Answer to Question #286377 in Microeconomics for mukti

Question

Years of Annual Sales

Salesperson Experience ($1000s)

1 1 80

2 3 97

3 4 92

4 4 102

5 6 103

6 8 111

7 10 119

8 10 123

9 11 117

10 13 136

a. Develop a scatter diagram for these data with years of experience as the independent

variable.

b. Develop an estimated regression equation that can be used to predict annual sales

given the years of experience.

c. Use the estimated regression equation to predict annual sales for a salesperson with

9 years of experience.

Answer 1

$a)$

A scatter diagram for these data is given below.

$b)$

The regression equation is of the form, $\hat{y}=ax+b$ where $a$ is the slope coefficient and $b$ is the y-intercept.

To begin with, we need to determine the values of $a$ and $b$ using the method of least squares as follows.

Now,

$n=10,\space \sum xy=8128,\space \sum x=70,\space \sum y=1080,\space \sum x^2=632,\space \bar{y}={\sum y\over n}={1080\over10}=108,\space \bar{x}={\sum x\over n}={70\over 10}=7$

$a={n\sum xy-\sum x\sum y\over n\sum x^2-(\sum x)^2}={81280-75600\over 6320-4900}={5680\over1420}=4$

The value of $b$ is given as,

$b=\bar{y}-a\bar{x}=108-(4\times 7)=108-28=80$

Therefore, our regression equation is given as,

$\hat{y}=4x+80$

$c)$

When the years of experience of a salesperson is 9, the annual sales will be, $\hat{y}=(4\times9)+80=116$

Therefore, the annual sales for a salesperson with 9 years of experience is 116 in ($1000s).