1. Psychology 202a Advanced Psychological Statistics
September 26, 2017

2. Complicated combining of probability rules
Developing the binomial distribution
Digression on combinatorics
The binomial distribution in R

3. The binomial distribution as a model for real world behavior
Describing a sequence of coin toss outcomes
A hypothetical game with consequences

4. Number of Heads
Probability
.001 1
.010 2
.044 3
.117 4
.205 5
.246 6 7 8 9 10

5. Sampling distributions
Recap our hierarchy of distribution types
Distributions
Probability distributions
A sampling distribution is a probability distribution for which the random variable happens to be a statistic.

6. Example of a sampling distribution
Imagine a change of emphasis in our coin toss experiment.
Instead of counting heads, we want to estimate the probability of heads.
If the number of heads were 0, what would we estimate p(heads) to be?
How about if the number were 1?

7. Est. Prob.
P(Est. Prob.)
0.0
.001
0.1
.010
0.2
.044
0.3
.117
0.4
.205
0.5
.246
0.6
0.7
0.8
0.9
1.0

8. Formalizing hypothesis testing
Statement of interest (research hypothesis)
Formal statement that nothing interesting is happening (null hypothesis)
Use of statements, theory, and assumptions to get sampling distribution
Definition of “unlikely”
Observation
Decision (reject or fail to reject null hypothesis)

9. Recap: sampling distributions
A sampling distribution is a special name we use for a probability distribution when the random variable happens to be a statistic.
One common example is the sampling distribution of the mean.

10. The central limit theorem
Part one:
For well-behaved variables…
The mean of the mean is the population mean.
i.e.,
The variance of the mean is the population variance divided by the sample size.

11. The central limit theorem
Part two:
If the variable is normally distributed, then the sample mean will also be normally distributed.
If the variable is not normally distributed, then the sample mean will approach normality as the sample size becomes large.
How large is large?
It depends.

12. Demonstrating the Central Limit Theorem
Demonstration in R

13. What if we don’t know the sampling distribution?
The bootstrap
Bradley Efron
Illustration in R