Independent Samples 1.Random Selection: Everyone from the Specified Population has an Equal Probability Of being Selected for the study (Yeah Right!) 2.Random Assignment: Every participant has an Equal Probability of being in the Treatment Or Control Groups

The Null Hypothesis Both groups from Same Population No Treatment Effect Both Sample Means estimate Same Population Mean Difference in Sample Means reflect Errors of Estimation of Mu X-Bar 1 + e 1 = Mu (Mu – X-Bar 1 = e 1 ) X-Bar 2 + e 2 = Mu (Mu – X-Bar 2 = e 2 ) Errors are Random and hence Unrelated

Expectation If Both Samples were selected from the Same Population: How much should the Sample Means Disagree about Mu? X-Bar 1 – X-Bar 2 Errors of Estimation decrease with N Errors of Estimation increase with Population Heterogeneity

The Expected Disagreement The Standard Error of a Difference: SE X-Bar1-X-Bar2 The Average Difference between two Sample Means The Expected Difference between two Sample Means When they are Estimating the Same Mu 68% chance of this much Or Less 95% chance of (this much x 2) Or Less Actually this much x 1.96, if you know sigma Rounded up to 2

Expectation: The Standard Error of the Difference The Expected Disagreement between two Sample Means (if H 0 true) T for Treatment Group C for Control Group  SEM for Treatment Group  SEM for Control Group Add the Errors and take the Square Root

Evaluation Compare the Difference you Got to the Difference you would Expect If H 0 true What you Got What you Expect ? df = n 1 + n 2 - 2

Evaluation Compare the Difference you Got to the Difference you would Expect If H 0 true What you Got What you Expect ? a) If they agree: Keep H 0 b)If they disagree: Reject H 0 Is TOO DAMN BIG!

Burn This!

Power The ability to find a relationship when it exists Errors of Estimation and Standard Errors of the Difference decrease with N Use the Largest sample sizes possible Errors of Estimation increase with Population Heterogeneity Run all your subjects under Identical Conditions (Experimental Control)

Power What if your data look like this? Everybody increased their score (X-bar 1 – X-Bar 2 ), but heterogeneity among subjects (SEM 1 & SEM 2 ) is large

Power Correlated Samples Designs: Natural Pairs: E.G.: Father vs. Son Measuring liberal attitudes Matched Pairs: Matching pairs of students on I.Q. One of each pair gets treatment (e.g., teaching with technology Repeated Measures: Measure Same Subject Twice (e.g., Pre-, Post-therapy) Look at differences between Pairs of Data Points, ignoring Between Subject differences

Correlated Samples  Same as  usual  Minus strength  of Correlation Smaller denominator Makes t bigger, hence More Power If r=0, denominator is the same, but df is smaller

Effect Size What are the Two Ts of Research? What is better than computing Effect Size? A weighted average of Two estimates of Sigma

Confidence Interval Use 2-tailed t-value at 95% confidence level With N 1 + N 2 –2 df N-1 df Does the Interval cross Zero?  Best Estimate 

Assumptions of the t-Test Both (if more than one) population(s): 1.Normally distributed 2.Equal variance Violations of Assumptions: Robust unless gross Transform scores (e.g. take log of each score)

Power Power = 1 – Beta Theoretical (Beta usually unknown) Reject H 0 : Decision is clear, you have a relationship Fail to reject H 0 : Decision is unclear, you may have failed to find a Relationship due to lack of Power

Power 1.Increases with Effect Size (Mu 1 – Mu 2 ) 2.Increases with Sample Size If close to p<0.05 add N 3.Decreases with Standard Error of the Difference (denominator) Minimize by Recording data correctly Use consistent criteria Maintain consistent experimental conditions (control) (Increasing N)