Parameter: A number describing a characteristic of the population (usually unknown) The mean gas price of regular gasoline for all gas stations in Maryland.

Using Simulations to understand the

Central Limit Theorem

Parameter: A number describing a characteristic of

the population (usually unknown)

The mean gas price of regular gasoline for all gas stations in Maryland

The mean gas price in Maryland is $______

Statistic: A number describing a characteristic of

a sample.

In Inferential Statistics we use the value of a sample

statistic to estimate a parameter value.

We want to estimate the mean height of MC students.

The mean height of MC students is 64 inches

Will x-bar be equal to mu?

What if we get another sample, will x-bar be the same?

How much does x-bar vary from sample to sample?

By how much will x-bar differ from mu?

How do we investigate the behavior of x-bar?

What does the x-bar distribution look like?

Graph the x-bar distribution, describe the shape and find the mean and standard deviation

Rolling a fair die and recording the outcome

Simulation

randInt(1,6)

Press MATHGo to PRBSelect 5:

randInt(1,6)

Rolling a die n times and finding the mean of the outcomes.

Mean(randInt(1,6,10)

Press 2nd STAT[list]Right to MATHSelect 3:mean(Press MATHRight to PRB5:randInt(

Let n = 2 and think on the range of the x-bar distribution

What if n is 10? Think on the range

Rolling a die n times and finding the mean of the outcomes.

The Central Limit Theorem in action

The Central Limit Theorem in action

• For the larger sample sizes, most of the x-bar values are quite close to the mean of the parent population mu. (Theoretical distribution in this case)  • This is the effect of averaging  • When n is small, a single unusual x value can result in an x-bar value far from the center  • With a larger sample size, any unusual x values, when averaged with the other sample values, still tend to yield an x-bar value close to mu.  • AGAIN, an x-bar based on a large will tends to be closer to mu than will an x-bar based on a small sample. This is why the shape of the x-bar distribution becomes more bell shaped as the sample size gets larger. 

Normal Distributions

The Central Limit Theorem in action

Closing stock prices ($)

Variability of sample means for samples of size 64

26 – 2.5 26 + 2.5 26 + 2*2.5

__|________|________|________X________|________|________|__18.5 21 23.5 26 28.5 31 33.5

20~ ( 26, 2.5

64x xx N

n

Closing stock prices ($)Variability of sample means for samples of

size 64

2.5% | 95% | 2.5% 26 – 2.5 26 + 2.5 26 + 2*2.5

__|________|________|________X________|________|________|__18.5 21 23.5 26 28.5 31 33.5

About 99.7% of samples of 64 closing stock prices have means that are within $7.50 of the population mean mu

20~ ( 26, 2.5

64x xx N

n

About 95% of samples of 64 closing stock prices have means that are within $5 of the population mean mu

We want to estimate the mean closing price of stocks by using a SRS of 64 stocks. Assume the standard deviation σ = $20.

X ~Right Skewed (μ = ?, σ = 20)

20~ ( 26, 2.5

64x xx N

n

__|________|________|________X________|________|________|__ μ-7.5 μ-5 μ-2.5 μ μ+2.5 μ+5 μ+7.5

We’ll be 95% confident that our estimate is within $5 from the population mean mu

We’ll be 99.7% confident that our estimate is within $7.50 from the population mean mu

SimulationRoll a die 5 times and record the number of ONES obtained: randInt(1,6,5)

Press MATHGo to PRBSelect 5: randInt(1,6,5)

Roll a die 5 times, record the number of ONES obtained. Do the process n times and find the mean number of ONES obtained.

The Central Limit Theorem in action

Use website APPLETS to simulate proportion

problems