The mean and variance of n observations are 8 and 16 , respectively. If the sum of the first (n-1) observations is 48 and the sum of squares of the first (n-1) observations is 496 , then the value of n is:

Step-by-step solution

Let the n -th observation be x_n . Given mean x = 8 , we have: _ i=1 ^ n-1 x_i + x_n n = 8 48 + x_n n = 8 x_n = 8n - 48 Given variance ^2 = 16 , we have: _ i=1 ^ n-1 x_i^2 + x_n^2 n - ( x )^2 = 16 496 + x_n^2 n - 64 = 16 496 + x_n^2 n = 80 496 + x_n^2 = 80n Substituting x_n = 8n - 48 : 496 + (8n - 48)^2 = 80n 496 + 64n^2 - 768n + 2304 = 80n 64n^2 - 848n + 2800 = 0 Dividing by 16 : 4n^2 - 53n + 175 = 0 4n^2 - 28n - 25n + 175 = 0 4n(n - 7) - 25(n - 7) = 0 (4n - 25)(n - 7) = 0 Since n must be an integer, n = 7 . Answe