Ch' 10, part II

1
Ch. 10, part II

• Analysis of Variance
• (ANOVA)

2
An Introduction to Analysis of Variance

• Analysis of Variance (ANOVA) can be used to test
  for the equality of three or more population
  means using data obtained from observational or
  experimental studies.
• We want to use the sample results to test the
  following hypotheses.
• H0 ?1???2???3??. . . ?k?
• Ha Not all population means are equal
• If H0 is rejected, we cannot conclude that all
  population means are different.
• Rejecting H0 means that at least two population
  means have different values.

3
Example Reed Manufacturing

• Analysis of Variance
• J. R. Reed would like to know if the mean
  number of hours worked per week is the same for
  the department managers at her three
  manufacturing plants (Buffalo, Pittsburgh, and
  Detroit).
• A simple random sample of 5 managers from each
  of the three plants was taken and the number of
  hours worked by each manager for the previous
  week is shown on the next slide.

4
Example Reed Manufacturing

• Analysis of Variance
• Plant 1 Plant 2 Plant 3
• Observation Buffalo Pittsburgh
  Detroit
• 1 48 73 51
• 2 54 63 63
• 3 57 66 61
• 4 54 64 54
• 5 62 74 56
• Sample Mean 55 68 57
• Sample Variance 26.0 26.5 24.5

5
Example Reed Manufacturing

• Analysis of Variance
• Hypotheses
• H0 ?1???2???3?
• Ha Not all the means are equal
• where
• ????1 mean number of hours worked per
  week by the managers at Plant 1
• ?2 mean number of hours worked per
  week by the managers at Plant 2
• ????3 mean number of hours worked per
  week by the managers at Plant 3

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Example Reed Manufacturing

• Two variables
• Average hours worked - Dependent or response
  variable.
• Plant location - Independent or factor variable.
• The values of a factor selected for investigation
  are referred to as the levels of the factor or
  treatments.
• Three treatments Buffalo, Pittsburgh, Detroit.

7
Assumptions for Analysis of Variance

• For each population, the response variable is
  normally distributed.
• The variance of the response variable, denoted ?
  2, is the same for all of the populations.
• The observations must be independent.

8
Conceptual Overview

• If the null hypothesis (H0 ?1???2???3) is
  true
• Each sample will have come from the same normal
  probability distribution with mean ? and variance
  ? 2.

9
Conceptual Overview

• If the null hypothesis (H0 ?1???2???3) is
  true
• Thus, we can think of each of the 3 sample means
  as values drawn at random from the following
  sampling distribution

?
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Conceptual Overview

• If the null hypothesis (H0 ?1???2???3) is
  true
• Thus, we can think of each of the 3 sample means
  as values drawn at random from the following
  sampling distribution

?
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Conceptual Overview

• If the null hypothesis (H0 ?1???2???3) is
  true
• And, we can use the mean and variance of the
  three values to estimate the mean and
  variance of the sampling distribution

?
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Conceptual Overview

• Estimate of the mean of the sampling distribution
  of
• Overall Sample Mean
• Estimate of the variance of the sampling
  distribution of
• Because , solving for ? 2 gives

Between-treatments estimate of ?2
Between-treatments estimate of ?2
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Conceptual Overview

• Between-treatments estimate of ?2 is based on the
  assumption that H0 is true (H0 ?1???2???3).
• If H0 is false, 2 or more samples will be from
  normal populations with different means.,
  resulting in 3 different sampling distributions.

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Conceptual Overview

• Between-treatments estimate of ?2 is based on the
  assumption that H0 is true (H0 ?1???2???3).
• If H0 is false, 2 or more samples will be from
  normal populations with different means.,
  resulting in 3 different sampling distributions.

x1
?1
15
Conceptual Overview

• Between-treatments estimate of ?2 is based on the
  assumption that H0 is true (H0 ?1???2???3).
• If H0 is false, 2 or more samples will be from
  normal populations with different means.,
  resulting in 3 different sampling distributions.

x1
x2
?1
?2
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Conceptual Overview

• Between-treatments estimate of ?2 is based on the
  assumption that H0 is true (H0 ?1???2???3).
• If H0 is false, 2 or more samples will be from
  normal populations with different means.,
  resulting in 3 different sampling distributions.

x1
x3
x2
?1
?
?3
?2
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Conceptual Overview

• Between-treatments estimate of ?2 is based on the
  assumption that H0 is true (H0 ?1???2???3).
• If H0 is false, 2 or more samples will be from
  normal populations with different means.,
  resulting in 3 different sampling distributions.
• Therefore, when the populations are not equal,
  the between-treatments estimate will overestimate
  ?2.

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Conceptual Overview
Within-Treatments Estimate of ?2

• Each s2 is a point-estimator of ?2
• Pooled or within-treatments estimate of ?2

• Within-treatments estimate is not affected by
  whether the population means are equal.
• If H0 is true, the between-treatments estimate
  and the within-treatments estimate will be close.
• If H0 is false, the between treatments estimate
  will overestimate ?2 and will be larger than the
  within-treatments estimate.

