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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

Name the Relation, Part II For each of the following statements, explain whether you think the variables will have positive correlation, negative correlation, or no correlation. Support your opinion. (a) Number of cigarettes smoked by a pregnant woman each week and birth weight of her baby (b) Years of education and annual salary (c) Number of doctors on staff at a hospital and number of administrators on staff (d) Head circumference and IQ (e) Number of movie goers and movie ticket price

31\. Putting It Together: A Tornado Model Is the width of a tornado related to the amount of distance for which the tornado is on the ground? Go to www.pearsonhighered.com/sullivanstats to obtain the data file \(4_{-} 3_{-} 31\) using the file format of your choice for the version of the text you are using. The data represent the width (yards) and length (miles) of tornadoes in the state of Oklahoma in \(2013 .\) (a) What is the explanatory variable? (b) Explain why this data should be analyzed as bivariate quantitative data. (c) Draw a scatter diagram of the data. What type of relation appears to exist between the width and length of a tornado? (d) Determine the correlation coefficient between width and length. (e) Is there a linear relation between a tornado's width and its length on the ground? (f) Find the least-squares regression line. (g) Predict the length of a tornado whose width is 500 yards. (h) Was the tornado whose width was 180 yards and length was 1.9 miles on the ground longer than would be expected? (i) Interpret the slope. (j) Explain why it does not make sense to interpret the intercept. (k) What proportion of the variability in tornado length is explained by the width of the tornado? (I) Plot residuals against the width. Does the residual plot suggest the two variables are linearly related? (m) Draw a boxplot of the residuals. Are there any outliers? (n) A major tornado was 4576 yards wide that had a length of 16.2 miles. Is this an influential tornado? Explain.

The time it takes for a planet to complete its orbit around the sun is called the planet's sidereal year. In 1618 , Johannes Kepler discovered that the sidereal year of a planet is related to the distance the planet is from the sun. The following data show the distances of the planets, and the dwarf planet Pluto, from the sun and their sidereal years. $$ \begin{array}{lcc} \text { Planet } & \begin{array}{l} \text { Distance from Sun, } x \\ \text { (millions of miles) } \end{array} & \text { Sidereal Year, } \boldsymbol{y} \\ \hline \text { Mercury } & 36 & 0.24 \\ \hline \text { Venus } & 67 & 0.62 \\ \hline \text { Earth } & 93 & 1.00 \\ \hline \text { Mars } & 142 & 1.88 \\ \hline \text { Jupiter } & 483 & 11.9 \\ \hline \text { Saturn } & 887 & 29.5 \\ \hline \text { Uranus } & 1785 & 84.0 \\ \hline \text { Neptune } & 2797 & 165.0 \\ \hline \text { Pluto } & 3675 & 248.0 \\ \hline \end{array} $$ (a) Draw a scatter diagram of the data treating distance from the sun as the explanatory variable. (b) Determine the correlation between distance and sidereal year. Does this imply a linear relation between distance and sidereal year? (c) Compute the least-squares regression line. (d) Plot the residuals against the distance from the sun. (e) Do you think the least-squares regression line is a good model? Why?

The wind chill factor depends on wind speed and air temperature. The following data represent the wind speed (in mph) and wind chill factor at an air temperature of \(15^{\circ}\) Fahrenheit. $$ \begin{array}{cc|cc} \begin{array}{l} \text { Wind } \\ \text { Speed, } x \\ (\mathrm{mph}) \end{array} & \begin{array}{l} \text { Wind Chill } \\ \text { Factor, } y \end{array} & \begin{array}{l} \text { Wind } \\ \text { Speed, } x \\ (\mathrm{mph}) \end{array} & \begin{array}{l} \text { Wind Chill } \\ \text { Factor, } y \end{array} \\ \hline 5 & 12 & 25 & -22 \\ \hline 10 & -3 & 30 & -25 \\ \hline 15 & -11 & 35 & -27 \\ \hline 20 & -17 & & \\ \hline \end{array} $$ (a) Draw a scatter diagram of the data treating wind speed as the explanatory variable. (b) Determine the correlation between wind speed and wind chill factor. Does this imply a linear relation between wind speed and wind chill factor? (c) Compute the least-squares regression line. (d) Plot the residuals against the wind speed. (e) Do you think the least-squares regression line is a good model? Why?

Professor Katula feels that there is a relation between the number of hours a statistics student studies each week and the student's age. She conducts a survey in which 26 statistics students are asked their age and the number of hours they study statistics each week. She obtains the following results: $$ \begin{array}{ll|ll|ll} \text { Age, } & \text { Hours } & \text { Age, } & \text { Hours } & \text { Age, } & \text { Hours } \\ \boldsymbol{x} & \text { Studying, } \boldsymbol{y} & \boldsymbol{x} & \text { Studying, } \boldsymbol{y} & \boldsymbol{x} & \text { Studying, } \boldsymbol{y} \\ \hline 18 & 4.2 & 19 & 5.1 & 22 & 2.1 \\ \hline 18 & 1.1 & 19 & 2.3 & 22 & 3.6 \\ \hline 18 & 4.6 & 20 & 1.7 & 24 & 5.4 \\ \hline 18 & 3.1 & 20 & 6.1 & 25 & 4.8 \\ \hline 18 & 5.3 & 20 & 3.2 & 25 & 3.9 \\ \hline 18 & 3.2 & 20 & 5.3 & 26 & 5.2 \\ \hline 19 & 2.8 & 21 & 2.5 & 26 & 4.2 \\ \hline 19 & 2.3 & 21 & 6.4 & 35 & 8.1 \\ \hline 19 & 3.2 & 21 & 4.2 & & \\ \hline \end{array} $$ (a) Draw a scatter diagram of the data. Comment on any potential influential observations. (b) Find the least-squares regression line using all the data points. (c) Find the least-squares regression line with the data point (35,8.1) removed. (d) Draw each least-squares regression line on the scatter diagram obtained in part (a). (e) Comment on the influence that the point (35,8.1) has on the regression line.
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