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Unformatted text preview: CHAPTER 9: ONEWAY ANALYSIS OF VARIANCE 9.1 THE FDISTRIBUTION Analysisofvariance procedures rely on a distribution called the Fdistribution , 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 Fdistribution if its distribution has the shape of a special type of rightskewed curve, called an Fcurve . There are infinitely many Fdistributions, and we identify an Fdistribution (and Fcurve) by stating its number of degrees of freedom, just as we did for tdistributions and chi square distributions. But, as shown below, an Fdistribution has two numbers of degrees of freedom instead of one. The first number of degrees of freedom ( df 1 ) for an Fcurve 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 FCURVES There are five major properties. Property 1 : The F distribution is a continuous probability distribution . The total area under an Fcurve equals 1 . Property 2 : The F distribution is asymptotic . An Fcurve 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 Fcurve is positively skewed . The long tail of the distribution is to the righthand 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 FVALUE HAVING A SPECIFIED AREA TO ITS RIGHT Dr. LOHAKA: QBA 2305  CHAPTER 9: ONEWAY ANALYSIS OF VARIANCE Page 24 Percentages (and probabilities), for a variable having an Fdistribution, are equal to areas under its associated Fcurve. To perform an analysis of variance test, we need to know how to find the Fvalue having a specified area to its right. The symbol F is used to denote the Fvalue having area to its right. EXAMPLE 9.1 Problem : a) For an Fcurve with df = (4, 12), find F 0.05 ; that is, find the Fvalue having area 0.05 to its right. b) For an Fcurve with df = (12, 4), find F 0.95 ; that is, find the Fvalue having area 0.95 to its right. Solution : a) To obtain the F 0.05value, 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 Fvalue; that is, for an Fcurve with df = (4, 12), the Fcurve 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
 Hulme

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