If we suppose that our data comes from a normally distribute

If we suppose that our data comes from a normally distributed population with standard deviation 2.6, what sample size is needed to ensure that with probability 99%, the mean of the sample will be in error by at most 0.25?

Solution

If X is a normal random variable with general mean (not necessarily 0) and standard deviation (not necessarily 1), then it can be converted to standard normal by way of Z = X / or equivalently X = + Z

P(Z) is 99% so

P[ -.25 Z .25 ] = P[ -0.25 Z 0 ] + P[0 Z .25 ] =2P[0 Z .25 ] = 99

so Z= 99/2 = approx 50