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Chapter 13: Problem 2

Explain the meaning of independent and dependent variables for a regression model.

Short Answer

Expert verified
In a regression model, independent variables are the factors one can control or change, which are hypothesized to be potentially causing changes or impacts on the dependent variable. The dependent variable is what you aim to predict or explain through the regression model.

Step by step solution

01

Clarify Key Terms

In a regression model, two key terms are essential. These are the independent variable(s) and the dependent variable. They are foundational in any statistical analysis setting.
02

Define Independent Variables

Independent variables, often designated as X, are values that can be changed or controlled in a given model or experiment. They are the factors or conditions that you manipulate in an effort to test their impact on the dependent variable.
03

Define Dependent Variables

Dependent variables, often represented by Y, are the values that result from the independent variables. They are the output or outcome of interest, which we aim to predict or explain through the model.
04

Explain Variables Role in Regression

In a regression model, one assumes that the dependent variable is directly influenced by the independent variables. This model attempts to illustrate this relationship mathematically so that a change in the independent variable would correspond in a certain change in the dependent variable.
05

Summary

To summarize, within a regression model, independent variables are the factors that are postulated to influence or produce changes in what's being studied鈥� the dependent variable. The dependent variable is, thus, what you aim to predict or explain.

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Most popular questions from this chapter

The following table gives information on the incomes (in thousands of dollars) and charitable contributions (in hundreds of dollars) for the last year for a random sample of 10 households. $$ \begin{array}{rc} \hline \text { Income } & \text { Charitable Contributions } \\ \hline 76 & 15 \\ 57 & 4 \\ 140 & 42 \\ 97 & 33 \\ 75 & 5 \\ 107 & 32 \\ 65 & 10 \\ 77 & 18 \\ 102 & 28 \\ 53 & 4 \\ \hline \end{array} $$ a. With income as an independent variable and charitable contributions as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y^{\circ}}\) b. Find the regression of charitable contributions on income. c. Briefly explain the meaning of the values of \(a\) and \(b\). d. Calculate \(r\) and \(r^{2}\) and briefly explain what they mean. e. Compute the standard deviation of errors. f. Construct a \(99 \%\) confidence interval for \(B\). g. Test at a \(1 \%\) significance level whether \(B\) is positive. h. Using a \(1 \%\) significance level, can you conclude that the linear correlation coefficient is different from zero?

The following data give the ages (in years) of husbands and wives for six couples. $$ \begin{array}{l|cccccc} \hline \text { Husband's age } & 43 & 57 & 28 & 19 & 35 & 39 \\ \hline \text { Wife's age } & 37 & 51 & 32 & 20 & 33 & 38 \\ \hline \end{array} $$ a. Do you expect the ages of husbands and wives to be positively or negatively related? b. Plot a scatter diagram. By looking at the scatter diagram, do you expect the correlation coefficient between these two variables to be close to zero, 1, or \(-1 ?\) c. Find the correlation coefficient. Is the value of \(r\) consistent with what you expected in parts a and b? d. Using a \(5 \%\) significance level, test whether the correlation coefficient is different from zero.

Briefly explain the assumptions of the population regression model.

The following data give the experience (in years) and monthly salaries (in hundreds of dollars) of nine randomly selected secretaries. $$ \begin{array}{l|rrrrrrrrr} \hline \text { Experience } & 14 & 3 & 5 & 6 & 4 & 9 & 18 & 5 & 16 \\ \hline \text { Monthly salary } & 62 & 29 & 37 & 43 & 35 & 60 & 67 & 32 & 60 \\\ \hline \end{array} $$ a. Find the least squares regression line with experience as an independent variable and monthly salary as a dependent variable. b. Construct a \(98 \%\) confidence interval for \(B\). c. Test at the \(2.5 \%\) significance level whether \(B\) is greater than zero.

For a sample data set, the linear correlation coefficient \(r\) has a positive value. Which of the following is true about the slope \(b\) of the regression line estimated for the same sample data? a. The value of \(b\) will be positive. b. The value of \(b\) will be negative. c. The value of \(b\) can be positive or negative.
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