This chapter introduces students to the design of experiments and analysis of variance. It covers one-way and two-way ANOVA, randomized block designs, and interaction. Students learn to compute and interpret results from one-way ANOVA, randomized block designs, and two-way ANOVA. They also learn about multiple comparison tests and when to use them to analyze differences between specific treatment means.
QUANTITATIVE TECHNIQUES, TIME SERIES, CROSS SECTIONAL ANALYSIS, TIME SERIES RESEARCH, CROSS SECTIONAL RESEARCH, COMPARISON BETWEEN TIME SERIES AND CROSS SECTIONAL ANALYSIS, QUANTITATIVE ANALYSIS, QUANTITATIVE RESEARCH, RESEARCH METHODS, ORGANIZATION'S STUDY, LIBCORPIO786, BUSINESS ADMINISTRATION, MANAGEMENT SCIENCE, EDUCATION AND LEARNING,
QUANTITATIVE TECHNIQUES, TIME SERIES, CROSS SECTIONAL ANALYSIS, TIME SERIES RESEARCH, CROSS SECTIONAL RESEARCH, COMPARISON BETWEEN TIME SERIES AND CROSS SECTIONAL ANALYSIS, QUANTITATIVE ANALYSIS, QUANTITATIVE RESEARCH, RESEARCH METHODS, ORGANIZATION'S STUDY, LIBCORPIO786, BUSINESS ADMINISTRATION, MANAGEMENT SCIENCE, EDUCATION AND LEARNING,
A Case Study on Research In Motion (now BlackBerry).
The case study is published by Amity Business School. Any kind of copyright infringement or plagiarism is strictly prohibited. Please respect the author and the extensive research that has been involved.
The analysis is purely for academic purposes only.
Experimental design is a way to carefully plan experiments in advance so that results are both objective and valid. Ideally, an experimental design should:
• Describe how participants are allocated to experimental groups. A common method is completely randomized design, where participants are assigned to groups at random. A second method is randomized block design, where participants are divided into homogeneous blocks (for example, age groups) before being randomly assigned to groups.
• Minimize or eliminate confounding variables, which can offer alternative explanations for the experimental results.
• Allows making inferences about the relationship between independent variables and dependent variables.
• Reduce variability, to make it easier to find differences in treatment outcomes.
Types of Experimental Design
1. Between Subjects Design.
2. Completely Randomized Design.
3. Factorial Design.
4. Matched-Pairs Design.
5. Observational Study
• Longitudinal Research
• Cross Sectional Research
6. Pretest-Posttest Design.
7. Quasi-Experimental Design.
8. Randomized Block Design.
9. Randomized Controlled Trial
10. Within subjects Design.
Here is a piece of detailed information about the experimental design used in the field of statistics. This also features some information on the three most widely accepted and most widely used designs.
UNIT 3
SUCCESS GUIDE
1 | GB 513 Unit 3 Success Guide v.6.13.17
UNIT 3 SUCCESS GUIDE
This unit is the other “most difficult” one. Hypothesis testing has two parts: setting-up
the hypotheses and calculating the critical values to determine results. They both
pose difficulty for a lot of students. The seminar will be on the first and the recorded
lecture will be on the second. You need to make sure you understand both,
otherwise you will not be able to get to the right conclusions.
1. As always, start by reading the chapters and studying the solved examples.
2. Watch the lecture video in document sharing. It focuses on why we do
hypothesis testing, how to do it with Excel and solves two sample problems.
3. Watch this from Khan Academy:
https://www.khanacademy.org/math/statistics-probability/significance-
tests-one-sample/tests-about-population-mean/v/hypothesis-testing-and-p-
values
This one talks more about how to write the null and alternative hypotheses
(which a lot of students get wrong) and also solves the problem using
formulas.
4. Watch the sample problem solutions in Course Resources.
5. If you still want more videos, search YouTube for “hypothesis testing.” Several
introductory level videos are available, such as
https://www.youtube.com/watch?v=HmMjS88eSVE and
https://www.youtube.com/watch?v=0zZYBALbZgg
Email your instructor if you find any of these links to be broken.
Avoid these mistakes!
GENERAL NOTES
RESOURCES
COMMON MISTAKES IN THE ASSIGNMENT
https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/hypothesis-testing-and-p-values
https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/hypothesis-testing-and-p-values
https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample/tests-about-population-mean/v/hypothesis-testing-and-p-values
http://www.youtube.com/watch?v=HmMjS88eSVE
http://www.youtube.com/watch?v=HmMjS88eSVE
http://www.youtube.com/watch?v=HmMjS88eSVE
http://www.youtube.com/watch?v=0zZYBALbZgg
http://www.youtube.com/watch?v=0zZYBALbZgg
http://www.youtube.com/watch?v=0zZYBALbZgg
2 | GB 513 Unit 3 Success Guide v.6.13.17
Students commonly get the null and alternative hypotheses reversed, or
get them completely wrong.
Students also commonly do not state the hypothesis fully. This is correct:
“null hypothesis: there is no difference between the average salary for
group 1 and the average salary of group 2.” This is not sufficient: “ho:
x1=x2”
Students sometimes compare the averages of the two groups and base
their determination on which one is greater, rather than properly doing a
hypothesis test.
Students sometimes do the calculations correctly, but do not write out
what the conclusion is. This is correct: “We therefore reject the null
hypothesis, which means we conclude that there i ...
Experiments
A Quick History of Design of Experiments
Why We Use Experimental Designs
What is Design of Experiment
How Design of Experiment contributes
Terminology
Analysis Of Variation (ANOVA)
Basic Principle of Design of Experiments
Some Experimental Designs
12/24/16, 11(23 AMModule 8: Mastery Exercise | Schoology
Page 1 of 3https://app.schoology.com/assignment/885059160/assessment
Statistics and SPSS: WINTER16-B-8-MIS445-1
Module 8: Mastery Exercise
Question 1 (1 point)
In a body weight loss trial, the calculated F-value was 5.91 and the tabulated F (0.95, 3, 16) = 3.2; what should be the conclusion?
a Since F-calculated, 5.91 is bigger than F-tabulated, 3.2, therefore, reject the null hypothesis that dietary treatments were
similar in reducing body weight.
b Accept the null hypothesis of no dietary treatments effects.
c Nothing can be calculated.
