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Sampling Distribution of Mean

Case I : σ Known

 

Consider a population having mean 1.1 and variance cr2. If a random sample of size n is taken from this population then the sample mean X is a random variable whose distribution has the mean µ.

If the population is infinite, then the variance of this distribution is σ2/n and the standard error    is defined as

If the population is finite of size N then the variance of this distribution is

σ2/n.(N-n)/(N-1)                                  and the standard error is defined as

provided the sample is drawn without replacement.

The factor   (N-n)/(N-1)     is called finite population correction factor.

Let us consider the standardized sample mean

Then we have the central limit theorem as follows :

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

If the samples come from a normal population then the sampling distribution of the mean is normal regardless of the size of the sample.

 

If the population is not normal then the sampling distribution of the mean is approximately normal for small size (n = 25) of the sample.

 

Example 2. A random sample of size 100 is taken from an infinite population having the mean

µ= 66 and the variance σ2 = 225. What is the probability of getting an x between 64 and 68 ?

Solution. Let

Required probability =  P[64 < x < 68]

=p [-1.33 < z < 1.33]

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

=0.8164.

 

 

Example 3. A random sample is of size 5 is drawn without replacement from a finite population consisting of 35 units. If the population standard deviation is 2.25. What is the standard error of sample mean ?

Solution. Here, n = 5, N = 35, σ = 2.25

S.E. of sample mean

 

Case II : σ Unknown

For small sample, the assumption of normal population gives fairly the sampling distribution of X. However the σ is replaced by sample standard deviation S. Then we have