**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

**Links of Previous Main Topic:-**

- Introduction to statistics
- Knowledge of central tendency or location
- Definition of dispersion
- Moments
- Bivariate distribution
- Theorem of total probability addition theorem
- Random variable
- Binomial distribution
- What is sampling
- Types sampling
- What is the use of random numbers
- Parameter and statistic

**Links of Next Statistics Topics:-**

- Estimation
- Statistical hypothesis and related terms
- Analysis of variance introduction
- Definition of stochastic process
- Introduction operations research
- Introduction and mathematical formulation in transportation problems
- Introduction and mathematical formulation
- Queuing theory introduction
- Inventory control introduction
- Simulation introduction
- Time calculations in network
- Introduction of game theory