Question 2 (1 point)
Some of the assumptions, for the data used in ANOVA are _________.
a data follows a normal distribution
b population means have similar variance (or standard deviation)
c samples are randomly selected and independent of one another
d all of the above
Question 3 (1 point)
Researchers wish to examine the effectiveness of a new weight-loss pill. A total of 200 obese adults are randomly assigned to one of
four conditions: weight-loss pill alone, weight-loss pill with a low-fat diet, placebo pill alone, or placebo pill with a low-fat diet. The
weight loss after six months of treatment is recorded in pounds for each subject. To analyze this data, you would use __________.
a a z-test
b a t-test
c an ANOVA F test
d a Chi-square test
Question 4 (1 point)
A medical research team is interested in determining whether a new drug has an effect on creatine kinase (CK), which is often assayed in
blood tests as an indicator of myocardial infarction. A random selection of 20 patients from a pool of possible subjects is selected, and
each subject is given the medication. The subjects’ CK levels are observed initially, after three (3) weeks, and again after six (6) weeks.
The purpose is to study the CK levels over time. Here is a summary of the findings:
Time (weeks) Mean CK level (U/L) Standard devia9on (U/L)
0 121 20.37
3 106 16.09
6 100 10.21
In this example, we notice that ____________.
a the data shows very strong evidence of a violation of the assumption that the three populations have the same standard
deviation
Questions 1-10 of 10 | Page 1 of 1
https://app.schoology.com/course/885058852
12/24/16, 11(23 AMModule 8: Mastery Exercise | Schoology
Page 2 of 3https://app.schoology.com/assignment/885059160/assessment
b ANOVA cannot be used on this data because the sample sizes are much too small
c the assumption that the data is independent for the three time points is unreasonable because the same subjects were
observed each time
d there is no reason not to use ANOVA in this situation
Question 5 (1 point)
The degree of freedom for the total number of observations in ANOVA will be ________.
a total number of observations less one
b total number of observations less two
c total number of observations plus one
d total number of observations plus two
Question 6 (1 point)
How much corn should be plante ...
Educational Psychology 565 Practice Quiz(use α = .05 unl.docxtoltonkendal
Educational Psychology 565 Practice Quiz
(use α = .05 unless otherwise stated).
1. A small school district wants to know what type of teaching/learning is most effective at helping students learn to read. Three methods are proposed (top-down, bottom-up, and interactive). It is believed that the gender of the teacher may also be important in student learning, so the study also aims to determine if gender of the teacher is important. There are 12 schools in the district, and each school has 1 second grade class (each class has 10 students). Two female teachers and two male teachers’ classrooms are randomly assigned to each of the three methods (all 12 teachers have just been hired in the district). At the end of the year, the students all took a 100 item standardized multiple-choice reading test called the “EZreading” test (note: the analysis was performed at the student level).
Coding:
teachgender = gender of teacher: 1= men, 2 = women
Teachmeth = teaching method (1=top-down, 2=bottom-up, 3=interactive)
EZread = scores on the Ezread reading test
Use SPSS output “SPSS printout for question 1”to help answer the parts below.
a. What is/are the independent variable(s) in this experiment (Be specific)? What level of measurement is/are the IV(s)? Explain why?
b. What is/are the dependent variable(s) in this experiment (Be specific)? What level of measurement is/are the DV(s)? Explain why?
c. State the null hypotheses and alternative hypotheses for the factors and the interaction in symbols and words.
d. Do you think the assumption of homogeneity of variance has been met? Support your answer.
e. Do you think the assumption of independence has been met? Support your answer.
f. Calculate Cohen’s d for the difference between the top-down and interactive methods. Explain what Cohen’s d means for this comparison.
g. Is the interaction of the two factors statistically significant? Explain your answer.
h. Report the results of the study along with an interpretation for the results. You do not need to write up the results like a results section; you can just report the findings with statements about each factor and the interaction of the two factors. Be sure to cite evidence from your analysis.
i. Based on the results of the study what would you recommend about teaching method and gender of teachers?
2. Answer the following questions.
Source
SS
df
MS
F
Between
100
20
Within
2
50
Total
200
7
a. Complete the ANOVA source table (fill in all blank spaces)
b. How many people are in this study. (hint: use degrees of freedom)
c. What is the critical F at α = .01? Would you reject the null hypothesis? Explain your answer.
d. What are the critical F at α = .05? Would you reject the null hypothesis? Explain your answer.
e. Why do the conclusions from items c and d differ? Explain your answer in terms of Type I and II errors.
3. A researcher wants to kn.
Researchers use several tools and procedures for analyzing quantitative data obtained from different types of experimental designs. Different designs call for different methods of analysis. This presentation focuses on:
T-test
Analysis of variance (F-test), and
Chi-square test
121
5t Tests
Owen Franken/Corbis
Chapter Learning Objectives
After reading this chapter, you should be able to do the following:
1. Explain the advantage of the one-sample t test over the z test.
2. Compare the one-sample t test to the independent t test.
3. Distinguish between one-sample and one-tailed t tests.
4. Explain hypothesis testing in statistical analysis.
5. Determine practical significance.
6. Construct a confidence interval for the difference between the means.
7. Discuss research applications for the t tests.
122
Section 5.1 Estimating the Standard Error of the Mean
Introduction
The z test (Chapter 4) involves more than just an expansion of the z score from individu-
als to groups. The z test also introduced us to statistical significance. Those who work with
quantitative data need to be able to distinguish between outcomes that probably occurred
by chance and those that are likely to emerge each time the data are gathered and analyzed.
For example, data indicate that a group of clients, each one grieving the loss of a loved one,
becomes more positive and peaceful with time. A therapist needs to know whether this would
have happened anyway with the passage of time, or whether it has something to do with the
treatment the therapist provided. The z test answers such questions.
The z test has important limitations, however. Its greatest difficulty is the requirement of a
value for the population standard error of the mean, σM. That value is not the type of infor-
mation that tends to come up just as a matter of course, and it may be inaccessible when the
researcher lacks access to population data, including the population standard deviation.
The z test’s second limitation is allowing only one type of comparison: a sample to a popula-
tion. What if two therapists want to compare their respective groups of grieving clients to see
if one type of grief counseling is better than the other? The z test does not allow that compari-
son, which is where William Sealy Gosset comes in.
Gosset worked for Guinness Brewing during the early part of the 20th century. Part of his
responsibility was quality control, and he studied ways to make sure that day-to-day brewing
remained consistent with Guinness’s standards. A man of remarkable ability, Gosset devised
procedures for quantifying product quality and then testing the consistency of the quality
over time. To help him in his analyses, he developed the t tests.
Gosset recognized that his t tests had application well beyond studying the quality of beer and
wanted to publish information about his tests so that others could benefit. A no-publishing
policy at Guinness—instituted after a previous employee published what the company con-
sidered trade secrets—presented a roadblock. Believing that his research would not compro-
mise Guinness, Gosset published anyway under the pseudonym “Student.” Traditional statis-
tics textbooks still contain references to “Student’s ...
SAP Sapphire 2024 - ASUG301 building better apps with SAP Fiori.pdfPeter Spielvogel
Building better applications for business users with SAP Fiori.
