Analysis of Variance

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Chapter 12

• Analysis of Variance

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Goals

1. List the characteristics of the F distribution
2. Conduct a test of hypothesis to determine whether
   the variances of two populations are equal
3. Discuss the general idea of analysis of variance
4. Organize data into a ANOVA table
5. Conduct a test of hypothesis among three or more
   treatment means

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F Distribution

• Used to test whether two samples are from
  populations having equal variances
• Applied when we want to compare several
  population means simultaneously to determine if
  they came from equal population
• ANOVA
• Analysis of variance
• In both situations
• Populations must be Normally distributed
• Data must be Interval-scale or higher

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Characteristics Of The F Distribution

• Family of F Distributions
• Each is determined by
• df in numerator
• comes from pop. 1 which has larger sample
  variation
• df in denominator
• comes from pop. 2 which has smaller sample
  variation
• F distribution is continuous
• F Value can assume an infinite number of values
  from 0 to 8
• Value for F Distribution cannot be negative
• Smallest value 0
• Positively skewed
• Long tail is always to right
• As of df increases in both the numerator and
  the denominator, the distribution approaches
  normal
• Asymptotic
• As X increases the F curve approaches the X-axis

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Why Do We Want To Compare To See If Two
Population Have Equal Variances?

• What if two machines are making the same part for
  an airplane?
• Do we want the parts to be identical or nearly
  identical? Yes!
• We would test to see if the means are the same
  Chapter 10 11
• We would test to see if the variation is the same
  for the two machines Chapter 12

• What if two stocks have similar mean returns?
• Would we like to test and see if one stock has
  more variation than the other?

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Why Do We Want To Compare To See If Two
Population Have Equal Variances?

• Remember Chapter 11 Assumptions for small sample
  tests of means
• Sample populations must follow the normal
  distribution
• Two samples must be from independent (unrelated)
  populations
• The variances standard deviations of the two
  populations are equal

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Conduct A Test Of Hypothesis To Determine Whether
The Variances Of Two Populations Are Equal

• To conduct a test
• Conduct two random samples
• List population 1 as the sample with the largest
  variance
• n1 of observations
• s12 sample variance
• n1 1 df1 degree of freedom (numerator for
  critical value lookup)
• List population 2 as the sample with the smaller
  variance
• n2 of observations
• s22 sample variance
• n2 1 df2 degree of freedom (denominator for
  critical value lookup)

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Step 1 State null and alternate hypotheses

• List the population with the suspected largest
  variance as population 1
• Because we want to limit the number of F tables
  we need to use to look up values, we always put
  the larger variance in the numerator and the
  smaller variance in the denominator
• This will force the F value to be at least 1
• We will only use the right tail of the F
  distribution
• Examples of Step 1

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Step 2 Select a level of significance

• Appendix G only lists significance levels .05
  and .01

Significance level .10 .10/2 .05 Use .05
table in Appendix G
Significance level .05 Use .05 table
in Appendix G
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Step 3 Identify the test statistic (F), find
critical value and draw picture

• Look up Critical value in Appendix G and draw
  your picture

If you have a df that is not listed in the
border, calculate your F by estimating a
value between two values. HW 5 df 11, use
value Between 10 12 Book says (3.143.07)/2
3.105 ? 3.10
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Step 4

• Step 4 Formulate a decision rule
• Example
• If our calculated test statistic is greater than
  3.87, reject Ho and accept H1, otherwise fail to
  reject Ho

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Step 5

• Step 5 Take a random sample, compute the test
  statistic, compare it to critical value, and make
  decision to reject or not reject null and
  hypotheses
• Test Statistic F
• Example Conclusion for a two tail test
• Fail to reject null
• The evidence suggests that there is not a
  difference in variation
• Reject null and accept alternate
• The evidence suggests that there is a difference
  in variation

Lets Look at Handout ?
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Example 1
Colin, a stockbroker at Critical Securities,
reported that the mean rate of return on a sample
of 10 software stocks was 12.6 percent with a
standard deviation of 3.9 percent.
The mean rate of return on a sample of 8 utility
stocks was 10.9 percent with a standard deviation
of 3.5 percent. At the .05 significance level,
can Colin conclude that there is more variation
in the software stocks?
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Example 1 continued
Step 1 The hypotheses are
Step 2 The significance level is .05.
Step 3 The test statistic is the F
distribution.
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Example 1 continued
Step 4 H0 is rejected if Fgt3.68 or if p lt .05.
The degrees of freedom are n1-1 or 9 in the
numerator and n1-1 or 7 in the denominator.
Step 5 The value of F is computed as follows.
H0 is not rejected. There is insufficient
evidence to show more variation in the software
stocks.
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ANOVAAnalysis Of Variance

• Technique in which we compare three or more
  population means to determine whether they could
  be equal
• Assumptions necessary
• Populations follow the normal distribution
• Populations have equal standard deviations (?)
• Populations are independent
• Why ANOVA?
• Using t-distribution leads to build up of type 1
  error
• Treatment different populations being examined

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Case Where Treatment Means Are Different
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Case Where Treatment Means Are The Same
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Example Of ANOVA Test To See If Four Treatment
Means Are The Equal

• 22 students earned the following grades in
  Professor Rads class. The grades are listed
  under the classification the student gave to the
  instructor
• Is there a difference in the mean score of the
  students in each of the four categories?
• Use significance level a .01

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Conduct A Test Of Hypothesis Among Four Treatment
Means

• Step 1 State H0 and H1
• H0 µ1 µ2 µ3 µ4
• H1 The Mean scores are not all equal (at least
  one treatment mean is different)
• Step 2 Significance Level?
• a .01

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Step 3 Determine Test Statistic And Select
Critical Value
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Step 4 State Decision Rule

• If our calculated test statistic is greater than
  we reject H0 and accept H1, otherwise we fail
  to reject H0

Now we move on to Step 5 Select the sample,
perform calculations, and make a decision Are
you ready for a lot of procedures?!!
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• The idea is If we estimate variation in two ways
  and use one estimate in the numerator and the
  other estimate in the denominator
• If we divide and get 1 or close to 1, the sample
  means are assumed to be the same
• If we get a number far from 1, we say that the
  means are assumed to be different
• The F critical value will determined whether we
  are close to 1 or not

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ANOVA Table So Far
Lets go calculate this!
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Calculation 1 Treatment Means and Overall Mean
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Calculation 2 Total Variation
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Calculation 2 Total Variation
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ANOVA Table So Far
Lets go calculate this!
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Calculation 3 Random Variation
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Calculation 3 Random Variation
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ANOVA Table So Far
Lets go calculate this!
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Calculation 4 Treatment Variation
Simple Subtraction!
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Calculation 5 Mean Square (Estimate of Variation)
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Calculation 6 F
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Step 5 Make A Decision

• Because is less than 5.09, we fail to
  reject H0
• The evidence suggests that the mean score of the
  students in each of the four categories are equal
  (no difference)

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Summarize Chapter 12

1. List the characteristics of the F distribution
2. Conduct a test of hypothesis to determine whether
   the variances of two populations are equal
3. Discuss the general idea of analysis of variance
4. Organize data into a ANOVA table
5. Conduct a test of hypothesis among three or more
   treatment means