*Cooper City High School
Broward Community College*

*
Statistics*

__About the Central Limit Theorem :__

:The Central Limit Theorem |

Draw an SRS of size n from any
population whatsoever with mean μ and finite standard deviation σ. When
n is large, the sampling distribution of the sample mean x
is close to the normal distribution N(μ, σ/√n)
with mean μ and standard deviation σ/√n. |

We are going to make several experiments, selecting samples with different n values. The samples will be selected from an right-skewed binomial distribution using the random generator randBin (5,0.1). Each experiment will be repeated 500 times.

First, let's have a look at the Binomial Distribution for number of trials =5 and p=0.1.

the parameters of this distribution are:
**μ=0.50, σ=0.67, and γ _{1}=1.19.
**So, it is a distribution skewed to the right.

Now,
let's perform the experiments. We will select random samples with **5, 10, 20,
and 30 **individuals in each. We repeat this experiment 500 times, and then we
calculate the mean value ** x**,
the standard deviation **S _{x}**, and the skewness

Sample Dimension |
Mean Value(Expected Value = 0.50) |
Standard Deviation(Expected value) |
Skewness |

5 |
0.532 |
0.307(0.299) |
0.688 |

10 |
0.484 |
0.207((0.212) |
0.516 |

20 |
0.501 |
0.140((0.149) |
0.411 |

30 |
0.509 |
0.120(0.122) |
0.155 |

as you can see from the table, when the dimension of the sample is greater than 20, the distribution tends to be symmetric ( the skewness is smaller than 0.4). I have included the graphs of the distribution for different dimensions of the sample.