• What is SAP Fiori and why it matters to you
• How a better user experience drives measurable business benefits
• How to get started with SAP Fiori today
• How SAP Fiori elements accelerates application development
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• How SAP Fiori paves the way for using AI in SAP apps
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
Welocme to ViralQR, your best QR code generator.ViralQR
Welcome to ViralQR, your best QR code generator available on the market!
At ViralQR, we design static and dynamic QR codes. Our mission is to make business operations easier and customer engagement more powerful through the use of QR technology. Be it a small-scale business or a huge enterprise, our easy-to-use platform provides multiple choices that can be tailored according to your company's branding and marketing strategies.
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Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
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1. Chapter 11: Analysis of Variance and Design of Experiments 1
Chapter 11
Analysis of Variance and
Design of Experiments
LEARNING OBJECTIVES
The focus of this chapter is learning about the design of experiments and the analysis of
variance thereby enabling you to:
1. Understand the differences between various experiment designs and when to use
them.
2. Compute and interpret the results of a one-way ANOVA.
3. Compute and interpret the results of a random block design.
4. Compute and interpret the results of a two-way ANOVA.
5. Understand and interpret interaction.
6. Know when and how to use multiple comparison techniques.
CHAPTER TEACHING STRATEGY
This important chapter opens the door for students to a broader view of statistics
than they have seen to this time. Through the topic of experimental designs, the student
begins to understand how they can scientifically set up controlled experiments in which
to test certain hypotheses. They learn about independent and dependent variables. With
the completely randomized design, the student can see how the t test for two independent
samples can be expanded to include three or more samples by using analysis of variance.
This is something that some of the more curious students were probably wondering about
in chapter 10. Through the randomized block design and the factorial designs, the
student can understand how we can analyze not only multiple categories of one variable,
but we can simultaneously analyze multiple variables with several categories each. Thus,
2. Chapter 11: Analysis of Variance and Design of Experiments 2
this chapter affords the instructor an opportunity to help the student develop a structure
for statistical analysis.
In this chapter, we emphasize that the total sum of squares in a given problem do
not change. In the completely randomized design, the total sums of squares are parceled
into between treatments sum of squares and error sum of squares. By using a blocking
design when there is significant blocking, the blocking effects are removed from the error
effects which reduces the size of the mean square error and can potentially create a more
powerful test of the treatment. A similar thing happens in the two-way factorial design
when one significant treatment variable siphons off sum of squares from the error term
that reduces the mean square error and creates potential for a more powerful test of the
other treatment variable.
In presenting the random block design in this chapter, the emphasis is on
determining if the F value for the treatment variable is significant or not. We have de
emphasized examining the F value of the blocking effects. However, if the blocking
effects are not significant, the random block design may be a less powerful analysis of the
treatment effects. If the blocking effects are not significant, even though the error sum of
squares is reduced, the mean square error might increase because the blocking effects
may reduce the degrees of freedom error in a proportional greater amount. This might
result in a smaller treatment F value than would occur in a completely randomized
design. We have shown the repeated measures design in the chapter as a special case of
the random block design.
In factorial designs, if there are multiple values in the cells, it is possible to
analyze interaction effects. Random block designs do not have multiple values in cells
and therefore interaction effects cannot be calculated. It is emphasized in this chapter
that if significant interaction occurs, then the main effects analysis are confounded and
should not be analyzed in the usual manner. There are various philosophies about how to
handle significant interaction but are beyond the scope of this chapter. The main factorial
example problem in the chapter was created to have no significant interaction so that the
student can learn how to analyze main effects. The demonstration problem has
significant interaction and these interactions are displayed graphically for the student to
see. You might consider taking this same problem and graphing the interactions using
Row effects along the x axis and graphing the Column means for the student to see.
There are a number of multiple comparison tests available. In this text, I selected
one of the more well-known tests, Tukey's HSD, in the case of equal sample sizes. When
sample sizes are unequal, a variation on Tukey’s HSD, the Tukey-Kramer test, is used.
MINITAB uses the Tukey test as one of its options under multiple comparisons and uses
the Tukey-Kramer test for unequal sample sizes. Tukey's HSD is one of the more
powerful multiple comparison tests but protects less for Type I errors than some of the
other tests.
3. Chapter 11: Analysis of Variance and Design of Experiments 3
CHAPTER OUTLINE
11.1 Introduction to Design of Experiments
11.2 The Completely Randomized Design (One-Way ANOVA)
One-Way Analysis of Variance
Reading the F Distribution Table
Using the Computer for One-Way ANOVA
Comparison of F and t Values
11.3 Multiple Comparison Tests
Tukey's Honestly Significant Difference (HSD) Test: The Case of Equal Sample
Sizes
Using the Computer to Do Multiple Comparisons
Tukey-Kramer Procedure: The Case of Unequal Sample Sizes
11.4 The Randomized Block Design
Using the Computer to Analyze Randomized Block Designs
11.5 A Factorial Design (Two-Way ANOVA)
Advantages of the Factorial Design
Factorial Designs with Two Treatments
Applications
Statistically Testing the Factorial Design
Interaction
Using a Computer to Do a Two-Way ANOVA
KEY TERMS
a posteriori Factors
a priori Independent Variable
Analysis of Variance (ANOVA) Interaction
Blocking Variable Levels
Classification Variables Multiple Comparisons
Classifications One-way Analysis of Variance
Completely Randomized Design Post-hoc
Concomitant Variables Randomized Block Design
Confounding Variables Repeated Measures Design
Dependent Variable Treatment Variable
Experimental Design Tukey-Kramer Procedure
F Distribution Tukey’s HSD Test
F Value Two-way Analysis of Variance
Factorial Design
4. Chapter 11: Analysis of Variance and Design of Experiments 4
SOLUTIONS TO PROBLEMS IN CHAPTER 11
11.1 a) Time Period, Market Condition, Day of the Week, Season of the Year
b) Time Period - 4 P.M. to 5 P.M. and 5 P.M. to 6 P.M.
