parameter: a number describing a characteristic of the population (usually unknown) the mean gas...
TRANSCRIPT
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