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### Lesson 13 - 1

Comparing Three or More Means ANOVA(One-Way Analysis of Variance)

Objectives

- Verify the requirements to perform a one-way ANOVA
- Test a claim regarding three or more means using one way ANOVA

Vocabulary

- ANOVA – Analysis of Variance: inferential method that is used to test the equality of three or more population means
- Robust – small departures from the requirement of normality will not significantly affect the results
- Mean squares – is an average of the squared values (for example variance is a mean square)
- MST – mean square due to the treatment
- MSE – mean square due to error
- F-statistic – ration of two mean squares

One-way ANOVA Test Requirements

- There are k simple random samples from k populations
- The k samples are independent of each other; that is, the subjects in one group cannot be related in any way to subjects in a second group
- The populations are normally distributed
- The populations have the same variance; that is, each treatment group has a population variance σ2

ANOVA Requirements Verification

- ANOVA is robust, the accuracy of ANOVA is not affected if the populations are somewhat non- normal or do not quite have the same variances
- Particularly if the sample sizes are roughly equal
- Use normality plots
- Verifying equal population variances requirement:
- Largest sample standard deviation is no more than two times larger than the smallest

ANOVA – Analysis of Variance

Computing the F-test Statistic

1. Compute the sample mean of the combined data set, x

- Find the sample mean of each treatment (sample), xi
- Find the sample variance of each treatment (sample), si2
- Compute the mean square due to treatment, MST
- Compute the mean square due to error, MSE
- Compute the F-test statistic:

mean square due to treatment MST F = ------------------------------------- = ---------- mean square due to error MSE

ni(xi – x)2 (ni – 1)si2

MST = -------------- MSE = -------------

k – l n – k

k

Σ

k

Σ

n = 1

n = 1

MSE and MST

- MSE -mean square due to error, measures how different the observations, within each sample, are from each other
- It compares only observations within the same sample
- Larger values correspond to more spread sample means
- This mean square is approximately the same as the population variance

- MST - mean square due to treatment, measures how different the samples are from each other
- It compares the different sample means
- Larger values correspond to more spread sample means
- Under the null hypothesis, this mean square is approximately the same as the population variance

Excel ANOVA Output

- Classical Approach:
- Test statistic > Critical value … reject the null hypothesis

- P-value Approach:
- P-value < α (0.05) … reject the null hypothesis

TI Instructions

- Enter each population’s or treatments raw data into a list
- Press STAT, highlight TESTS and select F: ANOVA(
- Enter list names for each sample or treatment after “ANOVA(“ separate by commas
- Close parenthesis and hit ENTER
- Example: ANOVA(L1,L2,L3)

Summary and Homework

- Summary
- ANOVA is a method that tests whether three, or more, means are equal
- One-Way ANOVA is applicable when there is only one factor that differentiates the groups
- Not rejecting H0 means that there is not sufficient evidence to say that the group means are unequal
- Rejecting H0 means that there is sufficient evidence to say that group means are unequal

- Homework
- pg 685-691; 1-4, 6, 7, 11, 13, 14, 19

Problem 19 TI-83 Calculator Output

- One-way ANOVA
- F=5.81095
- p=.013532
- Factor
- df=2
- SS=1.1675
- MS=0.58375

- Error
- df=15
- SS=1.50686
- MS=.100457
- Sxp=0.31695

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