Question

The sales manager of a large company selected a random sample of \(n=10\) salespeople and determined for each one the values of \(x=\) years of sales experience and \(y=\) annual sales (in thousands of dollars). A scatterplot of the resulting \((x, y)\) pairs showed a marked linear pattern. a. Suppose that the sample correlation coefficient is \(r=\) \(.75\) and that the average annual sales is \(\bar{y}=100\). If a particular salesperson is 2 standard deviations above the mean in terms of experience, what would you predict for that person's annual sales? b. If a particular person whose sales experience is \(1.5\) standard deviations below the average experience is predicted to have an annual sales value that is 1 standard deviation below the average annual sales, what is the value of \(r\) ?

Short answer

a. The predicted annual sales for the salesperson would be 1.5 standard deviations above the average annual sales. b. The correlation \(r\) would be 0.67.

Step-by-step solution

Predicting the Annual Sales

The sample correlation coefficient \(r\) is used to show the strength and direction of the linear relationship between the two variables. Here, \(r = 0.75\) suggests a strong positive relationship. The calculation implies that a deviation in sales experience leads to \(r = 0.75\) times deviation in the annual sales. Given that a salesperson is 2 standard deviations above the mean in terms of experience, then the predicted annual sales would be \(2r = 0.75*2 = 1.5\) standard deviations above the average annual sales.

Calculating the Annual Sales

If the average annual sales is \(\bar{y}=100\), and the deviation is 1.5 standard deviations above the mean, then you would need to determine the standard deviation of the annual sales. Since we lack this information, we can only provide an answer in terms of standard deviations, which is 1.5 standard deviations above the mean.

Finding the Value of \(r\)

Given a person whose sales experience is \(1.5\) standard deviations below the average experience is predicted to have an annual sales value that is \(1\) standard deviation below the average annual sales. The correlation \(r\) can thus be calculated by the ratio of predicted standard deviation in annual sales to the standard deviation in experience, which implies \(r = \frac{-1}{-1.5} = 0.67\).