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Conceptual Overview

• Between-treatments estimate of ?2 245
• Within-treatments estimate of ?2 25.67
• The ratio 245/25.67 9.5
• If H0 is true, the estimates will be similar
• and the ratio will be close to 1.
• If H0 is false, ratio will be large.
• How large must the ratio be to reject H0?

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Analysis of VarianceTesting for the Equality of
k Population Means
H0 ?1 ?2 ... ?k Ha Not all population
means are equal

• Between-Treatments Estimate of Population
  Variance
• Within-Treatments Estimate of Population Variance
• Comparing the Variance Estimates The F Test
• The ANOVA Table

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Between-Treatments Estimate of Population Variance

• A between-treatment estimate of ? 2 is called
  the mean square treatment and is denoted MSTR.
• Where
• k the number of treatments
• nj the number of observations in sample j
• the mean of sample j
• the overall sample mean

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Between-Treatments Estimate of Population Variance

• A between-treatment estimate of ? 2 is called
  the mean square treatment and is denoted MSTR.
• The numerator of MSTR is called the sum of
  squares treatment and is denoted SSTR.
• The denominator of MSTR represents the degrees of
  freedom associated with SSTR.

23
Example Reed Manufacturing

• Analysis of Variance
• Plant 1 Plant 2 Plant 3
• Observation Buffalo Pittsburgh
  Detroit
• 1 48 73 51
• 2 54 63 63
• 3 57 66 61
• 4 54 64 54
• 5 62 74 56
• Sample Mean 55 68 57
• Sample Variance 26.0 26.5 24.5

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Example Reed Manufacturing, MSTR
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Within-Treatments Estimate of Population Variance

• The estimate of ? 2 based on the variation of the
  sample observations within each sample is called
  the mean square error and is denoted by MSE.
• Where
• s2j The variance of sample j
• nT The sum of all sample sizes

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Within-Treatments Estimate of Population Variance

• The estimate of ? 2 based on the variation of the
  sample observations within each sample is called
  the mean square error and is denoted by MSE.
• The numerator of MSE is called the sum of squares
  error and is denoted by SSE.
• The denominator of MSE represents the degrees of
  freedom associated with SSE.

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Example Reed Manufacturing, MSE
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Comparing the Variance Estimates The F Test

• If the null hypothesis is true and the ANOVA
  assumptions are valid, the sampling distribution
  of MSTR/ MSE is an F distribution with MSTR d.f.
  equal to k - 1 and MSE d.f. equal to nT - k.
• If the means of the k populations are not equal,
  the value of MSTR/ MSE will be inflated because
  MSTR overestimates ? 2.
• Hence, we will reject H0 if the resulting value
  of MSTR/ MSE appears to be too large to have
  been selected at random from the appropriate F
  distribution.

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Test for the Equality of k Population Means

• Hypotheses
• H0 ?1???2???3??. . . ?k?
• Ha Not all population means are equal
• Test Statistic
• F MSTR/MSE
• Rejection Rule
• Reject H0 if F gt F?
• where the value of F?? is based on an F
  distribution with k - 1 numerator degrees of
  freedom and nT - k denominator degrees of freedom.

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Example Reed Manufacturing

• Analysis of Variance
• F - Test
• If H0 is true, the ratio MSTR/ MSE should
  be near
• 1 since both MSTR and MSE are estimating ?
  2. If
• Ha is true, the ratio should be
  significantly larger
• than 1 since MSTR tends to overestimate ?
  2.
• Rejection Rule
• Assuming ? .05, F.05 3.89 (2 d.f.
  numerator,
• 12 d.f. denominator). Reject H0 if F gt
  3.89
• Test Statistic
• F MSTR/ MSE 245/ 25.667 9.55

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Example Reed Manufacturing

• Analysis of Variance
• Conclusion
• F 9.55 gt F.05 3.89, so we reject H0.
  The mean
• number of hours worked per week by
  department
• managers is not the same at each plant.
• ANOVA Table
• Source of Sum of Degrees of
  Mean
• Variation Squares Freedom
  Square F
• Treatments 490 2 245
  9.55
• Error 308 12 25.667
• Total 798 14

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Using Excel to Test for the Equality of k
Population Means

• Excels Anova Single Factor Tool
• Step 1 Select the Tools pull-down menu
• Step 2 Choose the Data Analysis option
• Step 3 Choose Anova Single Factor
• from the list of Analysis Tools
• continued

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Using Excel to Test for the Equality of k
Population Means

• Excels Anova Single Factor Tool
• Step 4 When the Anova Single Factor dialog box
  appears
• Enter B1D6 in the Input Range box
• Select Grouped By Columns
• Select Labels in First Row
• Enter .05 in the Alpha box
• Select Output Range
• Enter A8 (your choice) in the Output
  Range box
• Select OK

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Using Excel to Test for the Equality of k
Population Means

• Value Worksheet (top portion)

35
Using Excel to Test for the Equality of k
Population Means

• Value Worksheet (bottom portion)

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Now You Try. Page 436, 29
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End of Chapter 10