Answer to Question #160047 in Statistics and Probability for Ayyan

Question

ZIMS store sells its products on installments in Karachi. The store decided to investigate the link between the life of the product and the cost. The data collected from store is shown in Table 1.1:

 

Table 1.1

LIFE OF PRODUCT (Years)

COST (Thousand Rupees)

3.0

25

3.5

28

4.0

32

4.5

34

5.0

34

3.0

30

5.5

40

4.5

40

4.0

42

3.5

42

6.0

50

7.5

50

8.0

53

8.5

55

     

a) Plot a scatter chart from the above data.

[3]

b) Find the equation of the least square regression line in the form 𝒚 = 𝒂 + 𝒃𝒙. Given the value of 𝒂 𝑎𝑛𝑑 𝒃 up to 3 decimal places. 

[3]

c) What are the exact interpretations of the estimated regression factors?

[3]

d) Using your answer of part (b). Find the cost predicted by the regression line for 𝟔. 𝟓 year’s life of a product. 

[3]

Answer 1

a)

b)

$$\bar{x}=\dfrac{70.5}{14}=5.0357, \bar{y}=\dfrac{555}{14}=39.6429$$

$$SS_{xx}=\sum x_i^2-n\bar{x}^2=43.7321$$

$$SS_{yy}=\sum y_i^2-n\bar{y}^2=1205.2143$$

$$SS_{xy}=\sum x_iy_i-n\bar{x}\bar{y}=200.6786$$

$$b=\dfrac{SS_{xy}}{SS_{xx}}=\dfrac{200.6786}{43.7321}=4.589$$

$$a=\bar{y}-b\bar{x}=39.6429-4.589(5.0357)=16.535$$

$$y=a+bx$$

$$y=16.535+4.589x$$

c)

$$r=\dfrac{SS_{xy}}{\sqrt{SS_{xx}}\sqrt{S_{yy}}}=\dfrac{200.6786}{\sqrt{43.7321}\sqrt{1205.2143}}=$$

$$=0.874$$

$$r^2=0.764$$

Strong positive correlation.

If life of product is 0 years, the cost is 16.636 thousand Rupees.

If life of product increases by 1 year, the cost increases by 4.589 thousand Rupees.

d)

$x=6.5$

$$y=16.535+4.589(6.5)=46.364$$

If life of product is 6.5 years, the cost is 46.364 thousand Rupees.