Problem 6 - College Algebra and Calculus: An Applied Approach

Question

Find the least squares regression line for the points. Use the regression capabilities of a graphing utility or a spreadsheet to verify your results. Then plot the points and graph the regression line. $$ (-3,0),(-1,1),(1,1),(3,2) $$

Step-by-Step Solution

Answer

The equation for the least squares regression line is \( y = 0.2x + 1 \).

Step 1: Calculate the mean of x and y

To begin, we first calculate the mean of the x-coordinates and the mean of the y-coordinates. This gives us \( \bar{x} = (-3-1+1+3)/4=0 \) and \( \bar{y} = (0+1+1+2)/4=1 \).

Step 2: Calculate the slope (b1) of the regression line

Next, we calculate the slope of the regression line, which is given by the formula: \( b1 = \sum_{i}^{n}(x_i-\bar{x}) * (y_i-\bar{y}) / \sum_{i}^{n}(x_i-\bar{x})^2 \). On substitution of the given values into the formula, the numerator gives 4 and the denominator gives 20. Hence, the slope \( b1 = 4/20 = 0.2 \).

Step 3: Calculate the y-intercept (b0) of the regression line

We calculate the y-intercept of the regression line using the formula: \( b0 = \bar{y} - b1 * \bar{x} \). Substituting the values calculated earlier, we get \( b0 = 1 - 0.2*0 = 1 \).

Step 4: Representation of the regression line

Combining the slope and y-intercept gives us the equation of the least squares regression line, \( y = 0.2x + 1 \). This line can be graphed on a graphing utility or spreadsheet, along with the provided points to verify the calculations.