1. Lecture 11: Chapter 9 David Wallace Croft
Statistics for Psychology
2005 Jun 17 Fri
Copyright 2005 David Wallace Croft
This work is licensed under the
Creative Commons Attribution License 2.0.

2. 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

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

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

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

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

7. 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.

8. 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

9. 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

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

11. 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

12. Exam 4 Prep
Do homework
Know theory
Know glossary terms

13. 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

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

15. 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