You do an experiment by taking random samples of size 50 from a population with a standard deviation of 13. You perform the experiment several times and find the mean of the sample means to be 87.
a) What is the population mean?
b) What is the standard deviation of the sample means?
a) Our estimate for the population mean is the mean of the sample means, ie 87. Each sample mean is an estimate of the population mean and we refine this estimate by averaging over samples. We do not know that this is the population mean, but it is our best guess given that we can only sample parts of the population and not observe the whole population at once.
b) We are told that the population standard deviation is `sigma=13` . Each sample is of size n=50. When we take the mean of that sample we calculate the value of `bar(X) = sum_i X_i/n` . Assuming that the observations `X_i` are iid (independently and identically distributed) the variance of this estimator is `V(bar(X)) = (V(X_1)+...V(X_n))/n^2` `= (nsigma^2)/n^2 = sigma^2/n` `= 169/50 = 3.38 `
Therefore the standard deviation associated with the sample mean estimator is `sqrt(3.38) = 1.83 `.
a) Our estimate for the population mean is the mean of the sample means, 87.
b) The standard deviation associated with the sample mean estimator is 1.83 to 3sf