Case 1: σ is unknown

When the sample is small, by making an assumption of a normal population, the sampling distribution of X is obtained. But then the σ is replaced by standard deviation S and gives

Where S²=

Case 2: σ is known

Consider a population with mean 1.1 and variance cr2. If a sample size n is randomly taken from it, then the sample X is a random variable with a distribution of mean µ.

With an infinite population, the variance is  and the standard error is

S.E. =

If the population is finite with a size N, then the variance is  and the standard error is

S.E= when the sample is drawn without replacement.

The factor  is known as finite population correction factor.

The central limit theorem:

If u is the mean of sample size n which is taken from a population, whose mean is µ and variance is σ², then

Z=  →N (0, 1) as n→∞

If the sample is from normal population, then sampling distribution of mean is normal irrespective of the size of the sample. However, if the population is not normal, then sampling distribution of mean is approximately normal for small size.

Example 1:

A random sample size of 100 is taken from an infinite population having mean µ= 66 and variance σ2 = 225. What is the probability of getting an x between 64 and 68?

Solution:

Z=  n= 100, µ=66 and σ= 15

Required probability = P [64 < x < 68]

=p [-1.33 < z < 1.33]

=2<1> (1.33) = 2 (0.4082)

=0.8164.

Example 2:

A random sample is of size 5 is drawn without replacement from a finite population consisting of 35 units. If the standard deviation of the population is 2.25, what is the standard error of sample mean?

Solution:

N= 35, n= 5, σ=2.25

S.E=

= S.E=  = 0.95 