Copyright © 2012 by Nelson Education Limited. Chapter 9 Hypothesis Testing III: The Analysis of Variance 9-1.

Slides:



Advertisements
Similar presentations
ANALYSIS OF VARIANCE (ONE WAY)
Advertisements

Chapter 10 Hypothesis Testing Using Analysis of Variance (ANOVA)
Chapter 12 ANALYSIS OF VARIANCE.
Hypothesis Testing IV Chi Square.
Analysis and Interpretation Inferential Statistics ANOVA
Chapter 10 Hypothesis Testing III (ANOVA). Basic Logic  ANOVA can be used in situations where the researcher is interested in the differences in sample.
Chapter Topics The Completely Randomized Model: One-Factor Analysis of Variance F-Test for Difference in c Means The Tukey-Kramer Procedure ANOVA Assumptions.
Chapter 9 Hypothesis Testing II. Chapter Outline  Introduction  Hypothesis Testing with Sample Means (Large Samples)  Hypothesis Testing with Sample.
Hypothesis Testing: Two Sample Test for Means and Proportions
Chapter 9 Hypothesis Testing II. Chapter Outline  Introduction  Hypothesis Testing with Sample Means (Large Samples)  Hypothesis Testing with Sample.
Hypothesis Testing and T-Tests. Hypothesis Tests Related to Differences Copyright © 2009 Pearson Education, Inc. Chapter Tests of Differences One.
Psy B07 Chapter 1Slide 1 ANALYSIS OF VARIANCE. Psy B07 Chapter 1Slide 2 t-test refresher  In chapter 7 we talked about analyses that could be conducted.
Statistics: A Tool For Social Research
HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Chapter 14 Analysis.
Chapter 13 – 1 Chapter 12: Testing Hypotheses Overview Research and null hypotheses One and two-tailed tests Errors Testing the difference between two.
Week 9 Chapter 9 - Hypothesis Testing II: The Two-Sample Case.
Hypothesis Testing II The Two-Sample Case.
Copyright © 2012 by Nelson Education Limited. Chapter 8 Hypothesis Testing II: The Two-Sample Case 8-1.
1 Tests with two+ groups We have examined tests of means for a single group, and for a difference if we have a matched sample (as in husbands and wives)
Week 10 Chapter 10 - Hypothesis Testing III : The Analysis of Variance
Week 8 Chapter 8 - Hypothesis Testing I: The One-Sample Case.
Chapter 8 Hypothesis Testing I. Chapter Outline  An Overview of Hypothesis Testing  The Five-Step Model for Hypothesis Testing  One-Tailed and Two-Tailed.
Chapter 9 Hypothesis Testing II: two samples Test of significance for sample means (large samples) The difference between “statistical significance” and.
Copyright © 2012 by Nelson Education Limited. Chapter 7 Hypothesis Testing I: The One-Sample Case 7-1.
Chapter 9: Testing Hypotheses
© Copyright McGraw-Hill CHAPTER 12 Analysis of Variance (ANOVA)
Copyright © 2012 by Nelson Education Limited. Chapter 10 Hypothesis Testing IV: Chi Square 10-1.
Chapter 10 Analysis of Variance.
Chapter 14 – 1 Chapter 14: Analysis of Variance Understanding Analysis of Variance The Structure of Hypothesis Testing with ANOVA Decomposition of SST.
One-Way Analysis of Variance
One-Way ANOVA ANOVA = Analysis of Variance This is a technique used to analyze the results of an experiment when you have more than two groups.
Chapter 12 Analysis of Variance. An Overview We know how to test a hypothesis about two population means, but what if we have more than two? Example:
PPA 501 – Analytical Methods in Administration Lecture 6a – Normal Curve, Z- Scores, and Estimation.
PPA 415 – Research Methods in Public Administration Lecture 7 – Analysis of Variance.
Chapter 14 – 1 Chapter 14: Analysis of Variance Understanding Analysis of Variance The Structure of Hypothesis Testing with ANOVA Decomposition of SST.
Lecture 9-1 Analysis of Variance
Chapter 13 - ANOVA. ANOVA Be able to explain in general terms and using an example what a one-way ANOVA is (370). Know the purpose of the one-way ANOVA.
Chapter 17 Comparing Multiple Population Means: One-factor ANOVA.
HYPOTHESIS TESTING FOR VARIANCE AND STANDARD DEVIATION Section 7.5.
Chapter 8 Hypothesis Testing I. Significant Differences  Hypothesis testing is designed to detect significant differences: differences that did not occur.
Chapter 9: Testing Hypotheses Overview Research and null hypotheses One and two-tailed tests Type I and II Errors Testing the difference between two means.
Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics S eventh Edition By Brase and Brase Prepared by: Lynn Smith.
Chapter 12 Introduction to Analysis of Variance PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick.
ANOVA P OST ANOVA TEST 541 PHL By… Asma Al-Oneazi Supervised by… Dr. Amal Fatani King Saud University Pharmacy College Pharmacology Department.
Chapter 10 Hypothesis Testing III (ANOVA). Chapter Outline  Introduction  The Logic of the Analysis of Variance  The Computation of ANOVA  Computational.
CHAPTER 12 ANALYSIS OF VARIANCE Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved.
The Analysis of Variance ANOVA
Statistics for Political Science Levin and Fox Chapter Seven
CHAPTER 10 ANOVA - One way ANOVa.
Statistical Analysis ANOVA Roderick Graham Fashion Institute of Technology.
Chapter 14: Analysis of Variance One-way ANOVA Lecture 9a Instructor: Naveen Abedin Date: 24 th November 2015.
Chapter 10 Section 5 Chi-squared Test for a Variance or Standard Deviation.
CHAPTER 7: TESTING HYPOTHESES Leon-Guerrero and Frankfort-Nachmias, Essentials of Statistics for a Diverse Society.
CHAPTER 10: ANALYSIS OF VARIANCE(ANOVA) Leon-Guerrero and Frankfort-Nachmias, Essentials of Statistics for a Diverse Society.
Chapter 12 Introduction to Analysis of Variance
DSCI 346 Yamasaki Lecture 4 ANalysis Of Variance.
Practice Questions for ANOVA
Statistics: A Tool For Social Research
10 Chapter Chi-Square Tests and the F-Distribution Chapter 10
Where we are Where we are going
CHAPTER 12 ANALYSIS OF VARIANCE
Hypothesis Testing: Two Sample Test for Means and Proportions
Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Spring 2017 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays.
INTEGRATED LEARNING CENTER
Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Spring 2018 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays.
Chapter 14: Analysis of Variance One-way ANOVA Lecture 8
Introduction to ANOVA.
Analysis of Variance (ANOVA)
Hypothesis Testing for Proportions
Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Spring 2019 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays.
Presentation transcript:

