Question

SOC A scale measuring support for increases in the national defense budget has been administered to a sample. The respondents have also been asked to indicate how many years of school they have completed and how many years, if any, they served in the military. Take "support" as the dependent variable. $$ \begin{array}{cccc} \hline \text { Case } & \text { Support } & \begin{array}{c} \text { Years of } \\ \text { School } \end{array} & \begin{array}{c} \text { Years of } \\ \text { Service } \end{array} \\ \hline \text { A } & 20 & 12 & 2 \\ \text { B } & 15 & 12 & 4 \\\ \text { C } & 20 & 16 & 20 \\ \text { D } & 10 & 10 & 10 \\ \text { E } & 10 & 16 & 20 \\ \text { F } & 5 & 8 & 0 \\ \text { G } & 8 & 14 & 2 \\\ \text { H } & 20 & 12 & 20 \\ \text { ? } & 10 & 10 & 4 \\ \text { J } & 20 & 16 & 0 \end{array} $$ a. Compute the partial correlation coefficient for the relationship between support \((Y)\) and years of school \((X)\) while controlling for the effect of years of service \((Z) .\) What effect does this have on the bivariate relationship? Is the relationship between support and years of school direct? b. Compute the partial correlation coefficient for the relationship between support ( \(Y\) ) and years of service \((X)\) while controlling for the effect of years of school \((Z) .\) What effect does this have on the bivariate relationship? Is the relationship between support and years of service direct? (HINT: You will need this partial correlation to compute the multiple correlation coefficient.) c. Find the unstandardized multiple regression equation with school \(\left(X_{1}\right)\) and service \(\left(X_{2}\right)\) as the independent variables. What level of support would be expected in a person with 13 years of school and 15 years of service? d. Compute beta-weights for each independent variable. Which has the stronger impact on turnout? e. Compute the multiple correlation coefficient \((R)\) and the coefficient of multiple determination \(\left(R^{2}\right)\). How much of the variance in support is explained

Short answer

a. Calculating the partial correlation coefficients and analyzing them shows the relationships' strengths. b. Same for years of service, controlling for school. c. Solve the multiple regression equation. d. Beta weights inform variable impact. e. R and R² indicate explained variance.

Step-by-step solution

- Define Variables

Let 'Support' be the dependent variable (Y), 'Years of School' be the first independent variable (X1), and 'Years of Service' be the second independent variable (X2).

- Calculate Means

Calculate the mean of 'Support', 'Years of School', and 'Years of Service'.

- Compute Covariances

Calculate the covariances between (Y and X1), (Y and X2), and (X1 and X2). Use the formula \[\text{Cov}(A, B) = \frac{1}{n-1} \sum (A_i - \bar{A})(B_i - \bar{B})\] for each pair.

- Compute Partial Correlation Coefficients

Use the formula for partial correlation coefficients \[\rho_{YX1|X2} = \frac{\rho_{YX1} - \rho_{YX2}\rho_{X1X2}}{\sqrt{(1-\rho_{YX2}^2)(1-\rho_{X1X2}^2)}}\] to find the partial correlation between Support and Years of School while controlling for Years of Service, and between Support and Years of Service while controlling for Years of School.

- Interpret Partial Correlation Coefficients

Interpret the results to determine the effect of controlling for the other variable. A direct relationship means that the partial correlation remains strong and significant after controlling for the other variable.

- Multiple Regression Equation

Using the unstandardized regression coefficients, compute the multiple regression equation \[ Y = b_0 + b_1X1 + b_2X2 \]. Find the coefficients using the normal equations.

- Predict Support

Use the multiple regression equation to predict the level of support for a person with 13 years of school and 15 years of service.

- Compute Beta Weights

Use the formula for standardized regression coefficients (beta weights) \[\beta_i = b_i \frac{s_{Xi}}{s_Y}\] to find which independent variable has a stronger impact on the dependent variable.

- Multiple Correlation Coefficient and Coefficient of Determination

Calculate the multiple correlation coefficient (R) and the coefficient of multiple determination (R²) to determine how much variance in support is explained by the independent variables combined.

Key concepts

Support for National Defense

In this exercise, we measure support for increases in the national defense budget. Support is the dependent variable (Y), reflecting how much respondents favor increased national defense spending. The data provides varying levels of support, with the assumption that different educational backgrounds and military experiences might influence these sentiments.
Understanding this concept is essential because it helps us to see how different factors, like years spent in school or military service, can relate to public opinions about defense spending.
This knowledge can be crucial for policymakers who wish to gauge public opinion or predict future trends in national defense support.

Years of School

Years of School (X1) is one of the independent variables in this exercise. It represents the total number of years an individual has spent in formal education. In the context of our analysis:

• We want to understand if there is a correlation between education and support for national defense.
• The notion being tested is whether more years in school might lead to higher or lower levels of support for defense spending.

In our multiple regression analysis, we will analyze this alongside years of military service to see how strong this influence is when isolating the effect of each variable.

Years of Military Service

Years of Military Service (X2) is another critical independent variable. This variable measures the total number of years a respondent has spent in the military. We are interested in understanding how military experience impacts support for national defense.
In our analysis:

• We look at whether more years in military service influence higher levels of support for national defense due to inherent biases developed during military service.
• Controlling for the years of schooling will help determine the true effect of military service on support for national defense.

We test this through calculating partial correlation coefficients and interpreting the direct and controlled relationships.

Multiple Regression Analysis

Multiple regression analysis helps us understand the relationship between one dependent variable and multiple independent variables. In this exercise:
We aim to create a multiple regression equation that predicts support for national defense based on years of school and years of military service. The general form of the equation is given by:
\[ Y = b_0 + b_1 X_1 + b_2 X_2 \]

• This method enables us to see how each year of schooling (X1) and each year of service (X2) contribute to support levels (Y).
• It also allows us to predict specific values, for instance, predicting support for someone with 13 years of school and 15 years of service.

Multiple regression analysis helps us see beyond simple bivariate correlations and understand the combined influence of multiple factors.