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Chapter 12: Analysis of Variance
12.1: One-Way ANOVA
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Chapter 10: Correlation and Regression
10.1: Correlation
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Chapter 10: Correlation and Regression
10.2: Regression
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Chapter 11: Goodness-of-Fit and Contingency Tables
11.1: Goodness of Fit Notation
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Chapter 9: Inferences from Two Samples
9.2: Two Means, Independent Samples
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Chapter 8: Hypothesis Testing
8.1: Basics of Hypothesis Testing
Solution to the practice test ch 10 correlation reg ch 11 gof ch12 anovaLong Beach City College
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Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
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Elementary Statistics Practice Test 1
Module 1: Chapters 1-3
Chapter 1: Introduction to Statistics.
Chapter 2: Exploring Data with Tables and Graphs.
Chapter 3: Describing, Exploring, and Comparing Data.
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Elementary Statistics Practice Test 2
Chapter 4: Probability
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Chapter 10: Correlation and Regression
10.1: Correlation
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Chapter 10: Correlation and Regression
10.2: Regression
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Chapter 11: Goodness-of-Fit and Contingency Tables
11.1: Goodness of Fit Notation
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Chapter 9: Inferences from Two Samples
9.2: Two Means, Independent Samples
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Chapter 8: Hypothesis Testing
8.1: Basics of Hypothesis Testing
Solution to the practice test ch 10 correlation reg ch 11 gof ch12 anovaLong Beach City College
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Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
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Elementary Statistics Practice Test 1
Module 1: Chapters 1-3
Chapter 1: Introduction to Statistics.
Chapter 2: Exploring Data with Tables and Graphs.
Chapter 3: Describing, Exploring, and Comparing Data.
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Elementary Statistics Practice Test 2
Chapter 4: Probability
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Elementary Statistics Practice Test 2 Solutions
Chapter 4: Probability
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Chapter 7: Estimating Parameters and Determining Sample Sizes
7.1: Estimating a Population Proportion
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Chapter 11: Goodness-of-Fit and Contingency Tables
11.2: Contingency Tables
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Chapter 8: Hypothesis Testing
8.4: Testing a Claim About a Standard Deviation or Variance
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Chapter 4: Probability
4.3: Complements and Conditional Probability, and Bayes' Theorem
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Chapter 5: Discrete Probability Distribution
5.2 - Binomial Probability Distributions
Analysis of variance (ANOVA) everything you need to knowStat Analytica
Most of the students may struggle with the analysis of variance (ANOVA). Here in this presentation you can clear all your doubts in analysis of variance with suitable examples.
Test statistic
General Formula for Test Statistic
Statistical Decision
Significance Level
p-values
Purpose of Hypothesis Testing
General Procedure of Testing a Hypothesis
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Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
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Elementary Statistics Practice Test 4
Module 4:
Chapter 8, Hypothesis Testing
Chapter 9: Two Populations
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Elementary Statistics Practice Test 2 Solutions
Chapter 4: Probability
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Chapter 7: Estimating Parameters and Determining Sample Sizes
7.1: Estimating a Population Proportion
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Chapter 11: Goodness-of-Fit and Contingency Tables
11.2: Contingency Tables
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Chapter 8: Hypothesis Testing
8.4: Testing a Claim About a Standard Deviation or Variance
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Chapter 4: Probability
4.3: Complements and Conditional Probability, and Bayes' Theorem
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Chapter 5: Discrete Probability Distribution
5.2 - Binomial Probability Distributions
Analysis of variance (ANOVA) everything you need to knowStat Analytica
Most of the students may struggle with the analysis of variance (ANOVA). Here in this presentation you can clear all your doubts in analysis of variance with suitable examples.
Test statistic
General Formula for Test Statistic
Statistical Decision
Significance Level
p-values
Purpose of Hypothesis Testing
General Procedure of Testing a Hypothesis
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Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
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Elementary Statistics Practice Test 4
Module 4:
Chapter 8, Hypothesis Testing
Chapter 9: Two Populations
In general, a factorial experiment involves several variables.
One variable is the response variable, which is sometimes called the outcome variable or the dependent variable.
The other variables are called factors.
