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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.

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Lesson 13 1

Lesson 13 - 1

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


Objectives
Objectives

  • Verify the requirements to perform a one-way ANOVA

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


Vocabulary
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
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 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
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 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
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
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 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
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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