Question

A sample is chosen randomly from a population that can be described by a Normal model. 1) What's the sampling distribution model for the sample mean? 2) Describe the shape, center and spread of the model you identified in #1. 3) If we choose a larger sample, what is the effect on the sampling distribution model?

Answer 1

A bit theory

Now here let us assume that be a sample is chosen randomly from a population that can be described by a Normal model that is for any arbitrary .

1)Then the sample mean

as here by the theory,

So the sampling distribution of the sample mean is Normal distribution with same mean of the population and standard deviation equal to the population standard deviation divided by square root of sample size (n).

(2)As the sample mean has the sampling distribution of Normal model hence its shape is of course bell shaped, symmetric aroundthe mean ( which is equal to the population mean) and on average the spread is inversely proportional to the square root of sample size.

(3) Note that the the standard deviation of sample mean which is , is inversely proportional to the sample size (more precisely to the square root of sample size). Hence as we choose larger sample, the standard deviation of sample mean decreases rapidly, and that implies the spread of the sampling distribution of sample mean decreases converging to the population mean (or the centre) , hence the peakedness of the sampling distribution model increases around the population mean.

Hence the answer...............

Thank you.............