Copyright © 2012 by Nelson Education Limited. Chapter 9 Hypothesis Testing III: The Analysis of Variance 9-1

Copyright © 2012 by Nelson Education Limited. The basic logic of hypothesis testing as applied to analysis of variance (ANOVA) Perform the ANOVA test using the five-step model Limitations of ANOVA In this presentation you will learn about: 9-2

Copyright © 2012 by Nelson Education Limited. ANOVA (analysis of variance) can be used in situations where the researcher is interested in the differences in sample means across three or more categories. Examples: ◦ How do urban, suburban, and rural families vary in terms of number of children? ◦ How do people with less than high school, high school, and post-secondary education vary in terms of income? ◦ How do younger, middle-aged, and older people vary in terms of frequency of religious service attendance? Basic Logic 9-3

Copyright © 2012 by Nelson Education Limited. Can think of ANOVA as extension of t test for more than two groups. ANOVA asks “are the differences between the samples large enough to reject the null hypothesis and justify the conclusion that the populations represented by the samples are different?” –The H 0 is that the population means are the same: H 0: μ 1 = μ 2 = μ 3 = … = μ k –If the H 0 is true, the sample means should be about the same value. Basic Logic (continued) 9-4

Copyright © 2012 by Nelson Education Limited. If H 0 is false, there should be substantial differences between the sample means of the categories, combined with relatively little difference within (sample standard deviations should be low in value) categories. When we reject the H 0, we are saying there are differences between the populations represented by the samples. Basic Logic (continued) 9-5

Copyright © 2012 by Nelson Education Limited. Students taking introductory biology at a large university were randomly assigned to one of three sections: 1.the first section was taught by traditional “lecture-lab” method 2.the second section by “all-lab” method 3.the third section by “videotaped lectures and labs” method.  At the end of the semester, random samples of final exam scores were collected from each section. Basic Logic: An Example 9-6

Copyright © 2012 by Nelson Education Limited. In Scenario 1 (Table 9.1) Means and standard deviations of the groups are very similar. These results would be quite consistent with the null hypothesis of no difference. Basic Logic: An Example (continued) 9-7

Copyright © 2012 by Nelson Education Limited. In Scenario 2 (Table 9.2) There are large differences in scores between groups (means) but small differences in scores within each group (standard deviations). These results would contradict the null hypothesis, and support the notion that final exam scores do vary by teaching method. Basic Logic: An Example (continued) 9-8

Copyright © 2012 by Nelson Education Limited. 1. Calculate total sum of squares (SST): OR Highlighted formula provides a quicker way to calculate the statistic. Six Steps in Computation of ANOVA 9-9

Copyright © 2012 by Nelson Education Limited. 2. Calculate sum of squares between (SSB): Six Steps in Computation of ANOVA (continued) 9-10

Copyright © 2012 by Nelson Education Limited. 3. Calculate sum of squares within (SSW): OR Highlighted formula provides a quicker way to calculate the statistic. Six Steps in Computation of ANOVA (continued) 9-11

