One-Way Analysis of Variance (ANOVA) Example Problem

Introduction

Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality of two

or more population (or treatment) means by examining the variances of samples that are taken.

ANOVA allows one to determine whether the differences between the samples are simply due to

random error (sampling errors) or whether there are systematic treatment effects that causes the

mean in one group to differ from the mean in another.

Most of the time ANOVA is used to compare the equality of three or more means, however

when the means from two samples are compared using ANOVA it is equivalent to using a t-test

to compare the means of independent samples.

ANOVA is based on comparing the variance (or variation) between the data samples to variation

within each particular sample. If the between variation is much larger than the within variation,

the means of different samples will not be equal. If the between and within variations are

approximately the same size, then there will be no significant difference between sample means.

Assumptions of ANOVA:

(i) All populations involved follow a normal distribution.

(ii) All populations have the same variance (or standard deviation).

(iii) The samples are randomly selected and independent of one another.

Since ANOVA assumes the populations involved follow a normal distribution, ANOVA falls

into a category of hypothesis tests known as parametric tests. If the populations involved did not

follow a normal distribution, an ANOVA test could not be used to examine the equality of the

sample means. Instead, one would have to use a non-parametric test (or distribution-free test),

which is a more general form of hypothesis testing that does not rely on distributional

assumptions.

Example

Consider this example:

Suppose the National Transportation Safety Board (NTSB) wants to examine the safety of

compact cars, midsize cars, and full-size cars. It...