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Using Simulations to understand the Central Limit TheoremPowerPoint Presentation

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 $______

Statistic: A number describing a characteristic of a sample.

In Inferential Statisticswe 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

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?

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

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)

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(

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

The Central Limit Theorem in action

The Central Limit Theorem in action the mean and standard deviation

- 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.

Normal Distributions quite close

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

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

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

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)

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

Use website APPLETS to simulate proportion problems stocks is µ = 26 and we

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