1. Introduction to Inference
Sampling Distribution of Means & Central Limit Theorem
Dr. Amjad El-Shanti
MD, PMH, Dr PH
University of Palestine
2016

2. Course Overview Exploring Data Inference Collecting Data
Probability Intro.
Inference
Comparing Variables
Relationships between Variables
Means
Proportions
Regression
Contingency Tables

3. Inference with a Single Observation?
Population
Parameter: 
Sampling
Inference
Observation Xi
Each observation Xi in a random sample is a representative of unobserved variables in population
How different would this observation be if we took a different random sample?

4. Normal Distribution
Last class, we learned normal distribution as a model for our overall population
Can calculate the probability of getting observations greater than or less than any value
Usually don’t have a single observation, but instead the mean of a set of observations

5. Inference with Sample Mean?
Population
Parameter: 
Sampling
Inference
Estimation
Sample
Statistic: x
Sample mean is our estimate of population mean
How much would the sample mean change if we took a different sample?
Key to this question: Sampling Distribution of x

6. Sampling Distribution of Sample Mean
Distribution of values taken by statistic in all possible samples of size n from the same population
Model assumption: our observations xi are sampled from a population with mean  and variance 2
Sample 1 of size n x
Sample 2 of size n x
Sample 3 of size n x
Sample 4 of size n x
Sample 5 of size n x
Sample 6 of size n x
Sample 7 of size n x
Sample 8 of size n x .
Distribution
of these
values?
Population
Unknown
Parameter: 

7. Mean of Sample Mean mean( X ) = μ
First, we examine the center of the sampling distribution of the sample mean.
Center of the sampling distribution of the sample mean is the unknown population mean:
mean( X ) = μ
Over repeated samples, the sample mean will, on average, be equal to the population mean
no guarantees for any one sample!

8. Variance of Sample Mean
Next, we examine the spread of the sampling distribution of the sample mean
The variance of the sampling distribution of the sample mean is
variance( X ) = 2/n
As sample size increases, variance of the sample mean decreases!
Averaging over many observations is more accurate than just looking at one or two observations

9. Comparing the sampling distribution of the sample mean when n = 1 vs
Comparing the sampling distribution of the sample mean when n = 1 vs. n = 10

10. Law of Large Numbers Remember the Law of Large Numbers:
If one draws independent samples from a population with mean μ, then as the number of observations increases, the sample mean x gets closer and closer to the population mean μ
This is easier to see now since we know that
mean(x) = μ
variance(x) = 2/n as n gets large

11. Example
Population: seasonal home-run totals for 7032 baseball players from 1901 to 1996
Take different samples from this population and compare the sample mean we get each time
In real life, we can’t do this because we don’t usually have the entire population!
Sample Size
Mean
Variance
100 samples of size n = 1
3.69
46.8
100 samples of size n = 10
4.43
100 samples of size n = 100
4.42
0.43
100 samples of size n = 1000
0.06
Population Parameter
 = 4.42

12. Distribution of Sample Mean
We now know the center and spread of the sampling distribution for the sample mean.
What about the shape of the distribution?
If our data x1,x2,…, xn follow a Normal distribution, then the sample mean x will also follow a Normal distribution!

13. Example Mortality in US cities (deaths/100,000 people)
This variable seems to approximately follow a Normal distribution, so the sample mean will also approximately follow a Normal distribution

14. Central Limit Theorem
What if the original data doesn’t follow a Normal distribution?
HR/Season for sample of baseball players
If the sample is large enough, it doesn’t matter! 14

15. Central Limit Theorem
If the sample size is large enough, then the sample mean x has an approximately Normal distribution
This is true no matter what the shape of the distribution of the original data! 

16. CENTRAL LIMIT THEOREM specifies a theoretical distribution
formulated by the selection of all possible random samples of a fixed size n
a sample mean is calculated for each sample and the distribution of sample means is considered

17. SAMPLING DISTRIBUTION OF THE MEAN
The mean of the sample means is equal to the mean of the population from which the samples were drawn.
The variance of the distribution is s divided by the square root of n. (the standard error.)

18. STANDARD ERROR
Standard Deviation of the Sampling Distribution of Means sx = s/ \/n

19. How Large is Large?
If the sample is normal, then the sampling distribution of will also be normal, no matter what the sample size.
When the sample population is approximately symmetric, the distribution becomes approximately normal for relatively small values of n.
When the sample population is skewed, the sample size must be at least 25 before the sampling distribution of becomes approximately normal.

20. Central Limit Theorem
CLT states that for randomly selected sample size (n should be at least 25, but the larger n , the better the approximation) with a mean (μ) and standard deviation (s ):
The mean of the distribution of sample means is equal to the mean of the population distribution (μ (x) = μ ).
The standard deviation of the distribution of sample means is equal to the standard deviation of the population divided by the square root of the sample size : (s (x) = s/√n).
For any selected sample for any population with mean μ and standard deviation s then the distribution of sample means is approximately normal regardless whether the population distribution is normal or not.

21. EXAMPLE
A certain brand of tires has a mean life of 25,000 miles with a standard deviation of 1600 miles. What is the probability that the mean life of 64 tires is less than 24,600 miles?

22. Example continued
The sampling distribution of the means has a mean of 25,000 miles (the population mean) m = mi. and a standard deviation (i.e.. standard error) of: 1600/8 = 200

23. Example continued
Convert 24,600 mi. to a z-score and use the normal table to determine the required probability. z = ( )/200 = -2 P(z< -2) = or 2.28% of the sample means will be less than 24,600 mi.

24. ESTIMATION OF POPULATION VALUES
Point Estimates
Interval Estimates

25. CONFIDENCE INTERVAL ESTIMATES for LARGE SAMPLES
The sample has been randomly selected
The population standard deviation is known or the sample size is at least 25.

26. Confidence Interval Estimate of the Population Mean-
X: sample mean
s: sample standard deviation
n: sample size

27. EXAMPLE
Estimate, with 95% confidence, the lifetime of nine volt batteries using a randomly selected sample where: --
X = 49 hours
s = 4 hours
n = 36

28. EXAMPLE continued
Lower Limit: 49 - (1.96)(4/6) 49 - (1.3) = 47.7 hrs Upper Limit: 49 + (1.96)(4/6) 49 + (1.3) = 50.3 hrs We are 95% confident that the mean lifetime of the population of batteries is between 47.7 and 50.3 hours.