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Chapter 1: Problem 29

If the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) lie on a straight line, what can you say about the regression line associated with these points?

Short Answer

Expert verified
The points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots, \left(x_{n}, y_{n}\right)\) lie on a straight line. The regression line, \(y = a + bx\), associated with these points will have the same slope and y-intercept as the given straight line, as it best fits the data points. Therefore, the regression line will coincide with the given straight line.

Step by step solution

01

Find the mean of x and y coordinates

To find the regression line, first calculate the mean of the x and y coordinates: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \quad \text{and} \quad \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i \]
02

Calculate the slope of the regression line

Find the slope, b, of the regression line using the following formula: \[ b = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2} \]
03

Calculate the y-intercept of the regression line

Find the y-intercept, a, of the regression line by using the mean values of x and y, and the calculated slope: \[ a = \bar{y} - b\bar{x} \]
04

Write down the equation of the regression line

Now that you have the slope and y-intercept, you can write down the equation of the regression line: \[ y = a + bx \]
05

Show that the given points lie on the regression line

Since all the given points lie on a straight line, subtracting any two points will give us the slope. Thus, for any i and j, we get: \[ \frac{y_i - y_j}{x_i - x_j} = \frac{y_j - y_k}{x_j - x_k} \] This condition guarantees that these points lie on the same straight line. Now plug the given points into the regression line equation: \[ y_i = a + bx_i \] If the points lie on the regression line, they must satisfy this equation. Since the slope, b, and y-intercept, a, already account for the relationship between the x and y coordinates, all the given points should satisfy this equation. Thus, the regression line associated with these points is the same as the given straight line.

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