### Introduction

So far we have dealt with situations in which we either had a single sample drawn from a population, or paired data whose differences were considered essentially as a single sample.

In this Section we shall look at the situations occurring when we have two random samples each drawn from
**
independent
**
populations. While the basic ideas involved will essentially repeat those already met, you will find that the calculations involved are more complex than those already covered. However, you will find as before that calculations do follow particular routines. Note that in general the samples will be of different sizes. Cases involving samples of the same size, while included, should be regarded as special cases.

#### Prerequisites

- be familiar with the normal distribution, $t$ -distribution, $F$ -distribution and chi-squared distribution

#### Learning Outcomes

- apply the ideas of hypothesis testing to a range of problems underpinned by a substantial range of statistical distributions and involving two samples of different sizes

#### Contents

1 Tests concerning two samples1.1 Two independent populations each with an unknown variance

1.2 The $F$ -test