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
In the t test for independent groups, ____.we estimate µ1 µ2.docxbradburgess22840
In the t test for independent groups, ____.
we estimate µ1 µ2
we estimate 2
we estimate X1-X2
df = N 1
Exhibit 14-1
A professor of women's studies is interested in determining if stress affects the menstrual cycle. Ten women are randomly sampled for an experiment and randomly divided into two groups. One of the groups is subjected to high stress for two months while the other lives in a relatively stress-free environment. The professor measures the menstrual cycle (in days) of each woman during the second month. The following data are obtained.
High stress
20
23
18
19
22
Relatively stress free
26
31
25
26
30
Refer to Exhibit 14-1. The obtained value of the appropriate statistic is ____.
tobt = 4.73
tobt = 4.71
tobt = 3.05
tobt = 0.47
Refer to Exhibit 14-1. The df for determining tcrit are ____.
4
9
8
3
Refer to Exhibit 14-1. Using = .052 tail, tcrit = ____.
+2.162
+2.506
±2.462
±2.306
Refer to Exhibit 14-1. Using = .052 tail, your conclusion is ____.
accept H0; stress does not affect the menstrual cycle
retain H0; we cannot conclude that stress affects the menstrual cycle
retain H0; stress affects the menstrual cycle
reject H0; stress affects the menstrual cycle
Refer to Exhibit 14-1. Estimate the size of the effect. = ____
0.8102
0.6810
0.4322
0.5776
A major advantage to using a two condition experiment (e.g. control and experimental groups) is ____.
the test has more power
the data are easier to analyze
the experiment does not need to know population parameters
the test has less power
Which of the following tests analyzes the difference between the means of two independent samples?
correlated t test
t test for independent groups
sign test
test of variance
If n1 = n2 and n is relatively large, then the t test is relatively robust against ____.
violations of the assumptions of homogeneity of variance and normality
violations of random samples
traffic violations
violations by the forces of evil
Exhibit 14-3
Five students were tested before and after taking a class to improve their study habits. They were given articles to read which contained a known number of facts in each story. After the story each student listed as many facts as he/she could recall. The following data was recorded.
Before
10
12
14
16
12
After
15
14
17
17
20
Refer to Exhibit 14-3. The obtained value of the appropriate statistic is ____.
3.92
3.06
4.12
2.58
Refer to Exhibit 14-3. What do you conclude using = 0.052 tail?
reject H0; the class appeared to improve study habits
retain H0; the class had no effect on study habits
retain H0; we cannot conclude that the class improved study habits
accept H0; the class appeared to improve study habits
Which of the following is (are) assumption(s) underlying the use of the F test?
the raw score populations are normally distributed
the variances of the raw score populations are the same
the mean of the populations differ
the raw score popul.
These are slides I use when teaching my second year undergraduate statistics course. They are designed more for conceptual understanding, and do not have syntax for programs like SPSS or R. So it is a more conceptual and mathematical review, rather than a "how-to" computer guide.
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Elementary Statistics Practice Test 4
Chapter 9: Inferences about Two Samples
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Elementary Statistics Practice Test 4
Chapter 8: Hypothesis Testing
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Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
Solution to the practice test ch 8 hypothesis testing ch 9 two populationsLong Beach City College
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Elementary Statistics Practice Test 4
Module 4:
Chapter 8, Hypothesis Testing
Chapter 9: Two Populations
Solution to the Practice Test 3A, Chapter 6 Normal Probability DistributionLong Beach City College
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Elementary Statistics Practice Test 3
Practice Test Chapter 6 (Normal Probability Distributions)
Chapter 6: Normal Probability Distributions
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Elementary Statistics Practice Test 3
Practice Test Chapter 6 (Normal Probability Distributions)
Chapter 6: Normal Probability Distributions
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Elementary Statistics Practice Test 1
Module 1: Chapters 1-3
Chapter 1: Introduction to Statistics.
Chapter 2: Exploring Data with Tables and Graphs.
Chapter 3: Describing, Exploring, and Comparing Data.
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Chapter 12: Analysis of Variance
12.2: Two-Way ANOVA
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Chapter 9: Inferences from Two Samples
9.4: Two Variances or Standard Deviations
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Chapter 9: Inferences from Two Samples
9.3 Two Means, Two Dependent Samples, Matched Pairs
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Chapter 9: Inferences from Two Samples
9.1: Inferences about Two Proportions
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Chapter 8: Hypothesis Testing
8.3: Testing a Claim About a Mean
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Chapter 8: Hypothesis Testing
8.2: Testing a Claim About a Proportion
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Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
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This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
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Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
2. Chapter 12: Analysis of Variance
12.1 One-Way ANOVA
12.2 Two-Way ANOVA
2
Objectives:
Use the ANOVA technique to determine if there is a significant
difference among three or more means.
