Shows the fraction of voters from minority groups in a precinct.
Beta Weights
Beta weights help us understand the relative importance of each independent variable on the dependent variable. Standardized beta weights allow comparison between variables that are measured on different scales.
To compute beta weights, we standardize the coefficients from the regression equation using the formula:
  • \( \beta_i = b_i \times \frac{S_{X_i}}{S_Y} \ \)
Here, \( b_i \) is the unstandardized coefficient, \( S_{X_i} \) is the standard deviation of the independent variable, and \[ S_Y \] is the standard deviation of the dependent variable. By comparing beta weights, we can determine which independent variable has a stronger effect on voter turnout.
R-squared
R-squared (\[R^2\]) is a statistical measure that represents the proportion of the variance for the dependent variable that's explained by the independent variables. It ranges from 0 to 1.
An \[R^2\] value close to 1 indicates that a large portion of the variance in the dependent variable is predictable from the independent variables. Conversely, a value near 0 suggests that the model does not explain much of the variance.
  • To compute \[R^2\]: \[R^2 = 1 - \frac{SS_{res}}{SS_{tot}}\] Where \(SS_{res}\) is the sum of squares of residuals and \(SS_{tot}\) is the total sum of squares.
This measure helps us judge the goodness of fit for our regression model.
Statistical Software
Using statistical software simplifies the complex calculations involved in multiple regression analysis. Software packages such as R, SPSS, and Python's statsmodels can compute regression coefficients, beta weights, \(R^2\), and other essential metrics quickly.
These tools take care of:
  • Data input and management
  • Calculations of coefficients
  • Performing diagnostics to ensure the validity of the model
By using statistical software, we ensure accurate results and can focus more on interpreting the data and the relationships it reveals.

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Most popular questions from this chapter

In Problem \(12.8\), you used the States database to analyze the bivariate relationships between region (South/Non-South), education (percentage of the population with a college degree), and infant mortality rates. Now, you will take education and region as the independent variables and look at their combined effects on infant mortality rates. \- Click Analyze \(\rightarrow\) Regression \(\rightarrow\) Linear. \- Move InfantMort to the "Dependent" window and College and SthDUMMY to the "Independent" window. \- Click Statistics and check "Descriptives" Click Continue to return to the "Linear Regression" screen. \- Click OK a. State the unstandardized multiple regression equation. $$ Y=+X_{1}+X_{2} $$ b. State the standardized multiple regression equation. What is the direction of each relationship? Which independent variable had the stronger effect on InfantMort? $$ Z y=Z_{1}+Z_{2} $$ c. Report the value of \(R^{2}\). What percentage of the variance in InfantMort is explained by the two independent variables combined? How does this compare to the amount of the variance explained by each independent variable alone? (HINTS: \(\mathrm{R}^{2}\) is in the "Model Summary" box and you can compute \(\mathrm{r}^{2}\) from the r's in the "Correlations" window of the output.)

Twelve families have been referred to a counselor. and she has rated each of them on a scale that measures family cohesiveness. Also, she has information on family income and number of children currently living at home. Take family cohesion as the dependent variable. $$ \begin{array}{lccc} \hline \hline \text { Family } & \begin{array}{c} \text { Cohesion } \\ \text { Score } \end{array} & \text { Income } & \begin{array}{c} \text { Number of } \\ \text { Children } \end{array} \\ \hline \text { A } & 10 & 30,000 & 5 \\ \text { B } & 10 & 70,000 & 4 \\ \text { C } & 9 & 35,000 & 4 \\ \text { D } & 5 & 25,000 & 0 \\ \text { E } & 1 & 55,000 & 3 \\ \text { F } & 7 & 40,000 & 0 \\ \text { G } & 2 & 60,000 & 2 \\ \text { H } & 5 & 30,000 & 3 \\ \text { I } & 8 & 50,000 & 5 \\ \text { J } & 3 & 25,000 & 4 \\ \text { K } & 2 & 45,000 & 3 \\ \text { L } & 4 & 50,000 & 0 \\ \hline \end{array} $$ a. Find the multiple regression equations (unstandardized). b. What level of cohesion would be expected in a family with an income of \(\$ 20,000\) and six children? c. Compute beta-weights for each independent variable and compare their relative effect on cohesion. Which was the more important factor? d. Compute \(R^{2}\). e. Write a paragraph summarizing your findings.

Load the GSS2012 data set and analyze the relationships between church attendance (the dependent or \(Y\) variable), and two independent variables: age and number of children. You analyzed some of these relationships in Problem \(12.7 .\) Now, we will use both partial correlation and multiple regression and correlation to further examine these relationships. \- Zero-order correlations. \- Click Analyze \(\rightarrow\) Correlate \(\rightarrow\) Bivariate. \- Enter attend, age, and childs in the Variables: window. \- Click OK. \- Partial Correlation Analysis: Is the relationship between attend and age affected by childs? \- Click Analyze \(\rightarrow\) Correlate \(\rightarrow\) Partial. \- Enter attend and age in the "Variables:" window and childs in the "Controlling for:" window. \- Click OK. \- Partial Correlation Analysis: Is the relationship between attend and childs affected by age? \- Click Analyze \(\rightarrow\) Correlate \(\rightarrow\) Partial. \- Enter attend and childs in the "Variables:" window and age in the "Controlling for:" window. \- Click OK. \- Multiple Regression and Correlation: What are the effects of childs and age on attend? \- Click Analyze \(\rightarrow\) Regression \(\rightarrow\) Linear. \- Move attend to the "Dependent" window and age and childs to the "Independent" window \- Click Statistics and check "Descriptives". Click Continue to retum to the "Linear Regression" screen. \- Click OK. a. Summarize the results for the partial correlation analysis in a paragraph in which you report the value of the zero-order and partial correlations for all relationships. Does the relationship between attend and age seem to be direct? How about the relationship between attend and childs? b. State the unstandardized multiple regression equation. (HINT: The values for a and b are in the "Coefficients" box of the output, under the column labeled B. The value in the first row is a and the values in the second and third rows are the slopes or b.) $$ Y=+X_{1}+X_{2} $$ c. State the standardized multiple regression equation. What is the direction of each relationship? Which independent variable had the stronger effect on attend? (HINT: The beta-weights are in the "Coefficients" box, under "Standardized Coefficients" and "Beta"). $$ Z_{y}=Z_{1}+Z_{2} $$ d. Report the value of \(R^{2}\). What percentage of the variance in attend is explained by the two independent variables combined? How does this compare to the amount of the variance explained by each independent variable alone? (HINTS: \(R^{2}\) is in the "Model Summary" box. You can compute \(r^{2}\) from the r's in the "Correlations" window of the output.)
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