ONE WAY ANOVA

One Way Anova



One Way Anova

Introduction

One-way ANOVA is used to compare means from at least three groups from one variable. The null hypothesis is that all the population group means are equal versus the alternative that at least one of the population means differs from the others. This may seem confusing as we call it Analysis of Variance even though we are comparing means. The reason is that the test statistic uses evidence about two types of variability. We will only consider the reason behind it instead of the complex formula used to calculate it (Bogartz, 2004, 55).

The dot plots present the minutes someone remained on hold before hanging up for three types of a recorded message. Say one got the sample data for A and someone else gathered data in B. Which case do you think gives stronger evidence that mean wait times for the three recording types are not all the same? That is, which gives stronger evidence against Ho: u1 = u2 = u3? In both sets of data the group means are the same.

That is, in both case A and B the mean wait time for Advertisement is 5.4 minutes; Classical 10.4 minutes and Muzak 2.8 minutes. What's the difference then? The variability between pairs of sample means is the same in each case because the sample means are the same. However, the variability within each sample is much smaller for B than for A. The sample SD is about 1 for each sample in B but in A the this SD is between 2.4 and 4.2 We will see that evidence against Ho is stronger when the variability within each sample is smaller. The evidence is also stronger when variability between sample means increases, i.e. sample means are farther apart.

Discussion and Analysis

For instance, say we were interested in studying if mean salaries for MLB, NFL and the NBA were the same. To answer this question we took a random sample of 30 players each for the three sports leagues. This gives us three levels, or sub-categories, for the overall variable Sports League. Since we want to compare means the question of interest is:

Ho: The means salaries for players across the three leagues are the equal

Ha: At least one of the mean salaries differs from the others (i.e. not all mean salaries are the same)

Source

DF

SS MS

F

P

Sport

2

202.7

101.4

6.08

0.03

Error

87

1450.7

16.7

 

 

Total

89

1652.8

Hypotheses Statements and Assumptions for One-Way ANOVA

The hypothesis test for analysis of variance for g populations:

Ho: µ1 = µ2 = ... = µg

Ha: not all µi (i = 1, ... g) are equal

Recall that when we compare the means of two populations for independent samples, we use a 2-sample t-test with pooled variance when the population variances can be assumed equal. For more than two populations, the test statistic is the ratio of between group sample variance and the within-group-sample variance. Under the null hypothesis, both quantities estimate the variance of the random error and thus the ratio should be close ...