Market Condition - Bull Market and Bear Market
Day of the Week - Monday, Tuesday, Wednesday, Thursday, Friday
Season of the Year - Summer, Winter, Fall, Spring
c) Volume, Value of the Dow Jones Average, Earnings of Investment Houses
11.2 a) Type of 737, Age of the plane, Number of Landings per Week of the plane, City
that the plane is based
b) Type of 737 - Type I, Type II, Type III
Age of plane - 0-2 y, 3-5 y, 6-10 y, over 10 y
Number of Flights per Week - 0-5, 6-10, over 10
City - Dallas, Houston, Phoenix, Detroit
c) Average annual maintenance costs, Number of annual hours spent on
maintenance
11.3 a) Type of Card, Age of User, Economic Class of Cardholder, Geographic Region
b) Type of Card - Mastercard, Visa, Discover, American Express
Age of User - 21-25 y, 26-32 y, 33-40 y, 41-50 y, over 50
Economic Class - Lower, Middle, Upper
Geographic Region - NE, South, MW, West
c) Average number of card usages per person per month,
Average balance due on the card, Average per expenditure per person,
Number of cards possessed per person
11.4 Average dollar expenditure per day/night, Age of adult registering the family,
Number of days stay (consecutive)
5. Chapter 11: Analysis of Variance and Design of Experiments 5
11.5 Source df SS MS F
Treatment 2 22.20 11.10 11.07
Error 14 14.03 1.00
Total 16 36.24
α = .05 Critical F.05,2,14 = 3.74
Since the observed F = 11.07 > F.05,2,14 = 3.74, the decision is to reject the null
hypothesis.
11.6 Source df SS MS F
Treatment 4 93.77 23.44 15.82
Error 18 26.67 1.48
Total 22 120.43
α = .01 Critical F.01,4,18 = 4.58
Since the observed F = 15.82 > F.01,4,18 = 4.58, the decision is to reject the null
hypothesis.
11.7 Source df SS MS F
Treatment 3 544.2 181.4 13.00
Error 12 167.5 14.0
Total 15 711.8
α = .01 Critical F.01,3,12 = 5.95
Since the observed F = 13.00 > F.01,3,12 = 5.95, the decision is to reject the null
hypothesis.
6. Chapter 11: Analysis of Variance and Design of Experiments 6
11.8 Source df SS MS F
Treatment 1 64.29 64.29 17.76
Error 12 43.43 3.62
Total 13 107.71
α = .05 Critical F.05,1,12 = 4.75
Since the observed F = 17.76 > F.05,1,12 = 4.75, the decision is to reject the null
hypothesis.
Observed t value using t test:
1 2
n1 = 7 n2 = 7
x 1 = 29 x 2 = 24.71
s1
2
= 3 s2
2
= 4.238
t =
7
1
7
1
277
)6)(238.4()6(3
)0()71.2429(
+
−+
+
−−
= 4.22
Also, t = 76.17=F = 4.214
11.9 Source SS df MS F
Treatment 583.39 4 145.8475 7.50
Error 972.18 50 19.4436
Total 1,555.57 54
7. Chapter 11: Analysis of Variance and Design of Experiments 7
11.10 Source SS df MS F
Treatment 29.64 2 4.82 3.03
Error 68.42 14 4.887
Total 98.06 16
F.05,2,14 = 3.74
Since the observed F = 3.03 < F.05,2,14 = 3.74, the decision is to fail to reject
the null hypothesis
11.11 Source df SS MS F
Treatment 3 .007076 .002359 10.10
Error 15 .003503 .000234
Total 18 .010579
α = .01 Critical F.01,3,15 = 5.42
Since the observed F = 10.10 > F.01,3,15 = 5.42, the decision is to reject the null
hypothesis.
11.12 Source df SS MS F
Treatment 2 180700000 90350000 92.67
Error 12 11699999 975000
Total 14 192400000
α = .01 Critical F.01,2,12 = 6.93
Since the observed F = 92.67 > F.01,2,12 = 6.93, the decision is to reject the null
hypothesis.
8. Chapter 11: Analysis of Variance and Design of Experiments 8
11.13 Source df SS MS F
Treatment 2 29.61 14.80 11.76
Error 15 18.89 1.26
Total 17 48.50
α = .05 Critical F.05,2,15 = 3.68
Since the observed F = 11.76 > F.05,2,15 = 3.68, the decison is to reject the null
hypothesis.
11.14 Source df SS MS F
Treatment 3 456630 152210 11.03
Error 16 220770 13798
Total 19 677400
α = .05 Critical F.05,3,16 = 3.24
Since the observed F = 11.03 > F.05,3,16 = 3.24, the decision is to reject the null
hypothesis.
11.15 There are 4 treatment levels. The sample sizes are 18, 15, 21, and 11. The F
value is 2.95 with a p-value of .04. There is an overall significant difference at
alpha of .05. The means are 226.73, 238.79, 232.58, and 239.82.
11.16 The independent variable for this study was plant with five classification levels
(the five plants). There were a total of 43 workers who participated in the study.
The dependent variable was number of hours worked per week. An observed F
value of 3.10 was obtained with an associated p-value of .02659. With an alpha
of .05, there was a significant overall difference in the average number of hours
worked per week by plant. A cursory glance at the plant averages revealed that
workers at plant 3 averaged 61.47 hours per week (highest number) while workers
at plant 4 averaged 49.20 (lowest number).
9. Chapter 11: Analysis of Variance and Design of Experiments 9
11.17 C = 6 MSE = .3352 α = .05 N = 46
q.05,6,40 = 4.23 n3 = 8 n6 = 7 x 3 = 15.85 x 6 = 17.2
HSD = 4.23
+
7
1
8
1
2
3352.
= 0.896
21.1785.1563 −=− xx = 1.36
Since 1.36 > 0.896, there is a significant difference between the means of
groups 3 and 6.
11.18 C = 4 n = 6 N = 24 dferror = N - C = 24 - 4 = 20
MSE = 2.389 q.05,4,20 = 3.96
HSD = q
n
MSE
= (3.96)
6
389.2
= 2.50
11.19 C = 3 MSE = 1.0 α = .05 N = 17
q.05,3,14 = 3.70 n1 = 6 n2 = 5 x 1 = 2 x 2 = 4.6
HSD = 3.70
+
5
1
6
1
2
00.1
= 1.584
6.4221 −=− xx = 2.6
Since 2.6 > 1.584, there is a significant difference between the means of
groups 1 and 2.
10. Chapter 11: Analysis of Variance and Design of Experiments 10
11.20 From problem 11.6, MSE = 1.48 C = 5 N = 23
n2 = 5 n4 = 5 α = .01 q.01,5,23 = 5.29
HSD = 5.29
+
5
1
5
1
2
48.1
= 2.88
x 2 = 10 x 4 = 16
161063 −=− xx = 6
Since 6 > 2.88, there is a significant difference in the means of
groups 2 and 4.
11.21 N = 16 n = 4 C = 4 N - C = 12 MSE = 14 q.01,4,12 = 5.50
HSD = q
n
MSE
= 5.50
4
14
= 10.29
x 1 = 115.25 x 2 = 125.25 x 3 = 131.5 x 4 = 122.5
x 1 and x 3 are the only pair that are significantly different using the
HSD test.
