Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Part 1 F Test as a Hypothesis Test with Two Variances or Standard Deviations

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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).

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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 = n 1 – 1 3. Denominator degrees of freedom = n 2 – 1 Test Statistic for Hypothesis Tests with Two Variances Where is the larger of the two sample variances

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Finding Critical F Values

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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 Creativity Scorens Red Background Blue Background

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Requirement Check: 1.The two populations are independent of each other. 2.Given that the design for the study was carefully planned, we assume the two samples can be treated as simple random samples. 3.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

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Step 4: The significance level is α = 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 n 1 – 1 = 34. The degrees of freedom for the denominator are n 2 – 1 = 35. Example

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Step 6: From Table A-5, the critical F value is between and Technology can be used to find the exact critical F value of The P-value can be determined using technology: P-value = Example

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Step 7: Since the P-value = 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

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Part 2 Alternative Methods – Two Methods that are not so Sensitive to Departures from Normality.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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