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Lecture Slides

This section provides an overview of the F-test for comparing two population variances, as well as alternative methods that are less sensitive to departures from normality. It covers notation, requirements, test statistics, critical values, properties of the F distribution, and examples illustrating the application of these concepts.

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Lecture Slides

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  1. Lecture Slides Elementary StatisticsTwelfth Edition and the Triola Statistics Series by Mario F. Triola

  2. Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations

  3. Key Concept This section presents the F test for comparing two population variances (or standard deviations). We introduce the F distribution that is used for the F test. Note that the F test is very sensitive to departures from normal distributions.

  4. Part 1 F Test as a Hypothesis Test with Two Variances or Standard Deviations

  5. Notation for Hypothesis Tests with Two Variances or Standard Deviations = larger of two sample variances = size of the sample with the larger variance = variance of the population from which the sample with the larger variance is drawn are used for the other sample and population

  6. Requirements 1. The two populations are independent. 2. The two samples are simple random samples. 3. The two populations are each normally distributed (very strict requirement, so be sure to examine the distributions of the two samples using histograms and normal quantile plots).

  7. Test Statistic for Hypothesis Tests with Two Variances Where is the larger of the two sample variances P-values: Automatically provided by technology or estimated using Table A-5. Critical values:Using Table A-5, we obtain critical F values that are determined by the following three values: 1. The significance level α. 2. Numerator degrees of freedom = n1 – 1 3. Denominator degrees of freedom = n2 – 1

  8. The F distribution is not symmetric. Values of the F distribution cannot be negative. The exact shape of the F distribution depends on the two different degrees of freedom. Properties of the F Distribution

  9. Finding Critical F Values

  10. If the two populations do have equal variances, then will be close to 1 because and are close in value. Properties of the F Distribution

  11. Consequently, a value of F near 1 will be evidence in favor of the conclusion that: But a large value of F will be evidence against the conclusion of equality of the population variances. Properties of the F Distribution

  12. Subjects with a red background were asked to think of creative uses for a brick, while other subjects were given the same task with a blue background. Creativity score responses were scored by a panel of judges. Test the claim that those tested with a red background have creativity scores with a standard deviation equal to the standard deviation for those tested with a blue background. Use a 0.05 significance level. Example

  13. Requirement Check: The two populations are independent of each other. Given that the design for the study was carefully planned, we assume the two samples can be treated as simple random samples. We have no knowledge of the distribution of the data, so we proceed assuming the samples are from normally distributed populations, but we note that the validity of the results depends on that assumption. Example

  14. Step 1: The claim of equal standard deviations can be expressed as: Step 2: If the original claim is false, then: Step 3: We can write the hypotheses as: Example

  15. Step 4: The significance level is α = 0.05. Step 5: Because this test involves two population variances, we use the F distribution. Step 6: The test statistic is The degrees of freedom for the numerator are n1 – 1 = 34. The degrees of freedom for the denominator are n2 – 1 = 35. Example

  16. Step 6: From Table A-5, the critical F value is between 1.8752 and 2.0739. Technology can be used to find the exact critical F value of 1.9678. The P-value can be determined using technology: P-value = 0.0129 Example

  17. Step 7: Since the P-value = 0.0129 is below the significance level of 0.05, we can reject the null hypothesis. Also, the figure below shows the test statistic is in the critical region, so reject. Example

  18. There is sufficient evidence to warrant rejection of the claim that the two standard deviations are equal. The variation among creativity scores for those with a red background appears to be different from the variation among creativity scores for those with a blue background. Example

  19. Part 2 Alternative Methods – Two Methods that are not so Sensitive to Departures from Normality.

  20. Count Five Method: If the two sample sizes are equal, and if one sample has at least five of the largest mean absolute deviations, then we conclude its population has the larger variance. Example

  21. Levene-Brown-Forsythe Test First, replace each x value with: Using the transformed values, conduct a t test of equality of means for independent samples. This t test for equality of means is actually a test comparing variation in the two samples. Example

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