1. STA 291 Spring 2008
Lecture 11
Dustin Lueker

2. Computer Simulation A new simulation would lead to different results
Reasoning why is same as to why different samples lead to different results
Simulation is merely more samples
However, the more samples we simulate, the closer the relative frequency distribution gets to the probability distribution (sampling distribution)
STA 291 Spring 2008 Lecture 11

3. Reduce Sampling Variability
The larger the sample size, the smaller the sampling variability
Increasing the sample size to 25…
10 samples
of size n=25
100 samples
of size n=25
1000 samples
of size n=25
STA 291 Spring 2008 Lecture 11

4. Interpretation
If you take samples of size n=4, it may happen that nobody in the sample is in AS/BE
If you take larger samples (n=25), it is highly unlikely that nobody in the sample is in AS/BE
The sampling distribution is more concentrated around its mean
The mean of the sampling distribution is the population mean
STA 291 Spring 2008 Lecture 11

5. Effect of Sample Size
The larger the sample size n, the smaller the standard deviation of the sampling distribution for the sample mean
Larger sample size = better precision
As the sample size grows, the sampling distribution of the sample mean approaches a normal distribution
Usually, for about n=30, the sampling distribution is close to normal
This is called the “Central Limit Theorem”
STA 291 Spring 2008 Lecture 11

6. Examples
If X is a random variable from a normal population with a mean of 20, which of these would we expect to be greater? Why?
P(15<X<25)
P(15< <25)
What about these two?
P(X<10)
P( <10)
STA 291 Spring 2008 Lecture 11

7. Mean of sampling distribution
Mean/center of the sampling distribution for sample mean/sample proportion is always the same for all n, and is equal to the population mean/proportion.
STA 291 Spring 2008 Lecture 11

8. Reduce Sampling Variability
The larger the sample size n, the smaller the variability of the sampling distribution
Standard Error
Standard deviation of the sample mean or sample proportion
Standard deviation of the population divided by
STA 291 Spring 2008 Lecture 11

9. Sampling Distribution of the Sample Mean
When we calculate the sample mean, , we do not know how close it is to the population mean
Because is unknown, in most cases.
On the other hand, if n is large, ought to be close to
STA 291 Spring 2008 Lecture 11

10. Parameters of the Sampling Distribution
If we take random samples of size n from a population with population mean and population standard deviation , then the sampling distribution of
has mean
and standard error
The standard deviation of the sampling distribution of the mean is called “standard error” to distinguish it from the population standard deviation
STA 291 Spring 2008 Lecture 11

11. Standard Error The example regarding students in STA 291
For a sample of size n=4, the standard error of is
For a sample of size n=25,
STA 291 Spring 2008 Lecture 11

12. Central Limit Theorem
For random sampling, as the sample size n grows, the sampling distribution of the sample mean, , approaches a normal distribution
Amazing: This is the case even if the population distribution is discrete or highly skewed
Central Limit Theorem can be proved mathematically
Usually, the sampling distribution of is approximately normal for n≥30
We know the parameters of the sampling distribution
STA 291 Spring 2008 Lecture 11

13. Example
Household size in the United States (1995) has a mean of 2.6 and a standard deviation of 1.5
For a sample of 225 homes, find the probability that the sample mean household size falls within 0.1 of the population mean
Also find
STA 291 Spring 2008 Lecture 11

14. Central Limit Theorem (Binomial Version)
For random sampling, as the sample size n grows, the sampling distribution of the sample proportion, , approaches a normal distribution
Usually, the sampling distribution of is approximately normal for np≥5, nq≥5
We know the parameters of the sampling distribution
STA 291 Spring 2008 Lecture 11

15. Example
Take a SRS with n=100 from a binomial population with p=.7, let X = number of successes in the sample
Find
Does this answer make sense?
Also Find
STA 291 Spring 2008 Lecture 11