Question

SOC Follow up on problem \(12.11\) by finding the regression coefficients for the relationship between cell phone use and urbanization. Use the Intl-Pop dataset and the Regression command. \- Load the Intl-Pop database. \- From the main menu bar, click Analyze, Regression, and Linear. \- On the "Linear Regression" screen, place CellPhones in the "Dependent" window and Urban in the "Independent" window. \- Click OK and results will be printed to the Output screen. \- The "Coefficients" box displays the value of \(a\) and \(b\) under "Unstandardized Coefficients." Look in the column labeled \(\mathrm{B}\). The value in the first row is \(a\) and the entry in the second row is \(b\). a. State the regression equation with the values for \(a\) and \(b\). b. Use the equation to predict cell phone use for nations that are \(25 \%\) urbanized \((X=25), 50 \%\) urbanized \((X=50), 75 \%\) urbanized \((X=75)\), and \(99 \%\) urbanized \((X=99)\). What happens to cell phone use as urbanization increases?

Short answer

Determine the regression coefficients \(a\) and \(b\) and use the equation \(Y = a + bX\) to predict cell phone use for given urbanization values.

Step-by-step solution

- Load the dataset

Load the Intl-Pop database. This can often be done by accessing the appropriate folder or using a data import function depending on the software being used.

- Access the Regression command

From the main menu bar, click on 'Analyze', then navigate to 'Regression', and select 'Linear'.

- Set up the regression parameters

On the 'Linear Regression' screen, place 'CellPhones' in the 'Dependent' window and 'Urban' in the 'Independent' window.

- Execute the analysis

Click 'OK' to run the regression analysis. The results will be printed to the output screen.

- Locate the coefficients

In the output screen, find the 'Coefficients' box which displays the values of the regression coefficients under 'Unstandardized Coefficients.' Look in the column labeled \(\text{B}\). The value in the first row is \(a\) (the intercept) and the value in the second row is \(b\) (the slope).

- State the regression equation

Using the values of \(a\) and \(b\), form the regression equation: \(Y = a + bX\). Here, \(Y\) represents cell phone use and \(X\) represents urbanization.

- Predict cell phone use

Substitute \(X = 25\), \(X = 50\), \(X = 75\), and \(X = 99\) into the regression equation to calculate the predicted values of \(Y\).

- Interpret the results

Based on the predictions, analyze how cell phone use changes as urbanization increases.

Key concepts

Understanding Regression Analysis

Regression analysis is a statistical method for examining the relationship between two or more variables. It allows us to understand how the dependent variable (response) changes when any one of the independent variables (predictors) is varied, while the other predictors are held fixed. For instance, in the context of the provided exercise, we are looking at how cell phone use (dependent variable) changes with urbanization (independent variable). This analysis involves creating a regression equation of the form: \[ Y = a + bX \]where \(Y\) is the dependent variable, \(X\) is the independent variable, \(a\) is the intercept, and \(b\) is the slope.

Dependent and Independent Variables

In regression analysis, it's crucial to distinguish between dependent and independent variables. The dependent variable is the outcome we aim to predict or explain, while independent variables are the predictors that influence the dependent variable.

• Dependent Variable: In the exercise, cell phone use is our dependent variable. It depends on the rate of urbanization.
• Independent Variable: Urbanization is the independent variable. We use it to predict changes in cell phone use.

Understanding this relationship is key to forming a meaningful regression equation. By knowing which variable influences which, we can make accurate predictions and better understand the underlying trends.

Predictive Modeling

Predictive modeling involves using statistical techniques to predict future outcomes based on historical data. In our exercise, we use regression analysis as a tool for predictive modeling. The regression equation \( Y = a + bX \) allows us to predict cell phone use for different levels of urbanization. **Example Predictions**:

• For a nation that is 25% urbanized, substitute \(X = 25\) into the equation.
• For 50% urbanization, use \(X = 50\).
• For 75% urbanization, substitute \(X = 75\).
• For 99% urbanization, use \(X = 99\).

These predictions help us see a pattern or trend. As urbanization increases, cell phone use may also increase, showing a positive correlation.

Using Statistical Software

Statistical software simplifies the process of conducting regression analysis. Popular software includes SPSS, R, and Python with libraries like statsmodels and scikit-learn. The exercise demonstrates using software to perform regression analysis efficiently.**Steps to Perform Regression**:

• Load the Dataset: Import the relevant data file.
• Access Regression Command: Usually found under the 'Analyze' menu, then 'Regression' and 'Linear'.
• Set Parameters: Define your dependent and independent variables.
• Run the Analysis: Click OK to execute the regression.
• View Results: Find the coefficients in the output screen under 'Unstandardized Coefficients' in the column labeled \( \text{B} \).

Using software not only speeds up calculations but also ensures accuracy and reliability of results.