11.22 n = 7 C = 2 MSE = 3.62 N = 14 N - C = 14 - 2 = 12
α = .05 q.05,2,12 = 3.08
HSD = q
n
MSE
= 3.08
7
62.3
= 2.215
x 1 = 29 and x 2 = 24.71
Since x 1 - x 2 = 4.29 > HSD = 2.215, the decision is to reject the null
hypothesis.
11. Chapter 11: Analysis of Variance and Design of Experiments 11
11.23 C = 4 MSE = .000234 α = .01 N = 19
q.01,4,15 = 5.25 n1 = 4 n2 = 6 n3 = 5 n4 = 4
x 1 = 4.03, x 2 = 4.001667, x 3 = 3.974, x 4 = 4.005
HSD1,2 = 5.25
+
6
1
4
1
2
000234.
= .0367
HSD1,3 = 5.25
+
5
1
4
1
2
000234.
= .0381
HSD1,4 = 5.25
+
4
1
4
1
2
000234.
= .0402
HSD2,3 = 5.25
+
5
1
6
1
2
000234.
= .0344
HSD2,4 = 5.25
+
4
1
6
1
2
000234.
= .0367
HSD3,4 = 5.25
+
4
1
5
1
2
000234.
= .0381
31 xx − = .056
This is the only pair of means that are significantly different.
12. Chapter 11: Analysis of Variance and Design of Experiments 12
11.24 α = .01 k = 3 n = 5 N = 15 N - k = 12 MSE = 975,000
HSD = q
n
MSE
= 5.04
5
000,975
= 2,225.6
x 1 = 28,400 x 2 = 36,900 x 3 = 32,800
21 xx − = 8,500
31 xx − = 4,400
32 xx − = 4,100
Using Tukey's HSD, all three pairwise comparisons are significantly different.
11.25 α = .05 C = 3 N = 18 N-C = 15 MSE = 1.26
q.05,3,15 = 3.67 n1 = 5 n2 = 7 n3 = 6
x 1 = 7.6 x 2 = 8.8571 x 3 = 5.83333
HSD1,2 = 3.67
+
7
1
5
1
2
26.1
= 1.706
HSD1,3 = 3.67
+
6
1
7
1
2
26.1
= 1.764
HSD2,3 = 3.67
+
6
1
7
1
2
26.1
= 1.621
31 xx − = 1.767 (is significant)
32 xx − = 3.024 (is significant)
13. Chapter 11: Analysis of Variance and Design of Experiments 13
11.26 α = .05 n = 5 C = 4 N = 20 N - C = 16 MSE = 13,798
x 1 = 591 x 2 = 350 x 3 = 776 x 4 = 563
HSD = q
n
MSE
= 4.05
5
798,13
= 212.75
21 xx − = 241 31 xx − = 185 41 xx − = 28
32 xx − = 426 42 xx − = 213 43 xx − = 213
Using Tukey's HSD = 212.75, only means 1 and 2 and means 2 and 3 are
significantly different.
11.27 α = .05 There were five plants and ten pairwise comparisons. The MINITAB
output revealed that the only pairwise significant difference was between plant 2
and plant 3. The reported confidence interval went from –22.46 to –0.18 which
contains the same sign indicating that 0 is not in the interval.
11.28 H0: µ1 = µ2 = µ3 = µ4
Ha: At least one treatment mean is different from the others
Source df SS MS F
Treatment 3 62.95 20.98 5.56
Blocks 4 257.50 64.38 17.07
Error 12 45.30 3.77
Total 19 365.75
α = .05 Critical F.05,3,12 = 3.49 for treatments
For treatments, the observed F = 5.56 > F.05,3,12 = 3.49, the decision is to
reject the null hypothesis.
14. Chapter 11: Analysis of Variance and Design of Experiments 14
11.29 H0: µ1 = µ2 = µ3
Ha: At least one treatment mean is different from the others
Source df SS MS F
Treatment 2 .001717 .000858 1.48
Blocks 3 .076867 .025622 44.10
Error 6 .003483 .000581
Total 11 .082067
α = .01 Critical F.01,2,6 = 10.92 for treatments
For treatments, the observed F = 1.48 < F.01,2,6 = 10.92 and the decision is to
fail to reject the null hypothesis.
11.30 Source df SS MS F
Treatment 5 2477.53 495.506 1.91
Blocks 9 3180.48 353.387 1.36
Error 45 11661.38 259.142
Total 59 17319.39
α = .05 Critical F.05,5,45 = 2.45 for treatments
For treatments, the observed F = 1.91 < F.05,5,45 = 2.45 and decision is to fail to
reject the null hypothesis.
11.31 Source df SS MS F
Treatment 3 199.48 66.493 3.90
Blocks 6 265.24 44.207 2.60
Error 18 306.59 17.033
Total 27 771.31
α = .01 Critical F.01,3,18 = 5.09 for treatments
For treatments, the observed F = 3.90 < F.01,3,18 = 5.09 and the decision is to
fail to reject the null hypothesis.
15. Chapter 11: Analysis of Variance and Design of Experiments 15
11.32 Source df SS MS F
Treatment 3 2302.5 767.5 15.66
Blocks 9 5402.5 600.3 12.25
Error 27 1322.5 49.0
Total 39 9027.5
α = .05 Critical F.05,3,27 = 2.96 for treatments
For treatments, the observed F = 15.66 > F.05,3,27 = 2.96 and the decision is to
reject the null hypothesis.
11.33 Source df SS MS F
Treatment 2 64.53 32.27 15.37
Blocks 4 137.60 34.40 16.38
Error 8 16.80 2.10
Total 14 218.93
α = .01 Critical F.01,2,8 = 8.65 for treatments
For treatments, the observed F = 15.37 > F.01,2,8 = 8.65 and the decision is to
reject the null hypothesis.
11.34 This a randomized block design with 3 treatments (machines) and 5 block levels
(operators). The F for treatments is 6.72 with a p-value of .019. There is a
significant difference in machines at = .05. The F for blocking effects is 0.22
with a p-value of .807. There are no significant blocking effects. The blocking
effects reduced the power of the treatment effects since the blocking effects were
not significant.
16. Chapter 11: Analysis of Variance and Design of Experiments 16
11.35 The p value for columns, .000177, indicates that there is an overall significant
difference in treatment means at alpha .001. The lengths of calls differ according
to type of telephone used. The p value for rows, .000281, indicates that there is
an overall difference in block means at alpha .001. The lengths of calls differ
according to manager. While the output does not include multiple comparisons,
an examination of the column means reveals that calls were longest on cordless
phones (a sample average of 4.112 minutes) and shortest on computer phones (a
sample average of 2.522 minutes). Manager 5 posted the largest average call
length in the sample with an average of 4.34 minutes. Manager 3 had the shortest
average call length in the sample with an average of 2.275.
