Question

1. The values, x in a sample of 15 are summarized as follows

Σ(x-c) = 72,Σ(x-c)2 = 499.6
where c is a constant. Given that the sample mean is 104.8.
(a) Find the value of constant c.
(b) Find the variance of x.​

Answer 1

(a) 100

(b) 10.27

Step-by-step explanation:

We are given

No of elements = 15

Σ(x-c) = 72,Σ(x-c)^2 = 499.6

,where c is a constant

and the sample mean is 104.8.

(a) lets take into account Σ(x-c) = 72

this means that we have the sum of the 15 elements of x and each element of x is subtracted by the constant c

so the equation becomes Σxi -15c = 72, ............(1)

where xi means the sum of the elements of x from 1 to 15.

we are given the mean as 104.8

this means that Σxi/15 = 104.8

Σxi = 15*104.8 = 1572 .............(2)

substituting (2) in (1)

we get

1572 - 15c = 72

15c = 1500

c = 100

(b) We will use the property that variance does not change when a constant value is added or subtracted to the elements. This we can observe in the given equation that c is a constant that has the value of 100.

so the variance is

σ^2 = Σ(x-c)^2/15 - (Σ(x-c)/15 )^2

= 499.6/15 - (72/15)^2

= 33.31 - 23.04

σ^2 = 10.27

Therefore the variance of the given problem is 10.27.