1. Sampling Distributions
A Lecture for the Intro Stat Course
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Professor Friedman's Statistics Course by H & L Friedman is licensed under a  Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 

2. Sampling Distribution of X̅
The sample mean, X̅, is a random variable.
There are a lot of different values of X̅
Every sample we collect has a different X̅
There is only one population mean, µ.
Sampling Distributions

3. Sampling Distribution of X̅
If each of you in the class collected your own data you would each get a different X̅
Approx 95% of your X̅’s would be close to µ, within ±2 s.d.
This is because X̅ is a normally distributed random variable – for large samples
X̅ follows a normal distribution centered about µ
This is known as the Central Limit Theorem
Sampling Distributions

4. Central Limit Theorem
Even if a population distribution is nonnormal, the sampling distribution of X̅ may be considered to be approximately normal for large samples.
What’s large? At least 30; some say 50.
Sampling Distributions

5. Central Limit Theorem (cont’d)
This “hypothetical” sampling distribution of the mean, as n gets large, has the following properties:
E(X̅) = μ. It has a mean equal to the population mean.
It has a standard deviation (called the standard error of the mean, ) equal to the population standard deviation divided by √n.
It is normally distributed.
Sampling Distributions

6. Central Limit Theorem (cont’d)
This means that, for large samples, the sampling distribution of the mean (X̅) can be approximated by a normal distribution with
Sampling Distributions

7. A Very, Very Small Example
Suppose that in a population consisting of 5 elements and one wishes to take a random sample of 2. There are 10 possible samples which might be selected.
Consider the Population (N=5): 1, 2, 3, 4, 5
Sampling Distributions

8. …Very Small Example…
Since we know the entire population, we can compute the population parameters:
μ = 3.0
σ = = √2 = 1.41
[Note: N, not n-1. This is the formula for computing the population standard deviation, σ.]
Sampling Distributions

9. …Very Small Example… All possible samples of size n=2:
The average of all the possible sample means is
E(X̅)=30.0 / 10 = 3.0.
So, E(X̅)=μ. This property is called unbiasedness.
Sampling Distributions

10. The Sampling Distribution of X̅
By the Central Limit Theorem, X̅ follows a normal distribution (for large n):
Sampling Distributions

11. The Sampling Distribution of X̅
Since this is a normal distribution, we can standardize it (transform to Z) just like any other normal distribution.
If n is large, say 30 or more, use s as an unbiased estimate of σ.
Sampling Distributions

12. Example- Steel Chains
Suppose you have steel chains with an average breaking strength of μ=200 lbs. with a σ=10 lbs., and you take a sample of n=100 chains.
What is the probability that the sample mean breaking strength will be 195 lbs. or less? This is the same as asking: What proportion of the sample means will be X̅ =195 lbs. or less?
Sampling Distributions

13. Example- Steel Chains Solution (Draw a picture!) Z = = -5
Ans: The probability is close to zero.
Sampling Distributions

14. Example- Bulbs
A manufacturer claims that its bulbs have a mean life of 15,000 hours and a standard deviation of 1,000 hours. A quality control expert takes a random sample of 100 bulbs and finds a mean of X̅ = 14,000 hours. Should the manufacturer revise the claim?
NOTE: We don’t know the distribution of the lifetimes of the bulbs. But we DO know the distribution of the mean life of a (large enough) sample of bulbs.
Sampling Distributions

15. Example- Bulbs Solution (DRAW A PICTURE!) Z = = = -10
That’s 10 standard deviations away from the mean! Yes, the claim should definitely be revised.
14,000
15,000 X
Sampling Distributions

16. Example- Hybrid Motors
In a large automobile manufacturing company, the life of hybrid motors is normally distributed with a mean of 100,000 miles and a standard deviation of 10,000 miles.
(a) What is the probability that a randomly selected hybrid motor has a life between 90,000 miles and 110,000 miles per year?
(b) If a random sample of 100 motors is selected, what is the probability that the sample mean will be below 98,000 miles per year?
Sampling Distributions

17. Example- Hybrid Motors
SOLUTION (Of course, DRAW A PICTURE!)
(a) Convert to Z using the formula: Z = (Xi − μ) / σ
Z = = −1
Z = = +1
Thus, we have to find how much area lies between −1 and +1 of the Z-distribution
Answer = .6826
Sampling Distributions

18. Example- Hybrid Motors
(b) SOLUTION (Of course, DRAW A PICTURE!)
(b) Here we are looking at the sampling distribution of the mean. Sample means follow a different distribution and to convert to Z, we use the following formula.
= −2
Ans: The probability the sample mean will be below 98,000/year is .5 − =
Sampling Distributions

19. Other Sampling Distributions
We have been looking at the relationship between X̅ and µ.
Of course, statisticians will often be interested in estimating other parameters such as the population proportion (P), the population standard deviation (σ), the population median, etc.
In each case we use a statistic from a sample to estimate the parameter. Each of these statistics has its own sampling distribution.
Sampling Distributions

20. Implications
The relationship between X̅ and µ is the foundation of statistical inference
Statistical inference includes estimation (of µ) and testing hypotheses (about µ)
Since there are so many X̅’s – as many as there are possible samples - we use the X̅ value we happened to get as a tool to make inferences about the only true mean, µ.
Without actually conducting a census we can never know µ with 100% certainty
Sampling Distributions

21. Importance This concept is what the whole rest of the course is about.
Sampling Distributions