The central limit theorem states:
If
isa random sample of size
drawnfrom any population with mean
and
variance
thenthe sample mean
hasexpected valueandexpected variance
Iflots of samples, all of sizearetaken from the population, then the distribution of the sample meansis approximately normally distributed,
andthe goodness of the fit improves with increasing
Theunderlying distribution of the population is arbitrary.
The central limit theorem is used for sampling when the samplesize is ‘large’, in practice over 50.
The central limit theorem can then be used to analyse the samplemeans, perfrom hypothesis testing and construct confidence intervalsusing the normal distribution.
A sample of size 100 is taken from a population with mean 40 andvariance 20. Find the
probability that the sample mean is larger than 45.
and
so
so
