1. Two Sample Tests When do use independent
When groups are inherently different
Normal controls vs. patient
Men vs. Women
When participation in one condition contaminates measurement of other condition
Conjunction fallacy (Earthquakes)
Reasoning problems

2. Two Sample T-test Independent Means
Two-tailed:
H0 : μ1 = μ2 H0 : μ1 μ2
One-tailed:
H0 : μ1 <= μ2 H1 : μ1 > μ2
H0 : μ1 >= μ2 H1 : μ1 < μ2 H1

3. Two Sample Tests Which distributions to use?
Related (dependent)
(Observed Value) – (Expected value under Null Hypothesis)
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Standard error of null hypothesis of mean differences
Unrelated (independent)
(Observed Value) – (Expected value under Null Hypothesis)
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Standard error of null hypothesis (comparison) distribution
= ????

4. Independent t-test Null hypothesis distribution
Comparing two different population means
Collecting two samples and asking:
Are these means from same population?
To answer this, we need to know:
What should a difference of these means be? 0.
But how much variability should there be between a difference of means? Hmmm..
Comparing a difference between these means requires a
distribution of the difference of means

5. Independent t-test Distribution of the difference of means
Three steps
Step 1: Find how much scores vary.
Step 2: Use central limit theorem to find out how much means vary.
Step 3: Find out how much difference of means vary

6. Independent t-test Finding how much scores vary
If we had one sample, we could estimate how population varies by using sample σ (same as sample estimate of population σ)
But we have two samples, so how do we combine two estimates of population standard deviation?
By using them both.
Should they both be used equally?
Are they both equally good estimates?
What is one is larger sample size?
Answer: Use larger sample size estimates more than smaller sample size estimates

7. Independent t-test Step 1: Pooled variance
Pooled variance is the combination of both sample standard variances weighted by the degrees of freedom (n-1).
Larger sample is weighted more. Smaller sample is weighted less.

8. Step 2: Variance of Means

9. Step 3: Variance of Difference of Means

10. Step 3: Variance of Difference of Means
Group 1 (California): n=36
Group 2 (USA): n = 9
Group 1 is 4 times as large, so mean distribution is twice as skinny.
Group 1 (California): n=36
Group 2 (USA): n = 9

12. Independent t-test
(Observed Value) – (Expected value under Null Hypothesis)
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13. __________________________________________________
Independent t-test
(μ1 - μ2 ) – (0)
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14. μ1 - μ2 __________________________________________________
Independent t-test
μ1 - μ2 __________________________________________________
Dftot = df1 + df2