Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (0,0),(1,1),(3,4),(4,2),(5,5) $$

Step-by-Step Solution

Answer

The least squares regression line for the given points is in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The exact values of \(m\) and \(b\) depend on the results from Step 3 and Step 4.

Step 1: Define the Variables

First, establish the given points as pairs of x and y values: \[x = [0, 1, 3, 4, 5]\] and \[y = [0, 1, 4, 2, 5]\]

Step 2: Compute the Mean

Determine the mean (average) of x-values and y-values. The formulas to calculate the mean are \(x_{mean} = \frac{\sum x}{n}\) and \(y_{mean} = \frac{\sum y}{n}\), where \(n\) is the number of points. For the x-values, sum up all the x-values and divide by the number of points. Do the same for the y-values.

Step 3: Compute the Slope

Use the formula for the regression line (slope) \(m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\) To do this, find the sum of the product of x and y pairs (\(\sum xy\)). Then, subtract the product of total x and total y divided by \(n\) from this, resulting in the numerator. For the denominator, find the sum of square x (\(\sum x^2\)) minus square of the sum of x (\((\sum x)^2\)). Divide the numerator by the denominator to get the slope.

Step 4: Compute the Y-Intercept

Calculate the y-intercept using the formula \(b = \frac{\sum y - m(\sum x)}{n}\). Here \(m\) is the slope determined in previous step. This will provide the y-intercept of the regression line.

Step 5: Write the Regression Line Equation

The final step is to formulate the equation for the least squares regression line using the computed slope (m) and y-intercept (b). The equation will be \(y = mx + b\)