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

• A. $\left(a\right)2,997;\left(b\right)19,147$
• B. $\left(a\right)2,997;\left(b\right)3,000$
• C. $\left(a\right)2,703;\left(b\right)18,853$
• D. $\left(a\right)2,703;\left(b\right)3,000$

Answer: (B)

Verified step by step guidance

1. 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 total sales generated. Then, use the regression equation to predict sales for x = 6 and x = 40.
2. 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.
3. Step 3: Compute the slope (m) of the regression line using the formula: m = (Σ(x_i - x̄)(y_i - ȳ)) / (Σ(x_i - x̄)^2), where x̄ and ȳ are the means of x and y, respectively. 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.
4. Step 4: Calculate the y-intercept (b) using the formula: b = ȳ - m * x̄. Substitute the values of the slope (m) and the mean of x and y into this formula to find the y-intercept.
5. Step 5: Use the regression equation y = mx + b to predict sales for x = 6 and x = 40. Substitute x = 6 and x = 40 into the equation to calculate the corresponding y values (predicted sales).