1. Psychology 202a Advanced Psychological Statistics September 29, 2015

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 HeadsProbability 0.001 1.010 2.044 3.117 4.205 5.246 6.205 7.117 8.044 9.010 10.001

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 0.6.205 0.7.117 0.8.044 0.9.010 1.0.001

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 Specific the decision rule (“alpha level”) 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. –i.e.,

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 http://www.chem.uoa.gr/applets/appletcent rallimit/appl_centrallimit2.htmlhttp://www.chem.uoa.gr/applets/appletcent rallimit/appl_centrallimit2.html