The variance of a sample was reported to be 144. The report indicated that = 7200. What has been the sample size?
A)49
B)50
C)51
D)52

Question

The variance of a sample was reported to be 144. The report indicated that  = 7200. What has been the sample size?
A)49 
B)50 
C)51 
D)52

Quizplus · Accepted Answer

We know that the sample variance (s^2) is equal to the sum of squared deviations (SS) divided by the degrees of freedom (df), which is one less than the sample size (n-1):

s^2 = SS/df

We also know that the sum of squared deviations is equal to the sum of squares minus the square of the sum:

SS = ∑(X - X̄)^2
SS = ∑X^2 - (ΣX)^2/n

Substituting this into the first equation and solving for n:

s^2 = [∑X^2 - (ΣX)^2/n]/(n-1)
(n-1)s^2 = ∑X^2 - (ΣX)^2/n
n(∑X^2 - (ΣX)^2/n) = (n-1)s^2
n∑X^2 - (ΣX)^2 = (n-1)s^2n
n∑X^2 - (ΣX)^2 = n(s^2)
n = [(ΣX)^2]/[∑X^2 - n(s^2)]

Plugging in the values given in the problem:

n = [(11eb0c4d_4233_fb6a_890e_c7d14eaadc01_TB2206_11)^2]/[∑X^2 - n(s^2)]
n = (7200)^2/[∑X^2 - 51(144)]
n = 51

Therefore, the sample size must be 51.

The Variance of a Sample Was Reported to Be 144