Sampling Distribution: Central Limit Theorem and z-value

Central Limit Theorem

If the population is normally distributes then the Central Limit Theorem (CLT)is applied. However,
if the population is not normally distributed normal and the sample size is large enough (greater than 30),
it is possible to apply the CLT:
In this case, the sample means from the population will be approximately normal and the sampling distribution will have the properties:

μ_X̄ = μ
σ_X̄ = σ / √ n

z-value for Sampling Distribution of X̄

Z-value for the sampling distribution of X̄ is given below:
z-function

where:

X̄sample mean
μ = population mean
σ = population standard deviation
n = sample size

Example

Suppose a population has mean μ = 7 and standard deviation σ = 3.
Suppose a random sample of size n = 38 is selected.
Question: What is the probability that the sample mean is between 6.8 and 7.2?
Solution:
Even if the population is not normally distributed, the CLT can be used since the sample size, n, > 30
Hence, the sampling distribution of X̄ is approximately normal with mean μ_X̄ = 7, and
σ_X̄ = σ / √ n = 3 / √ 38 = 0.4866
P(6.8 < μ_X̄ < 7.2) = P((6.8 − 7) / (3 / √ 38 )) < ((μ_X̄ − μ / (σ / ( √ n )) < ((7.2 - − 7 / (3 / √ 38 ))) = P(-0.41 < z < 0.41) = 0.31

For more details, please contact me here.
Date of last modification: March 25, 2019