Chapter 11.docx

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Western University
Statistical Sciences
Statistical Sciences 2244A/B
Jennifer Waugh

11.1 Overview ● We want to introduce a procedure for testing the hypothesis that three or more population means are equal, soa typical null hypothesis will be H : 0 =1μ = 2 etc3 ○ the alternative hypothesis then would be that at least one mean is different from the others ● In section 8-3 we already presented procedures for testing the hypothesis that two population means are equal but the methods of that section do not apply when three or more means are involved ● Analysis of variance (ANOVA) is a method of testing the equality of three or more population means by analyzing sample variances ● Why can’t we just test two samples at a time; why do we need a new procedure when we can test for equality of two means by using methods presented in chapter 8? ● For example, if we have four populations and wish to compare their means, why can’t we compare two means at a time (resulting in 6 total tests and 6 null hypotheses) ● This is because in general, as we increase the number of individual tests of significance, we increase the likelihood of finding a difference by chance alone (instead of a real difference in means) ● The risk of a type I error is far too high; the method ofANOVAhelps us avoid that particular pitfall by using one test for equality of several means F Distribution ● TheANOVAmethods of this chapter require the F distribution; it has the following important properties ○ the F distribution is not symmetric; it is skewed to the right ○ the values of F can be 0 or positive, but not negative ○ there is a different F distribution for each pair of degrees of freedom for the numerator and denominator ● ANOVAis based on a comparison of two different estimates of the variance common to the different populations ○ the estimates are the variance between samples and the variance within samples ● The term one-way is used because the sample data are separated into groups according to one characteristic or factor ● In section 11-3 we will introduce two way analysis of variance which allows us to compare populations separated into categories using two characteristics (or factors) ● We suggest that you begin section 11-2 by focusing on this key concept: we are using a procedure to test a claim that three or more means are equal; although the details of the calculations are complicated, the procedure will be easy because it is based on a P-value ○ if the P-value is small, reject equality of means; otherwise fail to reject equality of means 11­2 One­Way ANOVA ● In this section we consider tests of hypothesis that three or more population means are all equal, as in H0: μ1= μ 2 μ e3c. We recommended the following approach: ● Understand that a small P-value leads to rejection of the null hypothesis of equal means ● Develop an understanding of the underlying rationale by studying the example in this section ● Become acquainted with the nature of the sum of squares (SS) and mean square (MS) values and their role in determining the F test statistic ● The method we use in called one-way analysis of variance because we use a single property of characteristic for categorizing the populations; the characteristic is sometimes referred to as a treatment or factor ● Atreatment or factor is a property or characteristic that allows us to distinguish the different populations from one another Rationale ● The method of analysis of variance is based on this fundamental concept: with the assumption that the populations all have the same variance σ , we estimate the common value of σ using two different approaches ● The F test statistic is evidence against equal population means ○ a small F test, statistic means that the P-value is large, thus the sample means are all close and so we fail to reject the null hypothesis of equal means ○ a large F test statistic means that the P-value is small, thus at least one sample mean is very different so we reject the null hypothesis2of equal means ● The two approaches for estimating the common value of σ are as follows: ○ the variance between samples (variance due to treatment) is an estimate of common population variance σ that is based on the variation among the sample means ○ the variance within samples (variance due to error) is an estimate of the common population variance σ based on the sample variances ● Test Statistic for One-WayANOVA: ○ F = (variance between samples) / (variance within samples) ● The estimate of variance in the denominator depends on the sample variances and is not affected by differences among the sample means Calculations with Equal Sample Sizes n ● If data sets all have the sample sample size, the required calculations aren’t overwhelmingly difficult ● First find the variance between samples by evaluating ns x(bar) where s x(bar)s the variance of the sample means and n is the size of each of the samples ○ consider the sample means to be an ordinary set of values and calculate the variance ● Next estimate the variance within samples by calculating s which ip the pooled variance obtained by finding the mean of the sample variances ● The critical value of F is found by assuming a right tailed test because large F values correspond to significant differences among means ● With k sample each value n values, the number of degrees of freedom are as follows: ○ numerator degrees of freedom = k-1 ○ denominator degrees of freedom = k(n-1) ● The variance within a sample isn’t affected when we add a constant to every sample value; the change in the F test statistic and the P-value is attributable only to the change in x(bar) 1 ● This illustrates that the F test statistic is very sensitive to sample means, even though it is obtained through two different estimates of the common population variance Calculations with Unequal Sample Sizes ● While the calculations for cases with equal sample sizes are reasonable, they become c
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