Let  x1, x2, x3 …………xn  be  n observations such that  Σxi2  =  729  and  Σxi  =  81, then a possible value of  n among the following is:

The problem involves the sum of squares of observations and the sum of the observations. We can use the formula for the sum of squares, which is Σxi^2 = n*(mean^2 + standard deviation^2), and the formula for the sum of observations, which is Σxi = n*mean.

Given that Σxi^2 = 729 and Σxi = 81, we can set up the following equations:

1. n*(mean^2 + standard deviation^2) = 729
2. n*mean = 81

We can solve the second equation for mean = 81/n and substitute this into the first equation to get:

n*((81/n)^2 + standard deviation^2) = 729 n*(6561/n^2 + standard deviation^2) = 729 6561/n + n*(standard deviation^2) = 729

We can see that n must be a factor of 6561 to make the left side equal to 729. The factors of 6561 are 1, 3, 9, 27, 81, 243, 729, 2187, 6561.

We can check these values for n in the equation. If we find a value of n that makes the equation true, then that is a possible value of n.

For example, if we try n = 9, we get:

6561/9 + 9*(standard deviation^2) = 729 729 + 9*(standard deviation^2) = 729

This equation is true when the standard deviation is 0, which is possible if all observations are the same. Therefore, 9 is a possible value of n.

You can check the other factors of 6561 in the same way to find other possible values of n.