Lecture 11: Chapter 9 David Wallace Croft http://www.CroftSoft.com/people/david/ Statistics for Psychology 2005 Jun 17 Fri Copyright 2005 David Wallace Croft This work is licensed under the Creative Commons Attribution License 2.0. http://creativecommons.org/licenses/by/2.0/

Quiz Please mute your mobile phones Write your name on a blank piece of paper Quiz will begin at 09:00 When done, turn your paper over At 09:02, I will say, “Pens down” Writing after “Pens down” is cheating Pass your quizzes to your left

Outline Administrative Chapter 9 Slides Emphasis Example Assignment Exam 4 Prep Questions Brief Recess Homework Review

Administrative All quiz grades in Blackboard now Exam 3 not graded yet Free statistics tutoring UTD Learning Resources Center Located in the library

Chapter 9 Introduction to the t Test Aron and Aron, “Statistics for Psychology”, 3rd Ed., 2003.

Emphasis Why divide by N – 1? Why divide by N? Why df and t distribution?

Why Divide by N – 1? Why use S2 = SS / ( N – 1 )? Expected Value E ( X ) = ∑ ( x * px ) = μ S2 is unbiased estimator for σ2 E ( S2 ) = … = σ2 Proof on page 271 of Larsen, Richard J. and Morris L. Marx, “An Introduction to Probability and Its Applications”, Prentice-Hall, 1985.

Why Divide by N? Why does σM2 = σ2 / N? Yi ≡ Xi / N M = (∑Xi ) / N = ∑(Xi / N ) = ∑Yi E ( Yi ) = M / N Var ( Yi ) = E ( [ Yi – E ( Yi ) ]2 ) = E ( [ Xi / N – M / N ]2 ) = E ( [ Xi – M ]2 ) / N2 = σ2 / N2 Var ( ∑Yi ) = ∑ Var ( Yi ) (Yi’s independent) σM2 = Var ( M ) = Var ( ∑Yi ) = ∑ Var ( Yi ) = ∑ ( σ2 / N2 ) = N * ( σ2 / N2 ) = σ2 / N

Why df and t distribution? df = N – 1 t distribution thicker in tails Robust unless extreme skew Requires more extreme cutoffs t distribution → normal distribution as N →∞ compare p < 0.1 entry for df = ∞ to z table

Example Chapter 9, Set I, Problem 5, p334 Sign wrong in answers to Problems 5 and 6? df = N – 1 = 4 – 1 = 3 tcutoff = -4.541 (page 642) Differences: -7, -6, +1, -8 M = -20 / 4 = -5 SS = (-7- -5)2 + (-6 - -5)2 + (1 - -5)2 + (-8 - -5)2 = 4 + 1 + 36 + 9 = 50 S2 = SS / df = 50 / 3 = 16.7 SM = √ ( S2 / N ) = √ ( 16.7 / 4 ) = 2.04 t = ( M – μ ) / SM = ( -5 - 0 ) / 2.04 = -2.45 t(3) = -2.45, n.s., one-tailed

Assignment Monday Exam will cover Chapters 8 and 9 No calculators, closed book/closed notes Wednesday Read Chapter 10 before class Homework: Set I, Problem 3 Quiz at start of class

Exam 4 Prep Do homework Know theory Know glossary terms

Questions Don’t pack up to leave just yet Questions about lecture before we dismiss? Post additional questions to discussion electronic mailing list unless personal http://egroups.com/group/utd-statistics

Brief Recess 5 minute break Homework review when we return Attendance optional

Homework Review Chapter 9, Set I, Problem 1, p332 df = N – 1 Look up tcutoff from table (p642) Use lower df if not in table (conservative) SM = √ ( S2 / N ) t = ( M – μ ) / SM Decide whether to reject null hypothesis