Question

In an examination the number of candidates who secured marks between certain intervals was as follows:

Marks	0 - 19	20 - 39	40 - 59	60 - 79	80 - 99
No. of candidates	41	62	65	50	17

Estimate the number of candidates whose marks are less than 70.

Answer 1

Since the required mark is at the end of the table

We apply backward interpolation formula.

Let the marks be x and No. of candidates be y.

x	y	`Deltay`	`Delta^2y`	`Delta^3y`	`Delta^4y`
Below 20	41
		62
Below 40	103		3
		65		– 18
Below 60	168		– 15		0
		50		– 18
Below 80	218		– 33
		17
Below 100	235

`y_((x  =  x_0  +  "nh"))  = y_"n" + "n"/(1!) ∇y_"n" + ("n"("n" + 1))/(2!) ∇^2y_"n" + ("n"("n" + 1)("n" + 2))/(3!) Delta^3y_"n" + ..........`

To find y at x = 70

x = x₀ + nh

⇒ 70 = 100 + n(20)

70 – 100 = 20n

20n = – 30

⇒ n = `(- 30)/20`

n = – 1.5

`y_((x = 70)) = 235 + ((- 1.5))/(1!) (17) + ((-1.5)(- 1.5 + 1))/(2!) (- 33) + ((- 1.5)(-1.5 + 1)(-1.5 + 2))/(3!) (- 18) 6 ((1.5)(-1.5 + 1)(-1.5 + 2)(-1.5 + 3))/(4!) (0) +`

= `235 - 25.5 + ((-1.5)(-0.5)(-33))/2 + ((-1.5)(-0.5)(0.5))/6 (-18)`

= 235 – 25.5 – 12.375 – 1.125

= 235 – 39

= 196

∴ 196 candidates secured less than 70 marks