Good lecture Videos to watch: ANOVA
Concept: https://www.youtube.com/watch?v=ITf4vHhyGpc
Hand Calculation: https://www.youtube.com/watch?v=WUjsSB7E-ko
Visual Explanation: https://www.youtube.com/watch?v=JgMFhKi6f6Y
YouTube: https://www.youtube.com/watch?v=YbX-JUqD1so
Tukey Website + ANOVA:
https://astatsa.com/OneWay_Anova_with_TukeyHSD/
One-Way (Single Factor) ANOVA + Tukey
1. Get rid of the check mark
2. Choose the value of K (Number of treatment columns)
3. Proceed to enter your treatment columns
4. Enter Data
5. Calculate ANOVA with Tukey, Scheffe, Bonferroni & Holm
6. All is Ready
3. 3
12.1 One-Way ANOVA
One-way analysis of variance (ANOVA) is used for tests of hypotheses that three or more
populations have means that are all equal, as in H0: µ1 = µ2 = µ3 by analyzing sample variances.
One-way analysis of variance is used with data categorized with one factor (or treatment), so there
is one characteristic used to separate the sample data into the different categories.
• The F test, used to compare two variances, can also be used to compare three or more means.
• This technique is called analysis of variance or ANOVA.
• For three groups, the F test can only show whether or not a difference exists among the three means, not where
the difference lies. (The ANOVA test is right-tailed because only large values of the test statistic cause us to
reject equality of the population means.)
• Other statistical tests, Scheffé test and the Tukey test, are used to find where the difference exists.
• Although the t test is commonly used to compare two means, it should not be used to compare three or more.
There is a different F distribution for each different pair of degrees of freedom for numerator and denominator.
1. The F distribution is not symmetric. It is skewed right (positively skewed).
2. Values of the F distribution cannot be negative.
3. The exact shape of the F distribution depends on the two different degrees of freedom.
4. 4
12.1 One-Way ANOVA
In the F test, two different estimates of the population variance are made.
The first estimate is called the between-group variance, and it involves finding the variance of the means.
The second estimate, the within-group variance, is made by computing the variance using all the data
and is not affected by differences in the means.
If there is no difference in the means, the between-group variance will be approximately equal to the
within-group variance, and the F test value will be close to 1; do not reject null hypothesis.
However, when the means differ significantly, the between-group variance will be much larger than
the within-group variance; the F test will be significantly greater than 1; reject null hypothesis.
The following assumptions apply when using the F test to compare three or more means.
1. The populations from which the samples were obtained must be normally or approximately
normally distributed.
2. The samples must be independent of each other.
3. The populations have the same variance σ² (or standard deviation σ).
4. The samples are simple random samples of quantitative data.
5. The different samples are from populations that are categorized in only one way.
Step 1: H0 , H1, claim & Tails
Step 2: TS Calculate (TS)
Step 3: CV using α
Step 4: Make the decision to
a. Reject or not H0
b. The claim is true or false
c. Restate this decision: There is
/ is not sufficient evidence to
support the claim that…
5. 5
Example 1:
Three different techniques are used to lower the blood pressure of
individuals diagnosed with high blood pressure. The subjects are
randomly assigned to 3 groups; the first group takes medication, the
second group exercises, and the third group follows a special diet. After
two months, the reduction in each person’s blood pressure is recorded.
At α = 0.05, test the claim that there is no difference among the means.
Step 1: State the hypotheses and identify the claim.
H0: μ1 = μ2 = μ3 (claim)
H1: At least one mean is different from the others. (RTT)
Given:
Number of Groups (Factors): k = 3
Number of data in each group: n = 5
Total sample size: N = 3(5) = 15
Sum of Squares: SST = SSW + SSB
The total sum of squares: Find the squared distances
from the Grand Mean. (If we took the average, we
would have a variance.)
The within-group or within-cell sum of squares comes
from the distance of the observations to the cell
means. This indicates error.
The between-cells or between-groups sum of squares
tells of the distance of the cell means from the grand
mean. This indicates treatment (the column) effects.
