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Chapter 10: Problem 18

In regression analysis, the coefficient of determination: a. Is used to determine the proportion of the total variation in the dependent variable ( \(y\) ) explained by the independent variable \((X)\) b. Ranges between negative one and positive one. c. Is used to determine the expected value of the net income based on the regression line. d. Becomes smaller as the fit of the regression line improves.

Short Answer

Expert verified
In regression analysis, the coefficient of determination, denoted by \(R^2\), is used to determine the proportion of the total variation in the dependent variable (\(y\)) explained by the independent variable \((X)\). Thus, statement a is true. However, statements b, c, and d are false, as \(R^2\) ranges from 0 to 1, is not directly used to predict net income, and improves with larger values.

Step by step solution

01

Statement a

In regression analysis, the coefficient of determination is denoted by \(R^2\) and measures the proportion of the total variation in the dependent variable (\(y\)) that is explained by the independent variable \((X)\). This statement is true.
02

Statement b

The coefficient of determination, \(R^2\), ranges between 0 and 1. An \(R^2\) of 0 indicates that the independent variable \((X)\) does not explain any of the variation in the dependent variable (\(y\)), while an \(R^2\) of 1 indicates that the independent variable \((X)\) explains all of the variation in the dependent variable (\(y\)). Therefore, this statement is false.
03

Statement c

The coefficient of determination, \(R^2\), is not directly used to determine the expected value of the net income based on the regression line. However, \(R^2\) indicates how well the independent variable \((X)\) explains the variation in the dependent variable (\(y\)), which can be used to assess the quality of the prediction. If the task is to predict net income using regression, you'd use the regression equation rather than the coefficient of determination itself. Therefore, this statement is false.
04

Statement d

The fit of the regression line improves when the coefficient of determination (\(R^2\)) becomes larger, not smaller. An \(R^2\) closer to 1 indicates a better fit, as the independent variable \((X)\) explains a larger proportion of the variation in the dependent variable (\(y\)). Therefore, this statement is false. To sum up: - Statement a is true. - Statements b, c, and d are false.

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Key Concepts

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

Regression Analysis
Regression analysis is a fundamental statistical tool used to understand relationships between variables. It involves estimating the relationship between a dependent variable and one or more independent variables. The primary goal is to create a model that can predict or explain the behavior of the dependent variable based on known or observed changes in the independent variables.

In the context of the exercise, the coefficient of determination, denoted by \( R^2 \), plays a crucial role in regression analysis. It measures how well the independent variables explain the variation in the dependent variable. This metric provides insight into the strength and usefulness of the regression model. An \( R^2 \) value closer to 1 suggests a strong relationship, meaning the model does a good job of predicting the dependent variable.
Dependent Variable
The dependent variable, often represented as \( y \) in equations, is the outcome of interest in regression analysis. It is what the researcher is trying to predict or explain, and its values are assumed to depend on the independent variables.

Understanding the dependent variable is key when interpreting the coefficient of determination. If \( R^2 \) is high, it implies that the variation in the dependent variable can largely be explained by the independent variables included in the model. It's essential to choose a relevant dependent variable for the analysis because an inappropriate choice can lead to misleading conclusions.
Independent Variable
Independent variables, or predictors, are the inputs of the regression model. Represented as \( X \) in the regression equation, these variables are used to explain or predict changes in the dependent variable.

In the given exercise, the coefficient of determination\( R^2 \) assesses how well these independent variables explain the variability of the dependent variable. Each independent variable contributes a certain amount to this explanation, and the cumulative explanatory power is reflected in the \( R^2 \) value. When selecting independent variables for a model, it is important to consider their relevance and potential for producing a valid and reliable prediction.

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Most popular questions from this chapter

A regression equation is set up, where the dependent variable is total costs and the independent variable is production. A correlation coefficient of 0.70 implies that: a. The coefficient of determination is negative. b. The level of production explains \(49 \%\) of the variation in total costs c. There is a slightly inverse relationship between production and total costs. A correlation coefficient of 1.30 would produce a regression line with better fit to the data.

What two assumptions are frequently made when estimating a cost function?

Perfect Fit operates a chain of 10 retail department stores. Each department store makes its own purchasing deci sions. Carl Hart, assistant to the president of Perfect Fitt, is interested in better understanding the drivers of purchasing department costs. For many years, Perrect Fit has allocated purchasing department costs to products on the basis of the dollar value of merchandise purchased. A \(\$ 100\) item is allocated 10 times as many overnead costs associated with the purchasing department as a s10 itemm. Hart recently attended a seminar titled "Cost Drivers in the Retail Industry." In a presentation at the seminar, Kaliko Fabrics, a leading competitor that has implemented activity-based costing, reported num ber of purchase orders and number of suppliers to be the two most important cost drivers of purchasing department costs. The dollar value of merchandise purchased in each purchase order was not found to be a significant cost driver. Hart interviewed several members of the purchasing department at the Perfect fitt Hart collects the following data for the most recent year for Perfect fit's 10 retail department stores: Hart decides to use simple regression analysis to examine whether one or more of three variables (the last three columns in the table) are cost drivers of purchasing department costs. Summary results for these regressions are as follows: \\[\text { Regression } 1: \mathrm{PDC}=a+(b \times \mathrm{MPS})\\] 1\. Compare and evaluate the three simple regression models estimated by Hart. Graph each one. Also, use the format employed in Exhibit \(10-18\) (page 406 ) to evaluate the information. 2\. Do the regression results support the Kaliko Fabrics' presentation about the purchasing department's cost drivers? Which of these cost drivers would you recommend in designing an ABC system? 3\. How might Hart gain additional evidence on drivers of purchasing department costs at each of Perfect Fit's stores?

Describe the conference method for estimating a cost function. What are two advantages of this method?

A firm uses simple linear regression to forecast the costs for its main product line. If fixed costs are equal to \(\$ 235,000\) and variable costs are \(\$ 10\) per unit, how many units does it need to sell at \(\$ 15\) per unit to make a \(\$ 300,000\) profit? a. 21,400 b. 47,000 c. 60,000 d. 107,000
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