1. Sampling Distributions
Chapter 7
Sampling Distributions

2. Sampling Distributions
Sampling distribution of a statistic - the probability distribution of the statistic computed from all possible random samples using the same sample size from the sample population

3. Sampling Distributions (broken down)
Sample - subset of scores from all the scores (number of observations = n)
Random sample - each observation has an equal chance of being included
Statistic - quantitative characteristic of a sample ( M, s2, s, etc.)
Probability Distribution - represents scores and the likelihood of their occurrence (graphically, in tables)

4. Sampling Distributions
Shows the likelihood of obtaining the value of some statistic given n observations
What is the probability of getting a mean of 4, with 3 observations?

5. Constructing Sampling Distributions
Choose a sample size (n)
Randomly select (independently) all possible combinations of n scores
Compute the statistic of interest
Plot the relative frequency of the values of the statistics computed
Problem: with a population of 10 scores and an n of 3, there are 103 combinations (1000 independent random samples!)

6. Experiment 1 (The Smoking Study)
How many cigarettes do you smoke per day?

7. Experiment (The Smoking Study)
Score f p
1-10 19
11-20
146
21-30
284
31-40
111
41-50 35
51-60 9
61-70 3
71-80
81-90
91-100 1
Total (n)
608
How many cigarettes do you smoke per day?

8. Experiment (The Smoking Study)
How many cigarettes do you smoke per day?
Postive skew
Uni-modal
asymetric

9. Experiment (The Smoking Study)
How many cigarettes do you smoke per day?
Mo = 20
Md = 20
M = 22.44
Range = 95
s2 = 97.33
s = 9.87

10. Experiment (The Smoking Study)
How many cigarettes do you smoke per day?
This is the probability distribution of SCORES.

11. Experiment (The Smoking Study)
How many cigarettes do you smoke per day?
Randomly select 10
ALL 608
Scores 10 20 34 23 15 18 40 27

12. Experiment (The Smoking Study)
Return those 10
How many cigarettes do you smoke per day?
ALL 608
Scores 10 20 34 23 15 18 40 27

13. Experiment (The Smoking Study)
How many cigarettes do you smoke per day?
Randomly select 10 more
ALL 608
Scores 20 40 36 25 29 51 12 30 45

14. Experiment (The Smoking Study)
Return those 10
How many cigarettes do you smoke per day?
ALL 608
Scores 20 40 36 25 29 51 12 30 45

15. Experiment (The Smoking Study)
How many cigarettes do you smoke per day?
Randomly select 10 more
ALL 608
Scores X1 X2 X3 X4 X5 X6 X7 X8 X9
X10

16. Experiment (The Smoking Study)
Until Every Possible
Combination has been
Selected
How many cigarettes do you smoke per day?
ALL 608
Scores X1 X2 X3 X4 X5 X6 X7 X8 X9
X10

17. Experiment (The Smoking Study)
Or 1,765,877,296,986,850,000,000 times
How many cigarettes do you smoke per day?
ALL 608
Scores X1 X2 X3 X4 X5 X6 X7 X8 X9
X10

18. Experiment (The Smoking Study)
What if we do it 100 times?
How many cigarettes do you smoke per day?
ALL 608
Scores X1 X2 X3 X4 X5 X6 X7 X8 X9
X10

19. Experiment (The Smoking Study)
What if we do it 100 times?
How many cigarettes do you smoke per day?

20. Experiment (The Smoking Study)
What if we do it 1000 times?
How many cigarettes do you smoke per day?

21. Experiment (The Smoking Study)
What if we do it ∞ times?
How many cigarettes do you smoke per day?

22. Experiment (The Smoking Study)
Sampling Distribution
Of Sample Means
Distribution of Scores
Postive skew
Uni-modal
asymetric

23. Experiment (The Smoking Study)
Sampling Distribution
Of Sample Means
Distribution of Scores
Rectangular

24. Experiment (The Smoking Study)
Sampling Distribution
Of Sample Means
Distribution of Scores
Rectangular

25. Probability Theory to the Rescue
The mean of the sampling distribution of the sample mean is equal to the population mean
The standard deviation of the sampling distribution of the sample mean is equal to the standard deviation of the population divided by the square root of n
As the sample size increases, the sampling distribution of the sample mean approaches a normal distribution (Central Limit Theorem)
The Three
Amazing Facts

26. Probability vs. Sampling Distributions
All sampling distributions are probability distributions
all the possible levels of a sample statistic and their probabilities
But not all probability distributions are sampling distributions
The probability distribution of scores is what a sampling distribution is constructed from

27. Central Limit Theorem (CLT)
As sample size increases, the sampling distribution of the sample mean will approach a normal distribution
The shape of the the distribution of scores DOES NOT MATTER - the sampling distribution of the sample mean will approach normality, as sample size increases

28. Usefulness of the CLT
With large enough sample sizes, and the sampling distribution approaching normality, the probability of certain results can be found using z-score transformations and Table A

29. Example
A Psychology class with 25 students had a mean SAT verbal score of 535
What is the probability of having a class of this size with a mean SATv of 535 or higher?
We know that :

30. Example, cont.

31. Example (cont.)
Area with
M’s >535
z of 1.75 has a proportion of of the scores below it
95.99% of the means are at or below 535, we have a
= 4.01% chance of getting 535 or higher
.9599
+1.75

32. Review of Concepts Random Sampling Sampling Distributions
Central Limit Theorem
Find the probability of obtaining some mean, with z-scores and table A