Problem 5 - 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. $$ (-2,-1),(0,0),(2,3) $$

Step-by-Step Solution

Answer

The least squares regression line for the points (-2,-1),(0,0),(2,3) is \[y = 0.67 + 1.16x\]

Step 1: Calculate the Means of X and Y

First calculate the mean of x values and y values: \[mean_x = \frac{-2+0+2}{3} = 0\], \[mean_y = \frac{-1+0+3}{3} = 0.67\]

Step 2: Calculate the slope (b)

The formula for the slope (b) is: \[b = \frac{\sum{(x_i - mean_x) \cdot (y_i - mean_y)}}{\sum{(x_i - mean_x)^2}}\] Inserting the given values in this formula, we get: \[b = \frac{(-2-0) \cdot (-1-0.67) + (0-0)\cdot(0-0.67) + (2-0) \cdot (3 - 0.67)}{(-2-0)^2 + (0-0)^2 + (2-0)^2} = 1.16\]

Step 3: Calculate the y-intercept (a)

The y-intercept (a) is calculated using the formula: \[a = mean_y - b \cdot mean_x\] which resolves to: \[a = 0.67 - 1.16 \cdot 0 = 0.67\]

Step 4: Write the equation of the line

The equation of the regression line is: y = ax + b. So substituting our previously found a and b values we get the following equation: \[y = 0.67 + 1.16x\]

Step 5: Graph the points and the regression line

Plot the points: (-2,-1),(0,0),(2,3) on a graph as well as the line given by \[y = 0.67 + 1.16x\]. This step can be achieved using a graphing utility or manually.