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

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

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

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

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

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

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.

Step 2: Variance of Means

Step 3: Variance of Difference of Means

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

Independent t-test (Observed Value) – (Expected value under Null Hypothesis) __________________________________________________

__________________________________________________ Independent t-test (μ1 - μ2 ) – (0) __________________________________________________

μ1 - μ2 __________________________________________________ Independent t-test μ1 - μ2 __________________________________________________ Dftot = df1 + df2