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Chapter 1: Problem 33

Must the regression line pass through at least one of the data points? Illustrate your answer with an example.

Short Answer

Expert verified
In conclusion, the regression line does not necessarily have to pass through any of the data points. It can pass through one or more data points, but it depends on the distribution of the data points in the scatter plot. For example, consider the data points (1,3), (2,5), (3,7), and (4,9); the calculated regression line is \(y = 1.6x + 2\), which does not pass through any of the original data points.

Step by step solution

01

Understanding Regression Line

A regression line is a straight line that best represents the relationship between two variables in a scatter plot. It is used to predict the value of one variable based on the value of the other variable. The method used to create a regression line is called "least squares," which means that the line minimizes the sum of the squared distances between the data points and the line itself.
02

Analyzing the Relationship Between Data Points and Regression Line

The regression line is based on the overall pattern of the data points, which means that even though it may be close to some data points, it does not have to pass through any of them. It's possible for the regression line to pass through one or more data points, but that depends on the distribution of the data points in the scatter plot.
03

Example

Consider the following data points (x, y): \( (1, 3) \), \( (2, 5) \), \( (3, 7) \), \( (4, 9) \) For this data set, we can create a scatter plot and calculate the regression line that models the relationship between the x values and the y values:
04

Calculate the Means of X and Y

Mean of X: \((1+2+3+4)/4 = 2.5\) Mean of Y: \((3+5+7+9)/4 = 6\)
05

Calculate the Slope of the Regression Line

To calculate the slope of the regression line, we need to first calculate the covariance and variance: Covariance (\(cov(X, Y)\)): \(\sum\limits_{i=1}^{4}\frac{(x_i - \overline{X})(y_i - \overline{Y})}{4-1} =\frac{(1-2.5)(3-6) + (2-2.5)(5-6) + (3-2.5)(7-6) + (4-2.5)(9-6)}{3} = 2\) Variance (\(Var(X)\)): \(\sum\limits_{i=1}^{4}\frac{(x_i - \overline{X})^2}{4-1} =\frac{(1-2.5)^2 + (2-2.5)^2 + (3-2.5)^2 + (4-2.5)^2}{3} = 1.25\) Slope (\(m\)): \(\frac{cov(X, Y)}{Var(X)} = \frac{2}{1.25} = 1.6\)
06

Calculate the Y-Intercept of the Regression Line

To calculate the y-intercept (\(b\)) of the regression line, use the mean values: \(b = \overline{Y} - m\overline{X} = 6 - (1.6 \times 2.5) = 2\)
07

Write the Equation of the Regression Line

With the slope and y-intercept calculated, the equation of the regression line is: \(y = 1.6x + 2\)
08

Analyze the Result

Looking at the regression line equation, we see that none of the original data points are exactly on the line, which means the regression line does not pass through any of the data points. Even though the regression line does not pass through any of the data points, it still provides an adequate model for predicting the relationship between the x values and the y values. In conclusion, the regression line does not necessarily have to pass through any of the data points. It can pass through one or more data points, but it depends on the distribution of the data points in the scatter plot.

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