If the company wants to reduce the average phone call length, it would encourage
managers to use the computer for calls. However, this study might be
underscoring the fact that it is inconvenient to place calls using the computer. If
management wants to encourage more calling, they might make more cordless
phones available or inquiry as to why the other modes are used for shorter
lengths.
11.36 This is a two-way factorial design with two independent variables and one
dependent variable. It is 2x4 in that there are two row treatment levels and four
column treatment levels. Since there are three measurements per cell, interaction
can be analyzed.
dfrow treatment = 1 dfcolumn treatment = 3 dfinteraction = 3,
dferror = 16 dftotal = 23
11.37 This is a two-way factorial design with two independent variables and one
dependent variable. It is 4x3 in that there are four treatment levels and three
column treatment levels. Since there are two measurements per cell, interaction
can be analyzed.
dfrow treatment = 3 dfcolumn treatment = 2 dfinteraction = 6,
dferror = 12 dftotal = 23
17. Chapter 11: Analysis of Variance and Design of Experiments 17
11.38 Source df SS MS F
Row 3 126.98 42.327 3.46
Column 4 37.49 9.373 0.77
Interaction 12 380.82 31.735 2.60
Error 60 733.65 12.228
Total 79 1278.94
α = .05 Critical F.05,3,60 = 2.76 for rows
For rows, the observed F = 3.46 > F.05,3,60 = 2.76 and the decision is to reject the
null hypothesis.
Critical F.05,4,60 = 2.53 for columns
For columns, the observed F = 0.77 < F.05,4,60 = 2.53 and the decision is to fail
to reject the null hypothesis.
Critical F.05,12,60 = 1.92 for interaction
For interaction, the observed F = 2.60 > F.05,12,60 = 1.92 and the decision is to
reject the null hypothesis.
Since there is significant interaction, the researcher should exercise extreme
caution in analyzing the "significant" row effects.
11.39 Source df SS MS F
Row 1 1.047 1.047 2.40
Column 3 3.844 1.281 2.94
Interaction 3 0.773 0.258 0.59
Error 16 6.968 0.436
Total 23 12.632
α = .05 Critical F.05,1,16 = 4.49 for rows
For rows, the observed F = 2.40 < F.05,1,16 = 4.49 and decision is to fail to reject
the null hypothesis.
Critical F.05,3,16 = 3.24 for columns
For columns, the observed F = 2.94 < F.05,3,16 = 3.24 and the decision is to
fail to reject the null hypothesis.
18. Chapter 11: Analysis of Variance and Design of Experiments 18
Critical F.05,3,16 = 3.24 for interaction
For interaction, the observed F = 0.59 < F.05,3,16 = 3.24 and the decision is to
fail to reject the null hypothesis.
11.40 Source df SS MS F
Row 1 60.750 60.750 38.37
Column 2 14.000 7.000 4.42
Interaction 2 2.000 1.000 0.63
Error 6 9.500 1.583
Total 11 86.250
α = .01 Critical F.01,1,6 = 13.75 for rows
For rows, the observed F = 38.37 > F.01,1,6 = 13.75 and the decision is to reject
the null hypothesis.
Critical F.01,2,6 = 10.92 for columns
For columns, the observed F = 4.42 < F.01,2,6 = 10.92 and the decision is to fail to
reject the null hypothesis.
Critical F.01,2,6 = 10.92 for interaction
For interaction, the observed F = 0.63 < F.01,2,6 = 10.92 and the decision is to
fail to reject the null hypothesis.
11.41 Source df SS MS F
Row 3 5.09844 1.69948 87.25
Column 1 1.24031 1.24031 63.67
Interaction 3 0.12094 0.04031 2.07
Error 24 0.46750 0.01948
Total 31 6.92719
α = .05 Critical F.05,3,24 = 3.01 for rows
For rows, the observed F = 87.25 > F.05,3,24 = 3.01 and the decision is to reject the
null hypothesis.
19. Chapter 11: Analysis of Variance and Design of Experiments 19
Critical F.05,1,24 = 4.26 for columns
For columns, the observed F = 63.67 > F.05,1,24 = 4.26 and the decision is to
reject the null hypothesis.
Critical F.05,3,24 = 3.01 for interaction
For interaction, the observed F = 2.07 < F.05,3,24 = 3.01 and the decision is to fail
to reject the null hypothesis.
11.42 Source df SS MS F
Row 3 42.4583 14.1528 14.77
Column 2 49.0833 24.5417 25.61
Interaction 6 4.9167 0.8194 0.86
Error 12 11.5000 0.9583
Total 23 107.9583
α = .05 Critical F.05,3,12 = 3.49 for rows
For rows, the observed F = 14.77 > F.05,3,12 = 3.49 and the decision is to reject the
null hypothesis.
Critical F.05,2,12 = 3.89 for columns
For columns, the observed F = 25.61 > F.05,2,12 = 3.89 and the decision is to reject
the null hypothesis.
Critical F.05,6,12 = 3.00 for interaction
For interaction, the observed F = 0.86 < F.05,6,12 = 3.00 and fail to reject the null
hypothesis.
20. Chapter 11: Analysis of Variance and Design of Experiments 20
11.43 Source df SS MS F
Row 2 1736.22 868.11 34.31
Column 3 1078.33 359.44 14.20
Interaction 6 503.33 83.89 3.32
Error 24 607.33 25.31
Total 35 3925.22
α = .05 Critical F.05,2,24 = 3.40 for rows
For rows, the observed F = 34.31 > F.05,2,24 = 3.40 and the decision is to reject the
null hypothesis.
Critical F.05,3,24 = 3.01 for columns
For columns, the observed F = 14.20 > F.05,3,24 = 3.01 and decision is to reject the
null hypothesis.
Critical F.05,6,24 = 2.51 for interaction
For interaction, the observed F = 3.32 > F.05,6,24 = 2.51 and the decision is to
reject the null hypothesis.
11.44 This two-way design has 3 row treatments and 5 column treatments. There are 45
total observations with 3 in each cell.
FR =
49.3
16.46
=
E
R
MS
MS
= 13.23
p-value = .000 and the decision is to reject the null hypothesis for rows.
FC =
49.3
70.249
=
E
C
MS
MS
= 71.57
p-value = .000 and the decision is to reject the null hypothesis for columns.
FI =
49.3
27.55
=
E
I
MS
MS
= 15.84
p-value = .000 and the decision is to reject the null hypothesis for interaction.
Because there is significant interaction, the analysis of main effects is
confounded. The graph of means displays the crossing patterns of the line
segments indicating the presence of interaction.
21. Chapter 11: Analysis of Variance and Design of Experiments 21
11.45 The null hypotheses are that there are no interaction effects, that there are no
significant differences in the means of the valve openings by machine, and that
there are no significant differences in the means of the valve openings by shift.
