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CHAPTER 14 - CHAPTER 14 STATISTICAL INFERENCE OTHER...

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CHAPTER 14 – STATISTICAL INFERENCE: OTHER TWO-SAMPLE TEST STATISTICS 14.1 Introduction Learning about the F statistic and F sampling distribution that are used to test hypotheses about 2 variances and how to use a z statistic to test hypotheses about 2 population proportions 14.2 Two-Sample F Test and Confidence Interval for Variances Using Independent Samples F Test for 2 Variances (Independent Samples) Researcher might want to see if 2 populations differ in dispersion or they might want to test one of the assumptions of the t test for independent samples – that the 2 unknown population variances are equal F statistic for testing the hypotheses : = : : H0 σ12 σ22 H0 σ12 σ22 H0 σ12 σ22 : : < : > H1 σ12 σ22 H1 σ12 σ22 H1 σ12 σ22 is = F σlarger2σsmaller2 where σlarger2 and σsmaller2 denote, respectively, the larger and smaller sample variance and each sample variance is computed using = - - σ 2 Xi X2n 1 . The degrees of freedom for the numerator and denominator are, respectively, = - ν1 nlarger σ2 1 and = ν2 nsmaller - σ2 1 . Sampling distribution of the F derived by Ronald A. Fisher and G.W. Snedecor named it after him. Like the t distr., is a family of distributions whose shape depends on its d.f. Unlike the z and t distributions that are symmetrical, the F distribution is positively skewed Shape of the F distribution approaches the normal for large values of ν1 and ν2 . F is a ratio of non-negative numbers so it can take on values from 0 to ∞. Values around 1 are expected if the null hypothesis that = σ12 σ22 is true Assumptions for the F to test a null hypothesis: 1. Independent samples 2. Populations are normally distributed 3. Participants are random samples from the populations of interest or the participants have been randomly assigned to the conditions in the experiment. F is not robust to violation of the normality assumption like the t is regardless of how large your sample is. So, unless the normality assumption is fulfilled, the probability of making a Type 1 error will not equal the preselected value of α. Do NOT use the F unless you have good reason to believe the 2 variables X1 and X2 are normal.
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; , ν1 ν2 is the critical value of F that cuts off the upper α region of the sampling distribution for ν1 and ν2 degrees of freedom. The first ν denotes the df for the numerator of the F and the second ν denotes the df for the denominator of the F.
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