Aim: How do we test the difference between two variances?

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Aim How do we test the difference between two
variances?

• HW13 complete two questions on last slides

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Comparing Data

• Not only do we test the difference between two
  means (yesterdays aim) but statisticians are
  interested in comparing t two variances and
  standard deviations

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What test do we use?

• For comparison of two variances or standard
  deviations, an F test is used

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Characteristics of the F Distribution

1. The values of F cannot be negative, because
   variances are always positive or zero
2. The distribution is positively skewed
3. The mean value of F is approximately equal to 1
4. The F distribution is a family of curves based on
   the degrees of freedom of the variance of the
   numerator and the degrees of freedom of the
   variance of the denominator

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Shapes of F Distributions
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Formula for the F Test

• The larger of the two variances is placed in the
  numerator regardless of the subscripts

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Degrees of Freedom

• d.f.N ? degree of freedom for numerator
• d.f.D ? degree of freedom for denominator
• NEED TO USE H TABLE

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Example

• Find the critical value for a right-tailed F test
  when a0.05, the degrees of freedom for the
  numerator are 15, and the degrees of freedom for
  the denominator are 21.
• Since the test is right tailed with a0.05, use
  the 0.05 table. The d.f.N is listed across the
  top and the d.f.D is listed in the left column
• Solution 2.18

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Using F table

• When the degree of freedom values cannot be found
  in the table, closest values on the smaller side
  should be used
• Example If d.f.N 14, this value is between
  given table values of 12 and 15 therefore 12
  should be used, to be the safe side

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Testing the equality of two variances
Left-Tailed
Right-Tailed
Two-Tailed
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Steps

1. State the hypothesis and identify claim
2. Find the critical value
3. Compare the test values
4. Make the decision
5. Summarize the results

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Example

• A medical researcher wishes to see whether the
  variance of the heart rates (in beats per minute)
  of smokers is different from the variance of
  heart rates of people, who do not smoke. Two
  samples are selected, and the data are as shown.
  Using a0.05, is there enough evidence to support
  the claim?

Smoker Nonsmoker n 26 n 18 s2 36 s2
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Solution

• Critical value from table 2.56
• F 36/10 3.6
• 3.6 gt 2.56
• Summarize the results There is enough evidence
  to support the claim that the variance of the
  heart rates of smokers and nonsmokers is different

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Class Work 2

• Using the table H, find the P-value interval for
  each F-Test value
• F 2.97, d.f.N9, d.f.D14, right tailed
• F3.32, d.f.N6, d.f.D12, two tailed
• A researcher claims that the standard deviation
  of the ages of cats is smaller than the standard
  deviation of the ages of dogs who are owned by
  families in a large city. A randomly selected
  sample of 29 cats has a standard deviation of 2.7
  years and a random sample of 16 dogs has a
  standard deviation of 3.5 years. Is the
  researcher correct? Use a0.05. If there is a
  difference, suggest a reason for the difference.

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Homework
Question 1
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Homework
Question 2