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Chapter 4: Problem 3

True or False: The least-squares regression line always travels through the point \((\bar{x}, \bar{y})\)

Short Answer

Expert verified
True, the least-squares regression line always passes through the point \( (\bar{x}, \bar{y}) \).

Step by step solution

01

Understand the Slope-Intercept Form

The equation of the least-squares regression line is usually written in the form of \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.
02

Definition of Mean Values

Here, \( \bar{x} \) is the mean value of the x-coordinates (independent variable) and \( \bar{y} \) is the mean value of the y-coordinates (dependent variable).
03

Substitute Mean Values in Regression Equation

Substitute \( \bar{x} \) into the regression equation: \( \bar{y} = m\bar{x} + b \).
04

Simplify and Analyze

Since \( \bar{y} = m\bar{x} + b \), it confirms that the point \( (\bar{x}, \bar{y}) \) lies on the least-squares regression line because it satisfies the equation.
05

Conclusion

From the simplification and analysis, it is clear that the least-squares regression line does always pass through the point \( (\bar{x}, \bar{y}) \).

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Key Concepts

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

Regression Line
A regression line is a straight line that best fits the data points on a scatter plot. It shows the relationship between two variables, typically referred to as the independent variable (x) and the dependent variable (y). The purpose of the regression line is to predict the value of the dependent variable based on the value of the independent variable. The less the data points deviate from the regression line, the more accurate the predictions are. In the context of least-squares regression, the goal is to minimize the sum of the squared differences between the observed values and the values predicted by the line. This method ensures that the regression line is as close as possible to all the data points, which provides the best possible fit.
Mean Values
Mean values play a crucial role in determining the least-squares regression line. The mean, often represented by \(\bar{x}\) and \(\bar{y}\), is the average of all values in a dataset. For the independent variable x, we calculate the mean by summing all x-values and dividing by the number of values. The same is done for the y-values to get \(\bar{y}\).
The mean values help in simplifying the regression equation and analyzing data points. Specifically, the regression line always passes through the point (\bar{x}, \bar{y}) because these mean values represent the balanced central tendency of the data. This balance is foundational to the formula used in least-squares regression.
Slope-Intercept Form
The slope-intercept form of the equation of a line is written as \(y = mx + b\). In this equation, \(m\) represents the slope of the line, and \(b\) is the y-intercept.
The slope (m) describes the steepness of the line and indicates the rate of change of the dependent variable with respect to the independent variable. A positive slope means that as x increases, y also increases, and a negative slope means that as x increases, y decreases.
The y-intercept (b) is the value of y when x is zero. It indicates where the line crosses the y-axis.
When this form of the equation is used in the context of the least-squares regression line, we substitute the mean values (\(\bar{x}, \bar{y}\)) into the equation to confirm that the line indeed passes through that specific point, ensuring the best fit for the data.

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Most popular questions from this chapter

Consider the following data set: $$ \begin{array}{lllllllll} \hline x & 5 & 6 & 7 & 7 & 8 & 8 & 8 & 8 \\\ \hline y & 4.2 & 5 & 5.2 & 5.9 & 6 & 6.2 & 6.1 & 6.9 \\ \hline x & 9 & 9 & 10 & 10 & 11 & 11 & 12 & 12 \\ \hline y & 7.2 & 8 & 8.3 & 7.4 & 8.4 & 7.8 & 8.5 & 9.5 \\ \hline \end{array} $$ (a) Draw a scatter diagram with the \(x\) -axis starting at 0 and ending at 30 and with the \(y\) -axis starting at 0 and ending at 20 . (b) Compute the linear correlation coefficient. (c) Now multiply both \(x\) and \(y\) by 2 . (d) Draw a scatter diagram of the new data with the \(x\) -axis starting at 0 and ending at 30 and with the \(y\) -axis starting at 0 and ending at 20. Compare the scatter diagrams. (e) Compute the linear correlation coefficient. (f) Conclude that multiplying each value in the data set by a nonzero constant does not affect the correlation between the variables.

Attending Class The following data represent the number of days absent, \(x\), and the final grade, \(y,\) for a sample of college students in a general education course at a large state university. $$ \begin{array}{lllllllllll} \hline \begin{array}{l} \text { No. of } \\ \text { absences, } x \end{array} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \begin{array}{l} \text { Final } \\ \text { grade, } y \end{array} & 89.2 & 86.4 & 83.5 & 81.1 & 78.2 & 73.9 & 64.3 & 71.8 & 65.5 & 66.2 \\ \hline \end{array} $$ (a) Find the least-squares regression line treating number of absences as the explanatory variable and final grade as the response variable. (b) Interpret the slope and \(y\) -intercept, if appropriate. (c) Predict the final grade for a student who misses five class periods and compute the residual. Is the final grade above or below average for this number of absences? (d) Draw the least-squares regression line on the scatter diagram of the data. (e) Would it be reasonable to use the least-squares regression line to predict the final grade for a student who has missed 15 class periods? Why or why not?

In a recent Harris Poll, a random sample of adult Americans (18 years and older) was asked, "When you see an ad emphasizing that a product is 'Made in America,' are you more likely to buy it, less likely to buy it, or neither more nor less likely to buy it?" The results of the survey, by age group, are presented in the contingency table below. 3 $$\begin{array}{lrrrrr} & 18-34 & 35-44 & 45-54 & 55+ & \text { Total } \\ \hline \text { More likely } & 238 & 329 & 360 & 402 & 1329 \\\\\hline \text { Less likely } & 22 & 6 & 22 & 16 & 66 \\\\\hline \begin{array}{l}\text { Neither more } \\\\\text { nor less likely }\end{array} & 282 & 201 & 164 & 118 & 765 \\\\\hline \text { Total } & 542 & 536 & 546 & 536 & 2160\end{array}$$ (a) How many adult Americans were surveyed? How many were 55 and older? (b) Construct a relative frequency marginal distribution. (c) What proportion of Americans are more likely to buy a product when the ad says "Made in America"? (d) Construct a conditional distribution of likelihood to buy "Made in America" by age. That is, construct a conditional distribution treating age as the explanatory variable. (e) Draw a bar graph of the conditional distribution found in part (d). (f) Write a couple sentences explaining any relation between likelihood to buy and age.

(a) By hand, draw a scatter diagram treating \(x\) as the explanatory variable and y as the response variable. (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. (c) Graph the line found in part (b) on the scatter diagram. (d) By hand, determine the least-squares regression line. (e) Graph the least-squares regression line on the scatter diagram. (f) Compute the sum of the squared residuals for the line found in part (b). (g) Compute the sum of the squared residuals for the leastsquares regression line found in part (d). (h) Comment on the fit of the line found in part (b) versus the least-squares regression line found in part ( \(d\) ). $$ \begin{array}{rrrrrr} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 7 & 6 & 3 & 2 & 0 \\ \hline \end{array} $$

In Problems \(17-20,\) (a) draw a scatter diagram of the data, (b) by hand, compute the correlation coefficient, and \((c)\) determine whether there is a linear relation between \(x\) and \(y\). $$ \begin{array}{rrrrrr} \hline x & 2 & 3 & 5 & 6 & 6 \\ \hline y & 10 & 9 & 7 & 4 & 2 \\ \hline \end{array} $$
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