Since the p-value for interaction effects is .876, there are no significant interaction
effects which is good since significant interaction effects would confound that
study. The p-value for columns (shifts) is .008 indicating that column effects are
significant at alpha of .01. There is a significant difference in the mean valve
opening according to shift. No multiple comparisons are given in the output.
However, an examination of the shift means indicates that the mean valve
opening on shift one was the largest at 6.47 followed by shift three with 6.3 and
shift two with 6.25. The p-value for rows (machines) was .937 which is not
significant.
11.46 This two-way factorial design has 3 rows and 3 columns with three observations
per cell. The observed F value for rows is 0.19, for columns is 1.19, and for
interaction is 1.40. Using an alpha of .05, the critical F value for rows and
columns (same df) is F2,18,.05 = 3.55. Neither the observed F value for rows nor
the observed F value for columns is significant. The critical F value for
interaction is F4,18,.05 = 2.93. There is no significant interaction.
11.47 Source df SS MS F
Treatment 3 66.69 22.23 8.82
Error 12 30.25 2.52
Total 15 96.94
α = .05
Critical F.05,3,12 = 3.49
Since the treatment F = 8.82 > F.05,3,12 = 3.49, the decision is to reject the null
hypothesis.
For Tukey's HSD:
MSE = 2.52 n = 4 N = 16 N - k = 12 k = 4
q.05,4,12 = 4.20
HSD = q
n
MSE
= (4.20)
4
52.2
= 3.33
x 1 = 12 x 2 = 7.75 x 3 = 13.25 x 4 = 11.25
Using HSD of 3.33, there are significant pairwise differences between
means 1 and 2, means 2 and 3, and means 2 and 4.
22. Chapter 11: Analysis of Variance and Design of Experiments 22
11.48 Source df SS MS F
Treatment 6 68.19 11.365 0.87
Error 19 249.61 13.137
Total 25 317.80
11.49 Source df SS MS F
Treatment 5 210 42.000 2.31
Error 36 655 18.194
Total 41 865
11.50 Source df SS MS F
Treatment 2 150.91 75.46 16.19
Error 22 102.53 4.66
Total 24 253.44
α = .01 Critical F.01,2,22 = 5.72
Since the observed F = 16.19 > F.01,2,22 = 5.72, the decision is to reject the null
hypothesis.
x 1 = 9.200 x 2 = 14.250 x 3 = 8.714
n1 = 10 n2 = 8 n3 = 7
MSE = 4.66 C = 3 N = 25 N - C = 22
α = .01 q.01,3,22 = 4.64
HSD1,2 = 4.64
+
8
1
10
1
2
66.4
= 3.36
HSD1,3 = 4.64
+
7
1
10
1
2
66.4
= 3.49
HSD2,3 = 4.64
+
7
1
8
1
2
66.4
= 3.14
21 xx − = 5.05 and 32 xx − = 5.54 are significantly different at α = .01
23. Chapter 11: Analysis of Variance and Design of Experiments 23
11.51 This design is a repeated-measures type random block design. There is one
treatment variable with three levels. There is one blocking variable with six
people in it (six levels). The degrees of freedom treatment are two. The degrees
of freedom block are five. The error degrees of freedom are ten. The total
degrees of freedom are seventeen. There is one dependent variable.
11.52 Source df SS MS F
Treatment 3 20,994 6998.00 5.58
Blocks 9 16,453 1828.11 1.46
Error 27 33,891 1255.22
Total 39 71,338
α = .05 Critical F.05,3,27 = 2.96 for treatments
Since the calculated F = 5.58 > F.05,3,27 = 2.96 for treatments, the decision is to
reject the null hypothesis.
11.53 Source df SS MS F
Treatment 3 240.12 80.04 31.51
Blocks 5 548.71 109.74 43.20
Error 15 38.12 2.54
Total 23
α = .05 Critical F.05,3,5 = 5.41 for treatments
Since for treatments the calculated F = 31.51 > F.05,3,5 = 5.41, the decision is to
reject the null hypothesis.
For Tukey's HSD:
Ignoring the blocking effects, the sum of squares blocking and sum of squares
error are combined together for a new SSerror = 548.71 + 38.12 = 586.83.
Combining the degrees of freedom error and blocking yields a new dferror = 20.
Using these new figures, we compute a new mean square error, MSE =
(586.83/20) = 29.3415.
n = 6 C = 4 N = 24 N - C = 20
24. Chapter 11: Analysis of Variance and Design of Experiments 24
q.05,4,20 = 3.96
HSD = q
n
MSE
= (3.96)
6
3415.29
= 8.757
x 1 = 16.667 x 2 = 12.333 x 3 = 12.333 x 4 = 19.833
None of the pairs of means are significantly different using Tukey's HSD = 8.757.
This may be due in part to the fact that we compared means by folding the
blocking effects back into error. However, the blocking effects were highly
significant.
11.54 Source df SS MS F
Treatment 1 4 29.13 7.2825 1.98
Treatment 2 1 12.67 12.6700 3.45
Interaction 4 73.49 18.3725 5.00
Error 30 110.30 3.6767
Total 39 225.59
α = .05 Critical F.05,4,30 = 2.69 for treatment 1
For treatment 1, the observed F = 1.98 < F.05,4,30 = 2.69 and the decision is to
fail to reject the null hypothesis.
Critical F.05,1,30 = 4.17 for treatment 2
For treatment 2 observed F = 3.45 < F.05,1,30 = 4.17 and the decision is to
fail to reject the null hypothesis.
Critical F.05,4,30 = 2.69 for interaction
For interaction, the observed F = 5.00 > F.05,4,30 = 2.69 and the decision is to
reject the null hypothesis.
Since there are significant interaction effects, examination of the main effects
should not be done in the usual manner. However, in this case, there are no
significant treatment effects anyway.
25. Chapter 11: Analysis of Variance and Design of Experiments 25
11.55 Source df SS MS F
Row 3 257.889 85.963 38.21
Column 2 1.056 0.528 0.23
Interaction 6 17.611 2.935 1.30
Error 24 54.000 2.250
Total 35 330.556
α = .01 Critical F.01,3,24 = 4.72 for rows
For the row effects, the observed F = 38.21 > F.01,3,24 = 4.72 and the decision is to
reject the null hypothesis.
Critical F.01,2,24 = 5.61 for columns
For the column effects, the observed F = 0.23 < F.01,2,24 = 5.61 and the decision is
to fail to reject the null hypothesis.
Critical F.01,6,24 = 3.67 for interaction
For the interaction effects, the observed F = 1.30 < F.01,6,24 = 3.67 and the decision
is to fail to reject the null hypothesis.
