Question

The marks obtained by 10 candidates in English and Mathematics are given below:

Marks in English	20	13	18	21	11	12	17	14	19	15
Marks in Mathematics	17	12	23	25	14	8	19	21	22	19

Estimate the probable score for Mathematics if the marks obtained in English are 24.

Answer 1

Here n = 10.
Take the marks obtained in English and Mathematics as x and y respectively.
Let for x assumed mean be 17 and for v assumed mean be 19.
We construct the table as follows:

r = `(∑uv - 1/n ∑u ∑v)/(sqrt(∑u^2 - 1/n (∑u)^2) sqrt(∑v^2-1/n (∑v)^2)`

r = ` (135 - 1/10 (10 xx 10) )/(sqrt(120-1/10 (10)^2) sqrt(264-1/10 (10)^2)` 

r = `(125)/(sqrt110 . sqrt254) = 0.7478`

Now, `b_(yx) = (∑uv - 1/n ∑u ∑v)/(∑u^2 - 1/n (∑u)^2)  = (135 - 10)/(120 - 10) = (125)/(110) = (25)/(22) `

Here, `barx = 17 + ((-10))/10 = 16`

`bary = 19 + ((-10))/10 = 18`

Since marks obtained in English i.e., x = 24 is given
Now, using regression line of y on x : `(y - bary) = b_(yx) (x - barx)` 

y - 18 = `(25)/(22) (x - 16)`

⇒ 22y - 396 = 25x - 400

For 22y = 25x - 4 

⇒ 22y = 25 x 24 -4

y = `(596)/(22)`  = 27.09 = 27 marks approx

Probable marks of mathematics is 27, when marks obtained in English are 24.