Question

A company wants to estimate, how its monthly costs are related to its monthly output rate. The data

for a sample of nine months is tabulated below :

Using the data given above, perform following tasks:

(a) Calculate the best linear regression, where the monthly output is the dependent variable and

monthly cost is the independent variable.

(b) Use the regression line to predict the company’s monthly cost, if they decide to produce 4

tons per month.

Out Put (Tons) 1 2 4 8 6 5 8 9 7

Cost (Lakhs) 2 3 4 7 6 5 8 8 6

Answer 1

(a) Suppose$\ x_i$ denotes the output of the month and $y_i$ denotes the cost of $i^{th}$ month

Now,

$$S_{xx}=\sum (x_i)^2-n\bar x^2$$

Now from table we get that

$\bar x=\dfrac{\sum x_i}{n}=\dfrac{50}{9}\\\bar y=\dfrac{\sum y_i}{n}=\dfrac{49}{9}$

$\sum x_i^2=340\\\sum y_i^2=303\\and\ \ \sum x_iy_i=319$

Therefore we get that

$\beta_1=\dfrac{\sum x_iy_i-n\bar{x}\bar y}{\sum x_i^2-n\bar x^2}\\\ \\=\dfrac{9\times 319-50\times 49}{9\times 340-50^2}\\\ \\=\dfrac{421}{560}=319$

Corrrespondingly, we get

$\beta_o=\dfrac{49}{9}-(0.752)\cdot \dfrac{50}{9}\\\ \ \ \ \ =1.266$

Therefore the best linear line regression is

$y=1.266+(0.752)x$

(b) If the company produces 4 tons per month, then one can predict that its cost would be

$1.266+(0.752)\times 4=4.274$

Since the costs are measured in thousands of dollars , this means that the total cost would be expected to be $4274