Question

A researcher at a large company has collected data on both the beginning salary and the current salary of 48 randomly selected employees. The least‑squares regression equation for predicting their current salary from their beginning salary is y^ =−2532.7 + 2.12x.(a) The current salaries had a mean of $32,070 with a standard deviation of $15,300. The beginning salaries had a mean of $16,340 with a standard deviation of $5,970. What is the correlation between current and beginning salary?

(b) Mr. Joseph Keller started working for the company earning $22,000. What do you predict his current salary to be?(c) Kathy Jones started working for the company earning $19,000. She currently earns $40,000. What is the residual for Ms. Jones?

Answer 1

a. 0.83

b. 44107.3

c. 2252.7

Explanation:

a)

The correlation coefficient is calculated as

byx=rSy/Sx

rSy=byx*Sx

⇒r=byx*Sx/Sy

Sy=standard deviation of y=standard deviation of current salary=15300

Sx=standard deviation of x=standard deviation of beginning salary=5970

byx=2.12

r=2.12*5970/15300

r=12656.4/15300

r=0.83

There is a strong positive correlation between current and beginning salary.

b)

Beginning salary=$22000

As

y^ =−2532.7 + 2.12x.

Current salary=-2532.7+2.12*beginning salary

Current salary=-2532.7+2.12*22000

Current salary=-2532.7+46640

Current salary=$44107.3

The predicted current salary of Mr. Joseph Keller is $44107.3.

c)

Residual=y-y^=Observed-Predicted

Observed Current Salary of Kathy Jones=40000

Beginning salary of Kathy Jones=19000

y^ =−2532.7 + 2.12x.

Predicted Current salary=-2532.7+2.12*beginning salary

Predicted Current salary=-2532.7+2.12*19000

Predicted Current salary=-2532.7+40280

Predicted Current salary=$37747.3

Residual=Observed Current Salary-Predicted Current salary

Residual=40000-27747.3

Residual=$2252.7

The residual for Ms. Jones is $2252.7.