Null and Alternative Hypotheses

The best way to illustrate these concepts is by example. Suppose that we are interested in examining if a particular coin is biased. We start by establishing a null hypothesis, which represents the 'no effect' case, and therefore corresponds to the hypothesis that the coin is fair. That is we can define $H_{0} \equiv P($ heads) $=0.5$ (note that this is sort of an assertion that we will subsequently test, much like assuming an underlying value for some parameter of a population).

Our alternative hypothesis, on the other hand, is that the coin is biased, and therefore $H_{1} \equiv P($ heads $) \neq$ 0.5. Note that $H_{0}$ and $H_{1}$ share no outcomes, that is $H_{0} \cap H_{1}=\emptyset$.

The null hypothesis, $H_{0}$, represents the status quo - the 'no change', 'no effect' case.

The alternative hypothesis $\left(H_{1}\right)$, on the other hand, represents our suspicions about possible changes, differences or effects.

Examples of Null and Alternative Hypotheses

Reaction rates

Claim: New method alters the reaction rate $\theta$.

Null hypothesis: reaction rate is the same using both methods $H_{0}: \theta_{\text {old }}=\theta_{\text {new }}$

Alternative hypothesis: reaction rate is different $H_{1}: \theta_{\text {old }} \neq \theta_{\text {new }}$

2. Average male height

Claim: The average male height is $177.8 \mathrm{~cm}$

Null hypothesis: $H_{0}: T=177.8 \mathrm{~cm}$

Alternative hypothesis: $H_{1}: T \neq 177.8 \mathrm{~cm}$

3. Male vs. female height

Claim: The average male height is different from the average female height.

Null hypothesis: $H_{0}: T_{M}=T_{F}$

Alternative hypothesis: $H_{1}: T_{M} \neq T_{F}$

4. Women vs. men longevity

Claim: Women live longer than men, on average.

Null hypothesis: women live as long as men do $H_{0}: T_{\text {women }}=T_{\text {men }}$

Alternative hypothesis: women live longer than men $H_{1}: T_{\text {women }}>T_{\text {men }}$

Note: The last case is an example of a composite hypothesis, which illustrates the point that hypotheses need not be opposite (just mutually exclusive), but requires a slightly different statistical treatment than the other three cases (simple hypotheses). We discuss the peculiarities later in this chapter.