11.56 Source df SS MS F
Row 2 49.3889 24.6944 38.65
Column 3 1.2222 0.4074 0.64
Interaction 6 1.2778 0.2130 0.33
Error 24 15.3333 0.6389
Total 35 67.2222
α = .05 Critical F.05,2,24 = 3.40 for rows
For the row effects, the observed F = 38.65 > F.05,2,24 = 3.40 and the decision is to
reject the null hypothesis.
Critical F.05,3,24 = 3.01 for columns
For the column effects, the observed F = 0.64 < F.05,3,24 = 3.01 and the decision is
to fail to reject the null hypothesis.
26. Chapter 11: Analysis of Variance and Design of Experiments 26
Critical F.05,6,24 = 2.51 for interaction
For interaction effects, the observed F = 0.33 < F.05,6,24 = 2.51 and the decision is
to fail to reject the null hypothesis.
There are no significant interaction effects. Only the row effects are significant.
Computing Tukey's HSD for rows:
x 1 = 2.667 x 2 = 4.917 x 3 = 2.250
n = 12 k = 3 N = 36 N - k = 33
MSE is recomputed by folding together the interaction and column sum of
squares and degrees of freedom with previous error terms:
MSE = (1.2222 + 1.2778 + 15.3333)/(3 + 6 + 24) = 0.5404
q.05,3,33 = 3.49
HSD = q
n
MSE
= (3.49)
12
5404.
= 0.7406
Using HSD, there are significant pairwise differences between means 1 and 2 and
between means 2 and 3.
Shown below is a graph of the interaction using the cell means by row.
27. Chapter 11: Analysis of Variance and Design of Experiments 27
11.57 Source df SS MS F
Treatment 3 90.48 30.16 7.38
Error 20 81.76 4.09
Total 23 172.24
α = .05 Critical F.05,3,20 = 3.10
The treatment F = 7.38 > F.05,3,20 = 3.10 and the decision is to reject the null
hypothesis.
11.58 Source df SS MS F
Treatment 2 460,353 230,176 103.70
Blocks 5 33,524 6,705 3.02
Error 10 22,197 2,220
Total 17 516,074
α = .01 Critical F.05,2,10 = 4.10 for treatments
Since the treatment observed F = 103.70 > F.05,2,10 = 4.10, the decision is to
reject the null hypothesis.
11.59 Source df SS MS F
Treatment 2 9.4 4.7 0.46
Error 18 185.4 10.3
Total 20 194.8
α = .05 Critical F.05,2,18 = 3.55
Since the treatment F = 0.46 > F.05,2,18 = 3.55, the decision is to fail to reject the
null hypothesis.
Since there are no significant treatment effects, it would make no sense to
compute Tukey-Kramer values and do pairwise comparisons.
28. Chapter 11: Analysis of Variance and Design of Experiments 28
11.60 Source df SS MS F
Row 2 4.875 2.437 5.16
Column 3 17.083 5.694 12.06
Interaction 6 2.292 0.382 0.81
Error 36 17.000 0.472
Total 47 41.250
α = .05 Critical F.05,2,36 = 3.32 for rows
For rows, the observed F = 5.16 > F.05,2,36 = 3.32 and the decision is to reject the
null hypothesis.
Critical F.05,3,36 = 2.92 for columns
For columns, the observed F = 12.06 > F.05,3,36 = 2.92 and the decision is to reject
the null hypothesis.
Critical F.05,6,36 = 2.42 for interaction
For interaction, the observed F = 0.81 < F.05,6,36 = 2.42 and the decision is to fail
to reject the null hypothesis.
There are no significant interaction effects. There are significant row and column
effects at α = .05.
11.61 Source df SS MS F
Treatment 4 53.400 13.350 13.64
Blocks 7 17.100 2.443 2.50
Error 28 27.400 0.979
Total 39 97.900
α = .05 Critical F.05,4,28 = 2.71 for treatments
For treatments, the observed F = 13.64 > F.05,4,28 = 2.71 and the decision is to
reject the null hypothesis.
29. Chapter 11: Analysis of Variance and Design of Experiments 29
11.62 This is a one-way ANOVA with four treatment levels. There are 36 observations
in the study. An examination of the mean analysis shows that the sample sizes are
different with sizes of 8, 7, 11, and 10 respectively. The p-value of .045 indicates
that there is a significant overall difference in the means at α = .05. No multiple
comparison technique was used here to conduct pairwise comparisons. However,
a study of sample means shows that the two most extreme means are from levels
one and four. These two means would be the most likely candidates for multiple
comparison tests.
11.63 Excel reports that this is a two-factor design without replication indicating that
this is a random block design. Neither the row nor the column p-values are less
than .05 indicating that there are no significant treatment or blocking effects in
this study. Also displayed in the output to underscore this conclusion are the
observed and critical F values for both treatments and blocking. In both
cases, the observed value is less than the critical value.
11.64 This is a two-way ANOVA with 5 rows and 2 columns. There are 2 observations
per cell. For rows, FR = 0.98 with a p-value of .461 which is not significant. For
columns, FC = 2.67 with a p-value of .134 which is not significant. For
interaction, FI = 4.65 with a p-value of .022 which is significant at α = .05. Thus,
there are significant interaction and the row and column effects are confounded.
An examination of the interaction plot reveals that most of the lines cross
signifying verifying the significant interaction finding.
11.65 This is a two-way ANOVA with 4 rows and 3 columns. There are 3 observations
per cell. FR = 4.30 with a p-value of .014 is significant at α = .05. The null
hypothesis is rejected for rows. FC = 0.53 with a p-value of .594 is not
significant. We fail to reject the null hypothesis for columns. FI = 0.99 with a
p-value of .453 is not significant. We fail to reject the null hypothesis for
interaction effects.
11.66 This was a random block design with 5 treatment levels and 5 blocking levels.
For both treatment and blocking effects, the critical value is F.05,4,16 = 3.01. The
observed F value for treatment effects is MSC / MSE = 35.98 / 7.36 = 4.89 which
is greater than the critical value. The null hypothesis for treatments is rejected,
and we conclude that there is a significant different in treatment means. No
multiple comparisons have been computed in the output. The observed F value
for blocking effects is MSR / MSE = 10.36 /7.36 = 1.41 which is less than the
critical value. There are no significant blocking effects. Using random block
design on this experiment might have cost a loss of power.
30. Chapter 11: Analysis of Variance and Design of Experiments 30
11.67 This one-way ANOVA has 4 treatment levels and 24 observations. The F = 3.51
yields a p-value of .034 indicating significance at α = .05. Since the sample sizes
are equal, Tukey’s HSD is used to make multiple comparisons. The computer
output shows that means 1 and 3 are the only pairs that are significantly different
(same signs in confidence interval). Observe on the graph that the confidence
intervals for means 1 and 3 barely overlap.