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A BUSINESS STATISTICS COURSE Chapter 9 ANALYSIS OF VARIANCE_1

# A BUSINESS STATISTICS COURSE Chapter 9 ANALYSIS OF VARIANCE_1

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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 μ = - 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

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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 α is used to denote the F -value having area α 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, α = 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 α 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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A BUSINESS STATISTICS COURSE Chapter 9 ANALYSIS OF VARIANCE_1

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