Problem 1 - Basic Statistics for Business and Economics with Student

Question

The director of marketing at Reeves Wholesale Products is studying monthly sales. Three independent variables were selected as estimators of sales: regional population, per capita income, and regional unemployment rate. The regression equation was computed to be (in dollars): $$ \hat{Y}=64,100+0.394 X_{1}+9.6 X_{2}-11,600 X_{3} $$ a. What is the full name of the equation? b. Interpret the number 64,100 . c. What are the estimated monthly sales for a particular region with a population of \(796,000,\) per capita income of \(\$ 6,940,\) and an unemployment rate of 6.0 percent?

Step-by-Step Solution

Answer

The equation is a Multiple Linear Regression. The 64,100 is the intercept. The estimated monthly sales are $443,652.

Step 1: Understanding the Regression Equation

The equation given is a linear regression model. The formula, \( \hat{Y} = 64,100 + 0.394 X_1 + 9.6 X_2 - 11,600 X_3 \), relates the dependent variable \( \hat{Y} \) (estimated monthly sales) to the independent variables: \( X_1 \) (regional population), \( X_2 \) (per capita income), and \( X_3 \) (regional unemployment rate).

Step 2: Identify the Full Name of the Equation

The regression equation provided is known as a "Multiple Linear Regression Equation" because it predicts the outcome of a dependent variable based on more than one independent variable.

Step 3: Interpret the 64,100 Term

The number 64,100 in the regression equation is the intercept. It represents the estimated monthly sales when all the independent variables are equal to zero. In this context, it may have limited practical interpretation because it's unlikely for population, income, and unemployment to be exactly zero.

Step 4: Input Values into the Equation

To find the estimated sales for the given data, substitute the values into the regression equation. Here, \( X_1 = 796,000 \), \( X_2 = 6,940 \), and \( X_3 = 6 \% = 0.06 \).

Step 5: Calculate Estimated Sales

Use the equation to calculate \( \hat{Y} \): \[ \hat{Y} = 64,100 + 0.394 \times 796,000 + 9.6 \times 6,940 - 11,600 \times 0.06 \] Calculate each component: - \( 0.394 \times 796,000 = 313,624 \) - \( 9.6 \times 6,940 = 66,624 \) - \( 11,600 \times 0.06 = 696 \)

Step 6: Solve for Estimated Monthly Sales

Substitute the calculated components back into the equation:\[ \hat{Y} = 64,100 + 313,624 + 66,624 - 696 \].Summing these values yields: \[ \hat{Y} = 443,652 \].Thus, the estimated monthly sales for the specified region are \$443,652.

Key Concepts

Independent Variables

Independent variables are the key inputs in a multiple linear regression equation. In this context, they refer to the factors that you think might influence the outcome you are trying to predict. For the marketing analysis at Reeves Wholesale Products, three independent variables were chosen: the regional population, per capita income, and regional unemployment rate.

Why are these variables termed "independent"? It's because their values can change without being directly affected by the variable they are predicting—in this case, the monthly sales. They provide the data needed to form a logical estimation of the dependent variable based on observable trends and patterns.

When choosing independent variables, it's quite important to ensure that they have a reasonable predictive power for the outcome in question. This means they should logically relate to how changes might influence sales.

• **Regional Population,** which indicates how many potential customers are present in the area.
• **Per Capita Income,** reflecting the average purchasing power of people in the region.
• **Regional Unemployment Rate,** as a higher rate might suggest fewer people with money to spend, possibly lowering sales.

By evaluating these variables, we can better understand and forecast the dependent variable, which in this scenario is the monthly sales figure.

Dependent Variable

The dependent variable in regression analysis is the variable that we are trying to predict or estimate. It is dependent on the independent variables because its value is determined by them. In the exercise regarding Reeves Wholesale Products, the dependent variable is the monthly sales of their products within a particular region.

This variable is termed "dependent" because, unlike independent variables, it is directly influenced by changes in the other variables being analyzed. Imagine if the population suddenly increased; one might predict an increase in sales due to a larger customer base. Similarly, rises in per capita income could signal higher spending power, possibly boosting sales. When setting up a regression model:

• The dependent variable must be clearly defined and quantifiable.
• It represents the outcome or effect within the set parameters.

In practical terms, understanding the dynamics between independent and dependent variables is vital for predicting outcomes, forming business strategies, or testing new hypotheses. By using past data trends, companies can infer how changes in certain variables may influence future sales outcomes.

Regression Equation

A regression equation is a mathematical representation that helps us understand the relationship between one dependent variable and one or more independent variables. In the context of this exercise, the regression equation used is a multiple linear regression equation:\[ \hat{Y} = 64,100 + 0.394 X_1 + 9.6 X_2 - 11,600 X_3 \]This formula allows us to estimate the monthly sales \( \hat{Y} \) as a function of three factors—regional population \( X_1 \), per capita income \( X_2 \), and regional unemployment rate \( X_3 \). Each of these independent variables comes with a coefficient that represents its weight or significance in predicting the dependent variable.

Key Components of the Regression Equation:
- **Intercept (64,100):** It predicts the value of the dependent variable (monthly sales) when all independent variables are zero. While this may not always have practical significance, it is essential in forming the equation.
- **Coefficients (0.394, 9.6, -11,600):** These numbers indicate the change in the dependent variable for a one-unit change in each independent variable, assuming other variables remain constant. For instance, a 0.394 coefficient for population suggests that for each additional person, sales increase by 0.394 dollars.

• 0.394 for population \( X_1 \)
• 9.6 for income \( X_2 \)
• -11,600 for unemployment rate \( X_3 \)

Thus, regression equations provide a comprehensive tool to predict outcomes based on observed data by assessing how multiple factors contribute collectively to the variable of interest. This is crucial for businesses that rely on data-driven forecasts to strategize and make informed decisions.