1. Introduction to Hypothesis Testing: The Binomial Test
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2. ESP Your friend claims he can predict the future
You flip a coin 5 times, and he’s right on 4
Is your friend psychic?

3. Two Hypotheses Hypothesis Psychic Luck Hypothesis testing
A theory about how the world works
Proposed as an explanation for data
Posed as statement about population parameters
Psychic
Some ability to predict future
Not perfect, but better than chance
Luck
Random chance
Right half the time, wrong half the time
Hypothesis testing
A method that uses inferential statistics to decide which of two hypotheses the data support

4. Likelihood Likelihood Determining likelihood Strategy:
Probability distribution of a statistic, according to each hypothesis
If result is likely according to a hypothesis, we say data “support” or “are consistent with” the hypothesis
Determining likelihood
Psychic: hard to say; how psychic?
Luck: can work out exactly; 50/50 chance each time
Strategy:
Work out probabilities according to luck hypothesis
Calculate how likely outcome would be
Likely: luck can explain it
Unlikely: luck can't explain; must be ESP
“Innocent until proven guilty”
Assume luck unless compelling evidence to rule it out
Population
Sample
Probability
Inference
Hypotheses
Statistics
Likelihood
Support

5. Likelihood According to Luck
6/32 = 18.75%
Flip 1 2 3 4 5 Y N

6. Binomial Distribution
Binary data
A set of two-choice outcomes, e.g. yes/no, right/wrong
Binomial variable
A statistic for binary samples
Counts how many are “yes” / “right” / etc.
Binomial distribution
Probability distribution for a binomial variable
Gives probability for each possible value, from 0 to n
A family of distributions
Like Normal (need to specify mean and SD)
n: number of observations (sample size)
q: probability correct each time

7. Binomial Distribution
Formula (optional):
n = 20, q = .5
n = 5, q = .5
n = 20, q = .5
n = 20, q = .25

8. Binomial Test Hypothesis testing for binomial statistics
Null hypothesis
q = .5
Nothing interesting going on; blind chance
Alternative hypothesis
q equals something else
Find likelihood according to null hypothesis
p(count  c)
Special number, called p-value or p
Tells how good an explanation for the data the null hypothesis is
If p > .05
Null hypothesis can explain data
If p < .05
Null hypothesis cannot explain data
Rule out null and believe alternative hypothesis
.05 is arbitrary cut-off; more on this next time

9. Applications Any question of rate or proportion Population Parameter
Rates of outcomes: coins, other probabilities
Answers relative to chance: learning, memory, etc.
Composition of group: sex, politics, attitudes
Population
All people, events, etc.
Often infinite (probabilities)
Parameter
Proportion or probability in population
Sample
Set of binary outcomes
Answers for one student; category for each subject
Statistic
Proportion or count in sample

10. Testing for ESP Null hypothesis: Luck, q = .5
Alternative hypothesis: q > .5
Likelihood according to null: Binomial(5, .5)
p > .05
Retain null hypothesis
Luck can explain the data
.15625
.03125

11. Testing for ESP – Stronger Evidence
Nine correct out of ten
Null hypothesis: Luck, q = .5
Alternative hypothesis: q > .5
Likelihood according to null: Binomial(10, .5)
p < .05
Luck can’t explain the data
Reject null hypothesis
.010
.001

12. Other values of q Multiple-choice tests; guessing card suits
New null hypothesis, likelihood function
n = 20, q = .25
p = .014

13. Normal Approximation For large n, Binomial is nearly Normal
Consequence of Central Limit Theorem: count = mean  n
Mean = nq, Standard Deviation =
Convert cutoff to z-score .
Use Standard Normal to find p
Cutoff = 8.5 p
count n q
Mean SD
cutoff z
Approx
Exact 4 5 .5
2.5
1.1
3.5
.89
.186
.188 9 10
5.0
1.6
8.5
2.21
.013
.011