Verify that the mean of a random sample of size n256 is as r

Verify that the mean of a random sample of size n=256 is as reliable an estimate of the mean of a symmetric continuous population as the median of a random sample size of n=400.

Solution

When sample size exceeds 30, the means of random variable follow a normal distribution irrespective of the original distribution of random variable x

Hence here n = 256 very much large so x bar mean follows a normal distribution with estimated mean = sample mean

Since sample mean follows a normal, mean = median = mode in normal distribution

Hence all 3 will coincide and so median of sample can be taken as estimate of mean.