TESTING THE EQUALITY OF TWO VARIANCES: THE F TEST

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TESTING THE EQUALITY OF TWO VARIANCES THE F
TEST

• Application
• test assumption of equal variances that was made
  in using the t-test
• interest in actually comparing the variance of
  two populations

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The F-Distribution

• Assume we repeatedly select a random sample of
  size n from two normal populations.
• Consider the distribution of the ratio of two
  variances F s12/s12.
• The distribution formed in this manner
  approximates an F distribution with the following
  degrees of freedom
• v1 n1 - 1 and v1 n1 - 1

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Assumptions

• Random, independent samples from 2 normal
  populations
• Variability

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F-Table

• The F table can be found on the appendix of our
  text. It gives the critical values of the
  F-distribution which depend upon the degrees of
  freedom.

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Example 1

• Assume that we have two samples with
• n2 7 and n1 10
• df 7-1 6 and df 10-1 9
• Let v F(6,9)
• where 6 is the df from the numerator and 9 is the
  df of the denominator.
• Using the table with the appropriate df, we find
  
• P(v lt 3.37) 0.95.

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Example 2 Hypothesis Testto Compare Two
Variances

• 1. Formulate the null and alternate hypotheses.
• H0 s12 s12
• Ha s12gt s22
• Note that we might also use s12 lt s22 or s12 /
  s22
• 2. Calculate the F ratio.
• F s12/s12 where s1 is the largest or the
  two variances
• 3. Reject the null hypothesis of equal population
  variances if F(v1-1, v2-1) gt Fa
• or Fa/2 in the case of a two tailed test

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Example 2

• The variability in the amount of impurities
  present in a batch of chemicals used for a
  particular process depends on the length of time
  that the process is in operation.
• Suppose a sample of size 25 is drawn from the
  normal process which is to be compared to a
  sample of a new process that has been developed
  to reduce the variability of impurities.
• Sample 1 Sample 2
• n 25 25
• s2 1.04 0.51

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Example 2 continued

• H0 s12 s22
• Ha s12 gt s22
• F(24,24) s12/s22 1.04/.51 2.04
• Assuming a 0.05
• cv 1.98 lt 2.04
• Thus, reject H0 and conclude that the variability
  in the new process (Sample 2) is less than the
  variability in the original process.

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Try This

• A manufacturer wishes to determine whether there
  is less variability in the silver plating done by
  Company 1 than that done by Company 2.
  Independent random samples yield the following
  results. Do the populations have different
  variances?
• solution reject H0 since 3.14 gt 2.82