Analysis of Variance

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Chapter 12

• Analysis of Variance

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Section 12-1

• Many real-world problems involve a comparison
  among several groups.
• ? Which company will provide the best
  automobile insurance?

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• Comparing 2 means at a time would be too time
  consuming with numerous groups to compare, so we
  have a new method to use instead.
• Analysis of Variance (ANOVA) is a method of
  testing the equality of 3 or more population
  means by analyzing sample variances. Its
  purpose is to determine the existence (or not) of
  a statistically significant difference among
  group means.

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• To do the test, we will use a new distribution
  called the F Distribution
• Non symmetric, skewed to the right (similar to
  chi-square).
• The values for F are or 0, but not negative.
• There is a different F distribution for each pair
  of degrees of freedom for the numerator and
  denominator.
• The critical values are given in Table A-5.

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Three F distribution curves.
df (1 , 3)
df (7 , 6)
df (12 , 40)
F
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Section 12-2
One-Way ANOVA

• In this section, we will be testing the
  hypothesis that three or more population means
  are all equal.

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Assumptions

• The populations have distributions that are
  approximately normal.
• The populations have the same variance ?2 (or
  standard deviation ?).
• The samples are simple random samples.

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• 4. The samples are independent of each other.
• 5. The different samples are from populations
  that are categorized in only one way (hence the
  name one-way ANOVA).
• A treatment/factor is a property, or
  characteristic, that allows us to distinguish the
  different populations from one another.

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Definitions

• 1. Variance between samples
• ? estimates the common population variance that
  is based on the variability among sample means.
• Variance within samples
• ? estimates the common population
• variance based on sample variances.

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• 3. Test statistic (with same size samples)
• If HO is true, the variance between the samples
  and within the samples should be approximately
  the same, so this value should be approximately 1.

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• Degrees of Freedom with k samples of the same
  size n
• ?numerator degrees of freedom
• k 1
• ?denominator degrees of freedom
• k(n-1)

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• For unequal sample sizes, we would use the
  following test statistic
• 4.
• Since the calculations required for analysis of
  variance are lengthy, well use software and
  learn to interpret the output provided.

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• Procedure
• 1. Set up the hypotheses
• HO ?1 ?2 ?3
• H1 At least one of the means is
• different.
• 2. Use software to produce an ANOVA table.

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• 3. Identify the P-value from the display.
• 4. Make a proper initial conclusion.
• 5. Write a statement in the context of the
  problem.

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Example 1

• Chest deceleration data (g) from car crash
  experiments conducted by the National
  Transportation Safety Administration are given
  below. Use a 0.05 significance level to test the
  null hypothesis that the different weight
  categories have the same mean. Do the data
  suggest that larger cars are safer?

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• HO µ1 µ2 µ3 µ4
• H1 At least one of the means is different.
• Next,
• create the ANOVA table.

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