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Chapter 14

Chapter 14 Textbook Notes: Analysis of Variance

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MGMT 1050
Alan Marshall

Chapter 14: Analysis of Variance Analysis of variance  Determines whether differences exist between population means  Procedure works by analyzing the sample variance  Population means = treatment means 14.1: One-Way Analysis of Variance  Procedure that tests to determine whether differences exist between two or more population means  Technique analyzes the variance of the data to determine whether we can infer that the population means differ  Experimental design is a determinant in identifying the proper method to use One-way analysis of variance  Procedure to apply when the samples are independently drawn  Figure 14.1: Sampling Scheme for Independent Samples page 526  Confirm that the data are interval and that the problem objective is to compare populations  Parameters are the four* population means: and  Null hypothesis will state that there are no differences between the population means o  Analysis of variance determines whether there is enough statistical evidence to show that the null hypothesis is false  Alternative hypothesis will always specify the following: o  Next step is to determine the test statistic Response variable  The variable X is called the response variable Responses  The values of the response variable X Experimental unit  The unit we measure is called the experimental unit Factor  The criterion by which we classify the population  Each population is called a factor level Test Statistic  If null hypothesis is true, the population means would all be equal and we would then expect the sample means to be close to one another  If the alternative hypothesis is true, there would be large differences between some of the sample means Between-treatments variation  Measures the proximity of the sample means to each other  Denoted SST (sum of squares for treatments) Sum of squares for treatments (SST)  If the sample means are close to each other, all of the sample means would be close to the grand mean; as a result SST would be small  SST achieves its smallest value(zero) when all the sample means are equal  A small value of SST supports the null hypothesis Steps  Compute the sample means ( and the grand mean (X double bar)  Compute the sample sizes ( and the total of the sample sizes to get n  Then calculate SST  If large differences exist between the sample means, making SST a large value, it is reasonable to reject the null hypothesis  Important question becomes: how large does the statistic have to be for us to justify rejecting the null hypothesis?  To answer question, important to know how much variation exists in response variables Within-treatments variation  Provides a measure of the amount of variation in the response variable that is not caused by the treatments  Denoted by SSE (sum of squares for error) Sum of squares for error (SSE)  Each of the k components of SSE is a measure of the variability of that sample  If we divide each component by , we obtain the sample variances. We can express this be rewriting SSE as:  SSE is the combined or pooled variation of the k samples  A condition for SSE requires that the population variances be equal o Steps  Calculate the sample variances (  Calculate SSE  Next step is to compute quantities called the mean squares Sampling Distribution of the Test Statistic  Test statistic is F-distributed with k-1 and n-k degrees of freedom provided that the response variable is normally distributed Steps  Calculate degrees of freedom  Calculate MSE and MST  Calculate test statistic F Rejection Region and p-Value  Purpose of calculating the F-statistic is to determine whether the value of SST is large enough to reject the null hypothesis  If SST is large, F will be large  We reject the null hypothesis only if: o  Results of the analysis of variance are usually reported in an ANOVA table  Table 14.2: ANOVA Table for the One-Way Analysis of Variance page 532  If SST explains a significant portion of the total variation, we conclude that the population means differ Completely randomized design  When the data are obtained through a controlled experiment in the one-way analysis of variance, we call the experimental design the completely randomized design of the analysis of variance Checking the Required Conditions  The F-test of the analysis of variance requires that the random variable be normally distributed with equal variances  Normality requirement is easily checked graphically by producing the histograms for each sample  The equality of variances is examined by printing the sample standard deviations or variances  The similarity of sample variances allows us to assume that the population variances are equal Violation of the Required Conditions  If the data are not normally distributed, we can replace the one-way analysis of variance with its nonparametric counterpart, which is the Kruskal-Wallis Test  If the population variances are unequal, we can use several methods to correct the problem Can We Use the t-Test or the Difference between Two Means Instead of the Analysis of Variance?  Analysis of variance test determines whether there is evidence of differences between two or more population means  The t-test of determines whether there is evidence of a difference between two population means  The question arises, can we use t-tests instead of the analysis of variance? In other words, instead of testing all the means in one test as in the analysis of variance, why not test each pair of means?  There are two reasons why we don’t use multiple t-tests instead of one F-test: o We would have to perform many more calculations o Conducting multiple tests increases the probability of making Type I errors  When we want to compare more than two populations of interval data, we use the analysis of variance Can We Use the Analysis of Variance Instead of the t-Test of the Difference between Two Means?  If we want to determine whether is greater than we cannot use the analysis of variance because this technique allows us to test for a difference only  If we want to test to determine whether one population mean exceed the other, we must use the t-test  Moreover, the analysis of variance requires that the population variance are equal and if they are not, we must use the unequal variances test statistic Relationship between the F-Statistic and the t-Statistic  If we square the quantity of the t-statistic, the result is the F-statistic o Developing an Understanding of Statistical Concepts  The F-test of the independent samples’ single-factor analysis of variance is an extension o
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