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

1
Lesson 13 - 1

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

2
Objectives

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

3
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

4
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 s2
  

5
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

6
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
S
k
S
n 1
n 1
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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

8
ANOVA Analysis of Variance Table
Source of Variation Sum of Squares Degrees of Freedom Mean Squares F-testStatistic F Critical Value
Treatment S ni(xi x)2 k - 1 MST MST/MSE F a, k-1, n-k
Error S (ni 1)si2 n - k MSE
Total SST SSE n - 1
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Excel ANOVA Output

• Classical Approach
• Test statistic gt Critical value reject the null
  hypothesis
• P-value Approach
• P-value lt a (0.05) reject the null hypothesis

10
TI Instructions

• Enter each populations 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)

11
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

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Problem 19 TI-83 Calculator Output

• One-way ANOVA
• F5.81095
• p.013532
• Factor
• df2
• SS1.1675
• MS0.58375
• Error
• df15
• SS1.50686
• MS.100457
• Sxp0.31695