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Chapter 13: Problem 13

Explain the following. a. Population regression line b. Sample regression line c. True values of \(A\) and \(B\) d. Estimated values of \(A\) and \(B\) that are denoted by \(a\) and \(b\), respectively

Short Answer

Expert verified
The population regression line, denoted as \(Y = A + BX + u\), represents the true relationship between variables in the population where \(A\) and \(B\) are the true values of the intercept and slope, respectively. On the other hand, the sample regression line, \(y = a + bx + e\), is based on a sample from the population and is an estimation of the population regression line, where \(a\) and \(b\) are the estimated values of \(A\) and \(B\).

Step by step solution

01

Understanding Population Regression Line

The population regression line is a statistical tool that represents the relationship between two or more variables in a population. It is the 'true' regression line, and it is denoted as: \(Y = A + BX + u\), where \(Y\) is the dependent variable, \(A\) is the intercept, \(B\) is the slope (i.e. the effect that \(X\), the independent variable, has on \(Y\)), and \(u\) is the error term.
02

Understanding Sample Regression Line

The sample regression line is an estimated representation of the population regression line, based on a sample taken from the population. It is calculated using the least squares method to minimise the sum of the squared residuals. It is denoted as: \(y = a + bx + e\), where \(y\), \(a\), \(b\) and \(x\) are the sample counterparts of \(Y\), \(A\), \(B\) and \(X\), and \(e\) is the residual.
03

Understanding True Values of A and B

In the population regression line, \(A\) and \(B\) are called the true values (parameters) of the intercept and slope, respectively. They represent the exact mathematical relationship between the variables in the population. However, these true values are often unknown because we usually cannot observe the entire population.
04

Understanding Estimated Values of A and B denoted by a and b

In the Sample Regression Line, \(a\) and \(b\) are the estimated values of the population parameters \(A\) and \(B\). They are derived from our sample data and signify our best guess, based on the sample, of what \(A\) and \(B\) could be in the population.

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Most popular questions from this chapter

Explain the difference between a simple and a multiple regression model.

A population data set produced the following information. $$ \begin{aligned} &N=250, \quad \Sigma x=9880, \quad \Sigma y=1456, \quad \Sigma x y=85,080, \\ &\Sigma x^{2}=485,870, \text { and } \Sigma y^{2}=135,675 \end{aligned} $$ Find the values of \(\sigma_{e}\) and \(\rho^{2}\).

A population data set produced the following information. \(N=460, \quad \Sigma x=3920, \quad \Sigma y=2650, \quad \Sigma x y=26,570\) \(\Sigma x^{2}=48,530\), and \(\Sigma y^{2}=39,347\) Find the linear correlation coefficient \(\rho\).

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.

While browsing through the magazine rack at a bookstore, a statistician decides to examine the relationship between the price of a magazine and the percentage of the magazine space that contains advertisements. The data collected for eight magazines are given in the following table. $$ \begin{array}{l|rrrr} \hline \text { Percentage containing ads } & 37 & 43 & 58 & 49 \\ \hline \text { Price (\$) } & 5.50 & 6.95 & 4.95 & 5.75 \\ \hline \text { Percentage containing ads } & 70 & 28 & 65 & 32 \\ \hline \text { Price (\$) } & 3.95 & 8.25 & 5.50 & 6.75 \\ \hline \end{array} $$ a. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between the percentage of a magazine's space containing ads and the price of the magazine? b. Find the estimated regression equation of price on the percentage of space containing ads. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part b. d. Plot the estimated regression line on the scatter diagram of part a, and show the errors by drawing vertical lines between scatter points and the predictive regression line. e. Predict the price of a magazine with \(50 \%\) of its space containing ads. f. Estimate the price of a magazine with \(99 \%\) of its space containing ads. Comment on this finding.
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