𝑆𝑆𝑡𝑜𝑡 = 𝑆𝑆𝐵 + 𝑆𝑆𝑊 = (𝑋𝑖 − 𝑋𝐺)2
𝑆𝑆𝑊 = (𝑋𝑖 − 𝑋𝐴)2
= (𝑋1 − 𝑋1)2
+ (𝑋2 − 𝑋2)2
+ (𝑋3 − 𝑋3)2
+. . .
𝑆𝑆𝐵 = 𝑁𝐴(𝑋𝐴 − 𝑋𝐺)2
𝑂𝑅 = 𝑆𝑆𝑡𝑜𝑡 − 𝑆𝑆𝑊
𝑋𝑖: Each piece of Data,
𝑋𝐺: The Grand Mean, taken over all observations.
𝑋𝐴: The mean of any level of a treatment.
6. 6
Example 1: Continued
Step 2: Calculate the test statistic value.
a. Find the mean and variance of each sample (Given Table).
𝑥𝐺𝑀 =
𝑥
𝑁
=
10 + 12 + 9 + 15 + 13 + 6 + ⋯ + 4
15
𝑀𝑆𝐵 𝑂𝑅 𝑠𝐵
2
=
𝑆𝑆𝐵
𝑘 − 1
=
𝑛𝑖 𝑋𝑖 − 𝑋𝐺𝑀
2
𝑘 − 1
=
5 11.8 − 7.73 2
+ 5 3.8 − 7.73 2
+ 5 7.6 − 7.73 2
3 − 1
𝑀𝑆𝑤 𝑂𝑅 𝑠𝑊
2
𝑂𝑅 𝑠𝑝
2
=
𝑆𝑆𝑤
𝑁 − 𝑘
=
𝑛𝑖 − 1 𝑠𝑖
2
𝑛𝑖 − 1
=
4 5.7 + 4 10.2 + 4 10.3
4 + 4 + 4
𝐹 =
MS𝐵
MS𝑊
=
𝑠𝐵
2
𝑠𝑊
2
=
116
15
= 7.73
=
160.13
2
= 80.07
=
104.80
12
= 8.73
Given the means of each group: 11.8, 3.8 & 7.6
Calculate their weighted average and variance:
𝑏. . 𝑥𝐺𝑀 =
𝑥𝑖
𝑘
=
11.8 + 3.8 + 7.6
3
= 7.73
𝑐. 𝑠𝑥
2
=
𝑥𝑖 − 𝑥𝐺𝑀
2
𝑘 − 1
=
11.8 − 7.73 2
+ 3.8 − 7.73 2
+ 7.6 − 7.76 2
3 − 1
≈ 16.018
𝑆𝐵 = 𝜎𝑥 =
𝜎
𝑛
→ 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒: 𝜎2
= 𝑛𝑠𝑥
2
→ 𝑀𝑆𝐵 𝑂𝑅 𝑠𝐵
2
= 𝑛𝑠𝑥
2
= 5(16.018) = 80.09
b. Find the Grand Mean, the mean of all values in the samples.
c. Find the Between-group variance (Mean Square, MSB.)
d. Find the Within-group variance (Error)(Mean
of sample variances, or mean square, MSW)
e. Calculate the F value.
𝑑. 𝑠𝑊
2
𝑂𝑅 𝑠𝑝
2
=
𝑆1
2
+ 𝑆2
2
+ 𝑆3
2
3
=
5.7 + 10.2 + 10.3
3
= 8.733
Source Sum of
Squares
d.f. Mean
Squares
F
Between
Within (error)
SSB =160.13
SSW =104.80
k – 1= 2
N – k=12
MSB = 80.07
MSW = 8.73
9.17
Total 264.93 14
=
80.07
8.73
= 9.17
7. 7
Three different techniques are used to lower the blood pressure of individuals diagnosed
with high blood pressure. The subjects are randomly assigned to 3 groups; the first group
takes medication, the second group exercises, and the third group follows a special diet.
After two months, the reduction in each person’s blood pressure is recorded. At α = 0.05,
test the claim that there is no difference among the means.
Step 3: Find the critical value.
Number of Groups (Factors): k = 3, d.f.N. = k – 1 = 3 – 1 = 2,
Total sample size: N = 15, d.f.D. = N – k = 15 – 3 = 12
α = 0.05 → CV: 𝑭 = 3.8853
Step 4: Decision:
a. Reject H0
b. The claim is False
c. There is NOT sufficient evidence to support the claim that there is no difference among the
means and conclude that at least one mean is different from the others.
