12. Regression

Linear Regression & Least Squares Method

12. Regression

Linear Regression & Least Squares Method

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Multiple Choice

A regional sales manager records data on the number of clients a salesperson contacts in a week (x) and the total sales generated that week (y). The data from 10 salespeople is shown below. Find the equation of the regression line and use it to predict sales if the salesperson contacts (a) 6 clients; (b) 40 clients

$\left(a\right)2,997;\left(b\right)19,147$

$\left(a\right)2,997;\left(b\right)3,000$

$\left(a\right)2,703;\left(b\right)18,853$

$\left(a\right)2,703;\left(b\right)3,000$

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the equation of the regression line (y = mx + b) using the given data, where x represents the number of clients contacted and y represents the sales generated. Then, use the regression equation to predict sales for specific values of x (6 and 40).

Step 2: Calculate the mean of x and y. The mean of x (number of clients) is calculated as the sum of all x values divided by the number of data points. Similarly, the mean of y (sales) is calculated as the sum of all y values divided by the number of data points.

Step 3: Compute the slope (m) of the regression line using the formula: m = (Σ(xi - x̄)(yi - ȳ)) / (Σ(xi - x̄)^2), where xi and yi are individual data points, x̄ is the mean of x, and ȳ is the mean of y. This involves calculating the deviations of each x and y value from their respective means, multiplying these deviations, summing them, and dividing by the sum of squared deviations of x.

Step 4: Calculate the y-intercept (b) of the regression line using the formula: b = ȳ - m * x̄. Substitute the values of the mean of x, mean of y, and the slope (m) into this formula to find b.

Step 5: Write the regression equation in the form y = mx + b. Use this equation to predict sales for x = 6 and x = 40 by substituting these values into the equation.

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