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# Lesson 13 - 1 - PowerPoint PPT Presentation

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.

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### Lesson 13 - 1

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

• Verify the requirements to perform a one-way ANOVA

• Test a claim regarding three or more means using one way ANOVA

• 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

• 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 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

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 -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

• Classical Approach:

• Test statistic > Critical value … reject the null hypothesis

• P-value Approach:

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

• 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

• 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

• 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