Question

Suppose for 40 observations, the variance is 50. If all the observations are increased by 20, the variance of these increased observation will be

Select one:
a. 50
b. 70
c. 50/20
d. 40
e. 50-20=30



Note: Answer D is NOT the correct answer. Please find the correct answer. Any answer without justification will be rejected automatically.

Answer 1

a) 50

Explanation:

The variance will not change as all the observations are increased uniformly.

Proof:

Variance formula:

`s^(2) = (\sum x_i^(2) )/(n) -((\sum x_i)^(2) )/(n^(2) )`

When the obervations are inc by 20,

`s_1^(2) = (\sum (x_i + 20)^(2) )/(n) -((\sum (x_i + 20))^(2) )/(n^(2) )\\\\=(\sum(x_i^(2) + 2*20*x_i + 20^(2) ))/(n) - ((\sum x_i +20n)^(2) )/(n^(2) ) \\\\=(\sum x_i^(2) + 40\sum x_i + 20^(2)n )/(n) - ((\sum x_i)^(2) +2*20n\sum x_i + 20^(2) n^(2) )/(n^(2) ) \\\\= (\sum x_i^(2))/(n) - ((\sum x_i)^(2))/(n^(2) ) +(40\sum x_i)/(n) + 20^(2) - (40\sum x_i)/(n) - 20^(2)\\\\s_1^(2)= (\sum x_i^(2))/(n) - ((\sum x_i)^(2))/(n^(2) )\\\\=s^(2)`

Therefore variance doesn't change