Question

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (-2,2),(2,6),(3,7) $$

Short answer

The least squares regression line for the given points (-2, 2), (2, 6), and (3, 7) is \( y = \frac{35}{8} + \frac{5}{8}x \)

Step-by-step solution

Computing the Means

Firstly, find the mean of all x values and the mean of all y values.Mean of x values, \( \bar{x} = \frac{-2 + 2 + 3}{3} = 1 \) Mean of y values, \( \bar{y} = \frac{2 + 6 + 7}{3} = 5 \)

Computing the Slope

The slope of the line, \( b = \frac{\Sigma((x_i - \bar{x}) (y_i - \bar{y}))}{\Sigma((x_i - \bar{x})^2)} \)Substitute the given points into the formula:\( b = \frac{((-2-1)*(2-5) + (2-1)*(6-5) + (3-1)*(7-5))}{((-2-1)^2 + (2-1)^2 + (3-1)^2)} = \frac{5}{8} \)

Computing the y intercept

The y-intercept of the line, \( a = \bar{y} - b*\bar{x} \)Substitute the values of \( \bar{y} \), \( b \), and \( \bar{x} \) into the formula:\( a = 5 - \frac{5}{8}*1 = \frac{35}{8} \)

The Least Squares Regression Line

The least squares regression line or best fit line is \( y = a + bx \)So the equation of the line is \( y = \frac{35}{8} + \frac{5}{8}x \)