Central Limit Theorem

The CLT states that given certain conditions the sampling dsitribution will generally be bell shaped.

CLT tends to become more bell shaped as the same size increases, it helps the distribution become more accerate therefore become more central oriented (bell shaped).

Mu = \(\mu\)

Is an important symbol for CLT, \(\mu\) is refered to as the working hypothesis or as the null hypothesis. When \(\mu\) is close to 0 then the sample distribution will generally be bell shaped.

Central Limit Theorem

The Equation for CLT

t(\(\mu\)) = Y - \(\mu\) / SE(Y)

Low Sample Size

heights <- 65:80
sample.of.samples <- numeric()
for(i in 1:10){
  sample.of.samples[i] <- mean(sample(heights,10,replace=T))
} ; hist(sample.of.samples)

Notice how with the low sample size the graph tends to not be bell shaped.

Medium Sample Size

heights <- 65:80
sample.of.samples <- numeric()
for(i in 1:100){
  sample.of.samples[i] <- mean(sample(heights,10,replace=T))
} ; hist(sample.of.samples)

Now with a Larger sample size the graph begins to look more bell shaped.

Large Sample Size

heights <- 65:80
sample.of.samples <- numeric()
for(i in 1:1000){
  sample.of.samples[i] <- mean(sample(heights,10,replace=T))
} ; hist(sample.of.samples)

With a Large sample size the graph definitely looks bell shaped. (all Heights in all the graphs are in Inches)