### Null and Alternative Hypotheses

#### Definition

The null hypothesis $H_0$ is the hypothesis that is the default position.

The alternative hypothesis $H_1$ (sometimes denoted $H_A$) is the hypothesis that suggests that sample observations are influenced by a non-random cause.

The wording of the alternative hypothesis is important as it tells us which type of test to use. Look for phrases such as “greater than” or “less than” to indicate we need a one tailed test, whereas an alternative hypothesis simply suggesting that the default position is wrong would require a two-tailed test.

#### Worked Examples 1

###### Worked Example

Set up the null and alternative hypotheses for a hypothesis test to decide whether or not a coin is biased towards heads.

###### Solution

Let $p$ be the probability that flipping a coin produces heads. To perform a hypothesis test to check if the coin is biased towards heads, the null and alternative hypotheses would be:

\begin{align} H_0 &: p = \frac{1}{2}\text{,} \\ H_1 &: p > \frac{1}{2}. \end{align}

Here we have a greater than sign indicating a one-tailed test will be used.

#### Worked Example 2

###### Worked Example

Set up a hypothesis test to see if the lifetime of an energy saving light bulb is $60$ days as the manufacturer claims.

###### Solution

So we have

$H_0$ : The lifetime of an energy-saving light bulb is $60$ days.
$H_1$ : The lifetime of an energy-saving light bulb is not $60$ days.

Note the difference here, in the first example we only have to check if $p$ is greater than $\frac{1}{2}$ whereas in the second example we need to check both greater than or less then. This means that a two-tailed test is required.

#### Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.