Question

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.

Short answer

The expected cohesion score is derived from the regression equation. Beta weights indicate which variable has more impact. R² shows the model's explanatory power.

Step-by-step solution

- Organize the Data

Create a table with columns for Cohesion, Income, and Number of Children for each family.

- Define the Regression Model

Set up the multiple regression equation: \[ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \, \epsilon \]where Y is the Cohesion Score, \(X_1\) is the Income, and \(X_2\) is the Number of Children.

- Compute Coefficients

Use statistical software or calculations to find the coefficients \(\beta_0\), \(\beta_1\), and \(\beta_2\).

- Find Multiple Regression Equation

Substitute the computed coefficients back into the regression model from Step 2.

- Predict Cohesion Score for Given Income and Children

Substitute the given values (Income = 20000, Number of Children = 6) into the regression equation from Step 4 to find the expected cohesion score.

- Compute Beta Weights

Calculate the standardized beta coefficients for each independent variable \( X_1 \) and \( X_2 \) to compare their relative effect.

- Identify More Important Factor

Compare the magnitudes of the beta weights from Step 6 to determine which factor (Income or Number of Children) has a more significant impact.

- Compute R-squared (R²)

Use the regression output to find the R² value, which indicates how well the independent variables explain the variability in the cohesion scores.

- Summarize Findings

Write a brief summary interpreting the multiple regression equation, the expected cohesion for the given values, beta weights, and R² in the context of family cohesion.

Key concepts

Family Cohesion

Family cohesion refers to the emotional bonding that family members have towards one another. It represents the togetherness, support, and unity within the family unit.

In the context of this exercise, family cohesion is the dependent variable. The counselor has rated each family on a scale to measure their cohesiveness. These scores are what we are trying to predict and understand via multiple regression analysis. Knowing the level of family cohesion can help in devising strategies to improve family dynamics and provide better counseling services.

Independent Variables

Independent variables are the factors that are believed to influence the dependent variable. In this exercise, we have two independent variables: family income and the number of children currently living at home.

These variables will be used to predict the family cohesion score. Understanding their roles and how they interact with the cohesion score helps us analyze and interpret the influence of socioeconomic factors on family dynamics.

R-squared

R-squared (\( R^2 \)) is a statistical measure that explains the proportion of variance in the dependent variable that can be explained by the independent variables in the model.

For instance, an \( R^2 \) value of 0.8 would mean that 80% of the variability in the cohesion scores is explained by the income and the number of children. A higher \( R^2 \) value indicates a better fit for the model, meaning our predictions based on the independent variables are more accurate.

Beta Weights

Beta weights, or standardized coefficients, are used to compare the relative importance of each independent variable in the regression model.

In multiple regression analysis, the beta weight tells us how many standard deviations the dependent variable will change per standard deviation increase in the independent variable, keeping other variables constant. By comparing the beta weights of income and the number of children, we can determine which factor has a stronger influence on family cohesion. The variable with the largest beta weight is considered the most impactful.