1. Probability and the Sampling Distribution
Quantitative Methods in HPELS
HPELS 6210

2. Agenda Introduction Distribution of Sample Means
Probability and the Distribution of Sample Means
Inferential Statistics

3. Introduction Recall: Next step  convert sample mean into a Z-score
Any raw score can be converted to a Z-score
Provides location relative to µ and 
Assuming NORMAL distribution:
Proportion relative to Z-score can be determined
Z-score relative to proportion can be determined
Previous examples have looked at single data points
Reality  most research collects SAMPLES of multiple data points
Next step  convert sample mean into a Z-score
Why? Answer probability questions

4. Introduction Two potential problems with samples:
Sampling error
Difference between sample and parameter
Variation between samples
Difference between samples from same taken from same population
How do these two problems relate?

5. Agenda Introduction Distribution of Sample Means
Probability and the Distribution of Sample Means
Inferential Statistics

6. Distribution of Sample Means
Distribution of sample means = sampling distribution is the distribution that would occur if:
Infinite samples were taken from same population
The µ of each sample were plotted on a FDG
Properties:
Normally distributed
µM = the “mean of the means”
M = the “SD of the means”
Figure 7.1, p 202

8. Distribution of Sample Means
Sampling error and Variation of Samples
Assume you took an infinite number of samples from a population
What would you expect to happen?
Example 7.1, p 203

9. Assume a population consists of 4 scores (2, 4, 6, 8)
Collect an infinite number of samples (n=2)

11. Total possible outcomes: 16
p(2) = 1/16 = 6.25% p(3) = 2/16 = 12.5%
p(4) = 3/16 = 18.75% p(5) = 4/16 = 25%
p(6) = 3/16 = 18.75% p(7) = 2/16 = 12.5%
p(8) = 1/16 = 6.25%

12. Central Limit Theorem
For any population with µ and , the sampling distribution for any sample size (n) will have a mean of µM and a standard deviation of M, and will approach a normal distribution as the sample size (n) approaches infinity
If it is NORMAL, it is PREDICTABLE!

13. Central Limit Theorem
The CLT describes ANY sampling distribution in regards to:
Shape
Central Tendency
Variability

14. Central Limit Theorem: Shape
All sampling distributions tend to be normal
Sampling distributions are normal when:
The population is normal or,
Sample size (n) is large (>30)

15. Central Limit Theorem: Central Tendency
The average value of all possible sample means is EXACTLY EQUAL to the true population mean
µM = µ
If all possible samples cannot be collected?
µM approaches µ as the number of samples approaches infinity

16. µ = 2+4+6+8 / 4 µ = 5 µM = 2+3+3+4+4+4+5+5+5+5+6+6+6+7+7+8 / 16

17. Central Limit Theorem: Variability
The standard deviation of all sample means is denoted as M
M = /√n
Also known as the STANDARD ERROR of the MEAN (SEM)

18. Central Limit Theorem: Variability
SEM
Measures how well statistic estimates the parameter
The amount of sampling error between M and µ that is reasonable to expect by chance
The standard distance between the sample M and population µ

19. Central Limit Theorem: Variability
SEM decreases when:
Population  decreases
Sample size increases
Other properties:
When n=1, M =  (Table 7.2, p 209)
As SEM decreases the sampling distribution “tightens” (Figure 7.7, p 215)
M = /√n

21. Agenda Introduction Distribution of Sample Means
Probability and the Distribution of Sample Means
Inferential Statistics

22. Probability  Sampling Distribution
Recall:
A sampling distribution is NORMAL and represents ALL POSSIBLE sampling outcomes
Therefore PROBABILITY QUESTIONS can be answered about the sample relative to the population

23. Probability  Sampling Distribution
Example 7.2, p 209
Assume the following about SAT scores:
µ = 500
 = 100
n = 25
Population  normal
What is the probability that the sample mean will be greater than 540?
Process:
Draw a sketch
Calculate SEM
Calculate Z-score
Locate probability in normal table

24. Step 1: Draw a sketch
Step 2: Calculate SEM
SEM = M = /√n
SEM = 100/√25
SEM = 20
Step 3: Calculate Z-score
Z = 540 – 500 / 20
Z = 40 / 20
Z = 2.0
Step 4: Probability
Column C
p(Z = 2.0) =

25. Agenda Introduction Distribution of Sample Means
Probability and the Distribution of Sample Means
Inferential Statistics

26. Looking Ahead to Inferential Statistics
Review:
Single raw score  Z-score  probability
Body or tail
Sample mean  Z-score  probability
What’s next?
Comparison of means  experimental method

28. Textbook Assignment Problems: 14, 18, 24
In your words, explain the concept of a sampling distribution
In your words, explain the concept of the Central Limit Theorum