Copyright © 2012 by Nelson Education Limited. 4. Calculate degrees of freedom (Formulas 9.5 and 9.6): Six Steps in Computation of ANOVA (continued) 9-12

Copyright © 2012 by Nelson Education Limited. 5. Calculate the mean squares (Formulas 9.7 and 9.8): Six Steps in Computation of ANOVA (continued) 9-13

Copyright © 2012 by Nelson Education Limited. 6. Calculate F ratio (Formula 9.9): Six Steps in Computation of ANOVA (continued) 9-14

Copyright © 2012 by Nelson Education Limited. The computational routine for ANOVA can be summarized as: Six Steps in Computation of ANOVA: Summary 9-15

Copyright © 2012 by Nelson Education Limited. The grade point average of students in three (co-ed, all-male, and all-female) residences has been monitored by the administration of a university. The GPA from random samples of 14 students from each residence was collected. Computation of ANOVA: An Example 9-16

Copyright © 2012 by Nelson Education Limited. Does GPA vary significantly by type of residence? Computation of ANOVA: An Example (continued) Co-Ed All-Male All-Female

Copyright © 2012 by Nelson Education Limited. Co-Ed All-Male All-Female ΣX = = ΣX 2 = = 2.83 Computation of ANOVA: An Example (continued) 9-18

Copyright © 2012 by Nelson Education Limited. The difference in the means suggests that GPA does vary by type of residence. GPA seems to be highest in co-ed residence and lowest in all-male residence. Are these differences statistically significant? Computation of ANOVA: An Example (continued) 9-19

Copyright © 2012 by Nelson Education Limited. Six Steps in Computation of ANOVA: 1. SST (Formula 9.10) = ( )-(42)(2.83) 2 = (42)(8.01) = = Computation of ANOVA: An Example (continued) 9-20

Copyright © 2012 by Nelson Education Limited. 2. SSB (Formula 9.4) = 14( ) ( ) ( ) 2 = 14(0.26) + 14(0.31) + 14(0.0036) = = 8.03 Computation of ANOVA: An Example (continued) 9-21

Copyright © 2012 by Nelson Education Limited. 3. SSW (Formula 9.11) = 22.35– 8.03 = Degrees of freedom (Formulas 9.5 and 9.6) dfw= n - k = = 39 dfb= k - 1 = = 2 Computation of ANOVA: An Example (continued) 9-22

Copyright © 2012 by Nelson Education Limited. 5. Mean Squares (Formulas 9.7 and 9.8) MSW = SSW/dfw =14.32/39 = 0.37 MSB = SSB/dfb = 8.03 /2 = 4.02 Computation of ANOVA: An Example (continued) 9-23

Copyright © 2012 by Nelson Education Limited. Computation of ANOVA: An Example (continued) 6. F ratio (Formula 9.9) = 4.02 / 0.37 =

Copyright © 2012 by Nelson Education Limited. Independent Random Samples Level of Measurement is Interval-Ratio –The dependent variable (e.g., GPA) should be I-R to justify computation of the mean. Populations are normally distributed. Population variances are equal. *ANOVA will tolerate some deviation from its assumptions as long as sample sizes are roughly equal. Performing the ANOVA Test Using the Five-Step Model Step 1: Make Assumptions and Meet Test Requirements* 9-25

Copyright © 2012 by Nelson Education Limited. H 0 : μ 1 = μ 2 = μ 3 –The H 0 states that the population means are the same. H 1 : At least one population mean is different. Step 2: State the Null Hypothesis 9-26

Copyright © 2012 by Nelson Education Limited. Sampling Distribution = F distribution Alpha = 0.05 dfw = (n – k) = 39 dfb = k – 1 = 2 F(critical) = 3.32 (Note, the exact dfw (39) is not in the table but dfw = 30 and dfw = 40 are. Choose the larger F ratio as F critical). Step 3: Select Sampling Distribution and Establish the Critical Region 9-27

Copyright © 2012 by Nelson Education Limited. F (obtained) = Step 4: Calculate the Test Statistic 9-28

Copyright © 2012 by Nelson Education Limited. F (obtained) = F (critical) = 3.32 –The test statistic, F (obtained), falls in the critical region. W e reject the null hypothesis, H 0, of no difference. At least one of these residences is significantly different than the other residences. Step 5: Make Decision and Interpret Results 9-29

Copyright © 2012 by Nelson Education Limited. 1.Requires interval-ratio level measurement of the dependent variable 2.Statistically significant differences are not necessarily important. Limitations of ANOVA 9-30

Copyright © 2012 by Nelson Education Limited. 3.The alternative (research) hypothesis, H 1, is not specific. It only asserts that at least one of the population means differs from the others. –Thus, we must use other (e.g., post hoc) statistical techniques for more specific differences. Limitations of ANOVA (continued) 9-31