### Introduction

Problems in engineering often involve the exploration of the relationships between values taken by a variable under different conditions.
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HELM booklet
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41 introduced hypothesis testing which enables us to compare two population means using hypotheses of the general form

${H}_{0}:{\mu}_{1}={\mu}_{2}$

${H}_{1}:{\mu}_{1}\ne {\mu}_{2}$

or, in the case of more than two populations,

${H}_{0}:\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\mu}_{1}={\mu}_{2}={\mu}_{3}=\dots ={\mu}_{k}$

${H}_{1}:\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{H}_{0}$ is not true

If we are comparing more than two population means, using the type of hypothesis testing referred to above gets very clumsy and very time consuming. As you will see, the statistical technique called Analysis of Variance (ANOVA) enables us to compare several populations
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simultaneously
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. We might, for example need to compare the shear strengths of five different adhesives or the surface toughness of six samples of steel which have received different surface hardening treatments.

#### Prerequisites

- be familiar with the general techniques of hypothesis testing
- be familiar with the $F$ -distribution

#### Learning Outcomes

- describe what is meant by the term one-way ANOVA.
- perform one-way ANOVA calculations.
- interpret the results of one-way ANOVA calculations

#### Contents

1 One-way ANOVA1.1 Example of variance in data

1.2 Using the $F$ -test

1.3 ANOVA tables

1.4 Unequal sample sizes