Step 1: H0: μ1 = μ2 = μ3 (claim)
H1: At least one mean is different from the others. (RTT)
Step 2: Calculate the test statistic value: 𝐹 = 9.17
Example 1: Continued
CV: F = 3.8853 TS: 𝑭 = 9.17
0.05
Source Sum of
Squares
d.f. Mean
Squares
F
Between
Within (error)
SSB =160.13
SSW =104.80
k – 1= 2
N – k=12
MSB = 80.07
MSW = 8.73
9.17
Total 264.93 14
TI Calculator:
One Way ANOVA
1. ClrList 𝑳𝟏, 𝑳𝟐, …
2. Enter data 𝑳𝟏, 𝑳𝟐, . .
3. Stat
4. Tests
5. 𝑨𝑵𝑶𝑽𝑨 (𝑳𝟏, 𝑳𝟐 , . . )
6. Enter
8. 8
The between-group variance is sometimes
called the mean square, MSB.
• The numerator of the formula to
compute MSB is called the sum of
squares between groups, SSB.
• The within-group variance is sometimes
called the mean square, MSW.
• The numerator of the formula to
compute MSW is called the sum of
squares within groups, SSW.
Source Sum of
Squares
d.f. Mean
Squares = SS /df
𝑭 =
𝑴𝑺𝑩
𝑴𝑺𝒘
Squares = SS /df SSB =160.13
SSW =104.80
k – 1= 2
N – k=12
MSB = SSB / 2 = 80.07
MSW = SSW / 12 = 8.73
𝟖𝟎. 𝟎𝟕
𝟖. 𝟕𝟑
= 𝟗. 𝟏𝟕
Total SST = 264.93 14
12.1 One-Way ANOVA Summary Table
The Grand Mean,
taken over all
observations:
𝑥𝐺𝑀 =
𝑥
𝑁
=
𝑥𝑖
𝑘
Number of Groups (Factors): k
Number of data in each group: 𝒏𝟏, 𝒏𝟐,…
Total sample size: 𝑵 = 𝒏𝟏 + 𝒏𝟐 + ⋯
Each piece of data: 𝑿𝒊
Each Group (treatment) Mean: 𝒙𝒊
Between-group variance (Mean Square, MSB.)
𝑀𝑆𝐵 𝑂𝑅 𝑠𝐵
2
=
𝑆𝑆𝐵
𝑘 − 1
=
𝑛𝑖 𝑋𝑖 − 𝑋𝐺𝑀
2
𝑘 − 1
→ 𝑀𝑆𝐵 𝑂𝑅 𝑠𝐵
2
= 𝑛𝑠𝑥
2
, if same number of data in each
category (n times the variance for sample means)
Within-group variance (Error)(Mean of sample variances, or mean
square, MSW): 𝑀𝑆𝑤 𝑂𝑅 𝑠𝑊
2
𝑂𝑅 𝑠𝑝
2
=
𝑆𝑆𝑤
𝑁−𝑘
=
𝑛𝑖−1 𝑠𝑖
2
𝑛𝑖−1
𝑂𝑅 𝑠𝑝
2
=
𝑆1
2+𝑆2
2+𝑆3
2+⋯
𝑘
, if same number of data in each category
9. 9
Data shown represents the number of employees at the security gate for 3 companies.
At α = 0.05, can it be concluded that there is a significant difference in the average
number of employees at each interchange?
Example 2:
Company A Company B Company C
7 10 1
14 1 12
32 1 1
19 0 9
10 11 1
11 1 11
𝑥1 = 15.5,
𝑠1
2
= 81.9
𝑥2 = 4.0,
𝑠2
2
= 25.6
𝑥3 = 5.8,
𝑠3
2
= 29.0
Step 1: State the hypotheses and identify the claim.
H0: μ1 = μ2 = μ3
H1: At least one mean is different from the others. (RTT, claim)
Step 1: H0 , H1, claim & Tails
Step 2: TS Calculate (TS)
Step 3: CV using α
Step 4: Make the decision to
a. Reject or not H0
b. The claim is true or false
c. Restate this decision: There is
/ is not sufficient evidence to
support the claim that…
Given:
Number of Groups (Factors): k = 3
Number of data in each group: n = 6
Total sample size: N = 3(6) = 18
10. 10
Example 2: Continued
Step 2: Calculate the test statistic value.
a. Find the mean and variance of each sample (Given).
