Using simulations to understand the central limit theorem
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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 $______.

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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


Using simulations to understand the central limit theorem

The mean gas price in Maryland is $______

Statistic: A number describing a characteristic of a sample.


Using simulations to understand the central limit theorem

In Inferential Statisticswe use the value of a sample statistic to estimate a parameter value.


Using simulations to understand the central limit theorem

We want to estimate the mean height of MC students.

The mean height of MC students is 64 inches


Using simulations to understand the central limit theorem

Will x-bar be equal to µ?

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 µ?

How do we investigate the behavior of x-bar?




Using simulations to understand the central limit theorem

Simulation the mean and standard deviation

Rolling a fair die and recording the outcome

randInt(1,6)

Press MATH

Go to PRB

Select 5: randInt(1,6)


Using simulations to understand the central limit theorem

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

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

What if n is 10? Think on the range

Mean(randInt(1,6,10)

Press 2nd STAT[list]

Right to MATH

Select 3:mean(

Press MATH

Right to PRB

5:randInt(


Using simulations to understand the central limit theorem

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

The Central Limit Theorem in action


Using simulations to understand the central limit theorem

The Central Limit Theorem in action the mean and standard deviation


Using simulations to understand the central limit theorem

  • For the larger sample sizes, most of the x-bar values are quite close

  • to the mean of the parent population µ.

  • 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 sample will tends to be closer to µ 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.



Using simulations to understand the central limit theorem

The Central Limit Theorem in action quite close

Closing stock prices ($)

Variability of sample means for samples of size 64

26 – 2.526 + 2.5 26 + 2*2.5

__|________|________|________X________|________|________|__

18.5 21 23.5 26 28.5 31 33.5


Using simulations to understand the central limit theorem

Closing stock prices ($) quite close

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 95% of samples of 64 closing stock prices

have means that are within $5 of the population mean mu

About 99.7% of samples of 64 closing stock prices

have means that are within $7.50 of the population mean mu


Using simulations to understand the central limit theorem

Suppose we don’t know that the mean closing price of stocks is µ = 26 and we want to estimate it. Let’s say we use a SRS of 64 stocksan we can assume that σ = $20

X ~Right Skewed (μ = ?, σ = 20)

__|________|________|________X________|________|________|__

μ-7.5 μ-5 μ-2.5 μμ+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


Using simulations to understand the central limit theorem

Simulation stocks is µ = 26 and we

Roll a die 5 times and record the number of ONES obtained:

randInt(1,6,5)

Press MATH

Go to PRB

Select 5: randInt(1,6,5)


Using simulations to understand the central limit theorem

Roll a die 5 times, record the number of ONES obtained. stocks is µ = 26 and we

Do the process n times and find the mean number of ONES obtained.

The Central Limit Theorem in action