Transcript From Simulations to 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 $______
Statistic: A number describing
a characteristic of a sample.
In Inferential Statistics we use
the value of a sample statistic
to estimate a parameter value.
We want to estimate the mean
height of MC students.
The mean height of MC students is 64 inches
Will x-bar be equal to mu?
What if we get another sample, will x-bar be
the same?
How much does x-bar vary from sample to
sample?
By how much will x-bar differ from mu?
How do we investigate the behavior of x-bar?
What does the x-bar distribution look like?
Graph the x-bar distribution, describe the shape and
find the mean and standard deviation
Simulation
Rolling a fair die
and recording
the outcome
randInt(1,6)
Press MATH
Go to PRB
Select 5: randInt(1,6)
Rolling a die n times and finding the
mean of the outcomes.
Let n = 2 and think on the range of the x-bar distribution
What if n is 10? Think on the range
Mean(randInt(1,6,10)
Press 2nd STAT[list]
Right to MATH
Select 3:mean(
Press MATH
Right to PRB
5:randInt(
Rolling a die n times and finding the mean of the outcomes.
The Central Limit Theorem in action
The Central Limit Theorem in action
• For the larger sample sizes, most of the x-bar values are quite close
to the mean of the parent population mu.
(Theoretical distribution in this case)
• This is the effect of averaging
• When n is small, a single unusual x value can result in an x-bar
value far from the center
• With a larger sample size, any unusual x values, when averaged
with the other sample values, still tend to yield an x-bar value close
to mu.
• AGAIN, an x-bar based on a large will tends to be closer to mu than
will an x-bar based on a small sample. This is why the shape of the
x-bar distribution becomes more bell shaped as the sample size gets
larger.
Normal Distributions
The Central Limit Theorem in action
Closing stock prices ($)
Variability of sample means for samples of size 64
x ~ N (  x    26,  x 
26 – 2.5


20
n
26 + 2.5
 2.5
64
26 + 2*2.5
__|________|________|________X________|________|________|__
18.5
21
23.5
26
28.5
31
33.5
Closing stock prices ($)
Variability of sample means for samples of size 64
x ~ N (  x    26,  x 

2.5% |
n

20
 2.5
64
95%
26 – 2.5
|
26 + 2.5
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
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)
x ~ N (  x    26,  x 

n

20
 2.5
64
__|________|________|________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
Simulation
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.
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