𝑥𝐺𝑀 =
𝑥
𝑁
=
7 + 14 + 32 + ⋯ + 11
18
𝑠𝐵
2
=
𝑛𝑖 𝑋𝑖 − 𝑋𝐺𝑀
2
𝑘 − 1
=
6 15.5 − 8.4333 2 + 6 4 − 8.4333 2 + 6 5.8 − 8.4333 2
3 − 1
𝑠𝑊
2
𝑂𝑅 𝑠𝑝
2 =
𝑛𝑖 − 1 𝑠𝑖
2
𝑛𝑖 − 1
=
5 81.9 + 5 25.6 + 5 29.0
5 + 5 + 5
𝐹 =
𝑠𝐵
2
𝑠𝑊
2
=
152
18
= 8.444
=
459.18
2
= 229.59
=
682.5
15
= 45.5
Given the means of each group: 15.5, 4 & 5.8
Calculate their mean and variance:
𝑏. 𝑥 =
𝑥𝑖
𝑘
=
15.5 + 4 + 5.8
3
= 8.4333
𝑐. 𝑠𝑥
2
=
𝑥𝑖 − 𝑥𝐺𝑀
2
𝑘 − 1
=
15.5 − 8.433 2
+ 4 − 8.433 2
+ 5.8 − 8.433 2
3 − 1
≈ 38.263
𝑆𝐵 = 𝜎𝑥 = 𝜎/ 𝑛 → 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒: 𝜎2
= 𝑛𝑠𝑥
2
→ 𝑀𝑆𝐵 𝑂𝑅 𝑠𝐵
2
= 𝑛𝑠𝑥
2
= 6(38.263) = 229.580
b. Find the grand mean, the mean of all values in the samples.
c. Find the between-group variance
d. Find the within-group variance (Mean of sample variances),
e. Calculate the F value.
𝑑. 𝑠𝑊
2
𝑂𝑅 𝑠𝑝
2
=
𝑆1
2
+ 𝑆2
2
+ 𝑆3
2
3
=
81.9 + 25.6 + 29
3
= 45.5
Company A Company B Company C
7 10 1
14 1 12
32 1 1
19 0 9
10 11 1
11 1 11
𝑥1 = 15.5,
𝑠1
2
= 81.9
𝑥2 = 4.0,
𝑠2
2
= 25.6
𝑥3 = 5.8,
𝑠3
2
= 29.0
=
229.59
45.5
= 5.05
11. 11
Data shown represents the number of employees at the
security gate for 3 companies. At α = 0.05, can it be
concluded that there is a significant difference in the
average number of employees at each interchange?
Step 3: Find the critical value.
Number of Groups (Factors): k = 3, d.f.N. = k – 1 = 3 – 1 = 2,
Total sample size: N = 18, d.f.D. = N – k = 18 – 3 = 15
α = 0.05 → CV: 𝑭 = 3.6823
Step 4: Decision:
a. Reject H0
b. The claim is True
c. There is sufficient evidence to
support the claim that there is a
difference among the means and
conclude that at least one mean is
different from the others.
Step 1: H0: μ1 = μ2 = μ3
H1: At least one mean is different from the others. (RTT, claim)
Step 2: Test statistic: 𝐹 = 5.05
Example 2: Continued
0.05
Company A Company B Company C
7 10 1
14 1 12
32 1 1
19 0 9
10 11 1
11 1 11
𝑥1 = 15.5,
𝑠1
2
= 81.9
𝑥2 = 4.0,
𝑠2
2
= 25.6
𝑥3 = 5.8,
𝑠3
2
= 29.0
Source Sum of
Squares
d.f. Mean
Squares
F
Between
Within
(error)
459.18
682.5
2
15
229.59
45.5
5.05
Total 1141.68 17
TI Calculator:
One Way ANOVA
1. ClrList 𝑳𝟏, 𝑳𝟐, …
2. Enter data 𝑳𝟏, 𝑳𝟐, . .
3. Stat
4. Tests
5. 𝑨𝑵𝑶𝑽𝑨 (𝑳𝟏, 𝑳𝟐 , . . )
6. Enter
CV: F = 3.6823 TS: 𝑭 = 5.05