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Unformatted text preview: CHAPTER 9: ONE-WAY ANALYSIS OF VARIANCE 9.1 THE F-DISTRIBUTION Analysis-of-variance procedures rely on a distribution called the F-distribution , named in honor of Sir Ronald Fisher, who developed the initial techniques of the analysis of variance in the 1920s and 1930s. A variable is said to have an F-distribution if its distribution has the shape of a special type of right-skewed curve, called an F-curve . There are infinitely many F-distributions, and we identify an F-distribution (and F-curve) by stating its number of degrees of freedom, just as we did for t-distributions and chi- square distributions. But, as shown below, an F-distribution has two numbers of degrees of freedom instead of one. The first number of degrees of freedom ( df 1 ) for an F-curve is called the degrees of freedom for the numerator , and the second ( df 2 ) is called the degrees of freedom for the denominator . 9.1.1 BASIC PROPERTIES OF F-CURVES There are five major properties. Property 1 : The F distribution is a continuous probability distribution . The total area under an F-curve equals 1 . Property 2 : The F distribution is asymptotic . An F-curve starts at 0 on the horizontal axis and extends indefinitely to the right, approaching, but never touching, the horizontal axis as it does so. Property 3 : The F-curve is positively skewed . The long tail of the distribution is to the right-hand side. As the number of degrees of freedom increases in both the numerator and the denominator the F distribution approaches a normal distribution. Property 4 : The F distribution cannot be negative . The smallest value F can assume is 0. Property 5 : The mean of the F distribution is 2 2 2 df df =- for df 2 2, where df 2 is the degrees of freedom for the denominator. 9.1.2 FINDING THE F-VALUE HAVING A SPECIFIED AREA TO ITS RIGHT Dr. LOHAKA: QBA 2305 --- CHAPTER 9: ONE-WAY ANALYSIS OF VARIANCE Page 24 Percentages (and probabilities), for a variable having an F-distribution, are equal to areas under its associated F-curve. To perform an analysis of variance test, we need to know how to find the F-value having a specified area to its right. The symbol F is used to denote the F-value having area to its right. EXAMPLE 9.1 Problem : a) For an F-curve with df = (4, 12), find F 0.05 ; that is, find the F-value having area 0.05 to its right. b) For an F-curve with df = (12, 4), find F 0.95 ; that is, find the F-value having area 0.95 to its right. Solution : a) To obtain the F 0.05-value, we use the Fisher Table . In this case, = 0.05 , the degrees of freedom for the numerator is 4 , and the degrees of freedom for the denominator is 12 . We first go down the df column to 12 . Next, we go across the row for labeled 0.05 to the column headed 4 . The number in the body of the table there, 3.26, is the required F-value; that is, for an F-curve with df = (4, 12), the F-curve having area 0.05 to its right is 3.26: F (0.05, 4, 12) = 3.26 ....
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This note was uploaded on 08/04/2008 for the course QBA 2305 taught by Professor Hulme during the Spring '08 term at Baylor.
- Spring '08