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 γ1=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 Sx, and the skewness g1. The results are shown in the next table.

 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.

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