Chapter 12Analysis of Variance

An Overview

•We know how to test a hypothesis about two population means, but what if we have more than two?

•Example: teachers at a school devise three methods to teach math and they want to know if the three methods produce the same mean scores, in other words… μ₁ = μ₂ = μ₃?

12.1 The F Distribution

•Shape of curve depends on degrees of freedom

•There are two values for the F distribution for degrees of freedom and different combinations give different shapes

•Only non-negative values and skewed right like Chi-square

•df = (8, 14) where the first number is the numerator, and the second number is the denominator

Table VII

•Different tables for certain significance levels

•Only read for the right tail •Example: Find the F value for 8 degrees

of freedom for the numerator, 14 degrees of freedom for the denominator, and .05 area in the right tail of the F distribution.

•Answer: 2.70

df Formula

•df (numerator) = k - 1, where k is the number of different samples

•df (denominator) = n - k, where n is the number of values in all samples

12.2 One-Way Analysis of Variance•Null Hypothesis: the means of three or

more populations are the same •Alternative hypothesis: not all population

means are the same•Only testing one factor (one-way)•Use the same 5-step hypothesis test as

previous

Assumptions

•The following must hold true to use one-way ANOVA

1. The populations from which the samples are drawn are normally distributed

2. The populations from which the samples are drawn have the same variance (or standard deviation)

3. The samples drawn from different populations are random and independent

Test Statistic with Calculator

•Steps:1. Store the lists in your calculator under L₁,

L₂, L₃2. Select Stat>Tests>ANOVA(3. Enter the names of the lists, separated by

commas, and then type the right parenthesis

4. Press Enter5. The test results include the F statistic as

well as other data

Example:• From time to time, unknown to its employees, the

research department at Post Bank observes various employees for their work productivity. Recently this department wanted to check whether the four tells at a branch of this bank serve, on average, the same number of customers per hour. The research manager observed each of the four tellers for a certain number of hours. The following table gives the number of customers served by the four tellers during each of the observed hours. At the 5% significance level, test the null hypothesis that the mean number of customers served per hour by each of these four tellers is the same. Assume the requirements hold true.

Prem Mann, Introductory Statistics, 6/ECopyright 2007 John Wiley & Sons. All rights reserved.

Table 12.5 (p. 542)

Answers:

1. H₀: μ₁ = μ₂ = μ₃ = μ₄ H₁: Not all four population means are equal

2. F distribution because we are testing for equality of four means

3. Critical Value: df (n) = 3, df (d) = 18, α = .05, F = 3.164. Test Statistic: F = 9.695. Since the test statistic falls in the rejection region, we

can state that the mean number of customers served per hour of each of the four tellers is not the same.

Homework

•12.1 – pg #535, #1-4, 8•12.2 – pg #543, 10-12, 16, 17