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Here, we would like to discuss some common hypothesis testing problems. We assume that we have a random sample $X_1$,$X_2$,...,$X_n$ from a distribution and our goal is to make inference about the mean of the distribution $\mu$. We consider three hypothesis testing problems. The first one is a test to decide between the following hypotheses:

$\quad$ $H_0$: $\mu=\mu_0$,

$\quad$ $H_1$: $\mu \neq \mu_0$.

In this case, the null hypothesis is a simple hypothesis and the alternative hypothesis is a two-sided hypothesis (i.e., it includes both $\mu \lt \mu_0$ and $\mu>\mu_0$). We call this hypothesis test a two-sided test. The second and the third cases are one-sided tests. More specifically, the second case is

$\quad$ $H_0$: $\mu \leq \mu_0$,

$\quad$ $H_1$: $\mu > \mu_0$.

Here, both $H_0$ and $H_1$ are one-sided, so we call this test a one-sided test. The third case is very similar to the second case. More specifically, the third scenario is

$\quad$ $H_0$: $\mu \geq \mu_0$,

$\quad$ $H_1$: $\mu \lt \mu_0$.

In all of the three cases, we use the sample mean \begin{align}%\label{} \overline{X}=\frac{X_1+X_2+...+X_n}{n} \end{align} to define our statistic. In particular, if we know the variance of the $X_i$'s, $\textrm{Var}(X_i)=\sigma^2$, then we define our test statistic as the normalized sample mean (assuming $H_0$): \begin{align}%\label{} W(X_1,X_2, \cdots,X_n)=\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}. \end{align} If we do not know the variance of the $X_i$'s, we use \begin{align}%\label{} W(X_1,X_2, \cdots,X_n)=\frac{\overline{X}-\mu_0}{S / \sqrt{n}}, \end{align} where $S$ is the sample standard deviation, \begin{align}%\label{} S=\sqrt{\frac{1}{n-1} \sum_{k=1}^n (X_k-\overline{X})^2}=\sqrt{\frac{1}{n-1} \left(\sum_{k=1}^n X^2_k-n\overline{X}^2\right)}. \end{align} In any case, we will be able to find the distribution of $W$, and thus we can design our tests by calculating error probabilities. Let us start with the first case.

Two-sided Tests for the Mean:

Here, we are given a random sample $X_1$,$X_2$,...,$X_n$ from a distribution. Let $\mu=EX_i$. Our goal is to decide between

$\quad$ $H_0$: $\mu=\mu_0$,

$\quad$ $H_1$: $\mu \neq \mu_0$.

Example 8.22, which we saw previously is an instance of this case. If $H_0$ is true, we expect $\overline{X}$ to be close to $\mu_0$, and so we expect $W(X_1,X_2, \cdots,X_n)$ to be close to $0$ (see the definition of $W$ above).

Therefore, we can suggest the following test. Choose a threshold, and call it $c$. If $|W| \leq c$, accept $H_0$, and if $|W|>c$, accept $H_1$. How do we choose $c$? If $\alpha$ is the required significance level, we must have

\begin{align} P(\textrm{type I error}) &= P(\textrm{Reject }H_0 \; | \; H_0) \\ &= P(|W| > c \; | \; H_0) \leq \alpha. \end{align} Thus, we can choose $c$ such that $P(|W| > c \; | \; H_0) = \alpha$. Let us look at an example.

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Example
Let $X_1$,$X_2$,...,$X_n$ be a random sample from a $N(\mu,\sigma^2)$ distribution, where $\mu$ is unknown but $\sigma$ is known. Design a level $\alpha$ test to choose between

$\quad$ $H_0$: $\mu=\mu_0$,

$\quad$ $H_1$: $\mu \neq \mu_0$.

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Relation to Confidence Intervals: It is interesting to examine the above acceptance region. Here, we accept $H_0$ if \begin{align} \left|\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}} \right| \leq z_{\frac{\alpha}{2}}. \end{align} We can rewrite the above condition as \begin{align} \mu_0 \in \left[\overline{X}- z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} , \overline{X}+ z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}}\right]. \end{align} The above interval should look familiar to you. It is the $(1-\alpha)100\%$ confidence interval for $\mu_0$. This is not a coincidence as there is a general relationship between confidence interval problems and hypothesis testing problems.

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Example
For the above example (Example 8.24), find $\beta$, the probability of type II error, as a function of $\mu$.

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Unknown variance: The above results (Example 8.25) can be extended to the case when we do not know the variance using the $t$-distribution. More specifically, consider the following example.

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Example
Let $X_1$,$X_2$,...,$X_n$ be a random sample from a $N(\mu,\sigma^2)$ distribution, where $\mu$ and $\sigma$ are unknown. Design a level $\alpha$ test to choose between

$\quad$ $H_0$: $\mu=\mu_0$,

$\quad$ $H_1$: $\mu \neq \mu_0$.

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Example
The average adult male height in a certain country is $170$ cm. We suspect that the men in a certain city in that country might have a different average height due to some environmental factors. We pick a random sample of size $9$ from the adult males in the city and obtain the following values for their heights (in cm): \begin{equation} 176.2 \quad 157.9 \quad 160.1 \quad 180.9 \quad 165.1 \quad 167.2 \quad 162.9 \quad 155.7 \quad 166.2 \end{equation} Assume that the height distribution in this population is normally distributed. Here, we need to decide between

$\quad$ $H_0$: $\mu=170$,

$\quad$ $H_1$: $\mu \neq 170$.

Based on the observed data, is there enough evidence to reject $H_0$ at significance level $\alpha=0.05$?

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What if the sample is not from a normal distribution? In the case that $n$ is large, we can say that \begin{align}%\label{} W(X_1,X_2, \cdots,X_n)=\frac{\overline{X}-\mu_0}{S / \sqrt{n}} \end{align} is approximately standard normal. Therefore, we accept $H_0: \mu=\mu_0$ if \begin{align} \left|\frac{\overline{X}-\mu_0}{S / \sqrt{n}} \right| \leq z_{\frac{\alpha}{2}}, \end{align} and reject it otherwise (i.e., accept $H_1: \mu \neq \mu_0$).

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Let us summarize what we have obtained for the two-sided test for the mean.

Case	Test Statistic	Acceptance Region
$X_i \sim N(\mu, \sigma^2)$, $\sigma$ known	$W=\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}$	$|W| \leq z_{\frac{\alpha}{2}}$
$n$ large, $X_i$ non-normal	$W=\frac{\overline{X}-\mu_0}{S / \sqrt{n}}$	$|W| \leq z_{\frac{\alpha}{2}}$
$X_i \sim N(\mu, \sigma^2)$, $\sigma$ unknown	$W=\frac{\overline{X}-\mu_0}{S / \sqrt{n}}$	$|W| \leq t_{\frac{\alpha}{2},n-1}$

One-sided Tests for the Mean:

We can provide a similar analysis when we have a one-sided test. Let's show this by an example.

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Example
Let $X_1$,$X_2$,...,$X_n$ be a random sample from a $N(\mu,\sigma^2)$ distribution, where $\mu$ is unknown and $\sigma$ is known. Design a level $\alpha$ test to choose between

$\quad$ $H_0$: $\mu \leq \mu_0$,

$\quad$ $H_1$: $\mu > \mu_0$.

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The above analysis can be repeated for other cases. More generally, suppose that we are given a random sample $X_1$,$X_2$,...,$X_n$ from a distribution. Let $\mu=EX_i$. Our goal is to decide between

$\quad$ $H_0$: $\mu \leq \mu_0$,

$\quad$ $H_1$: $\mu > \mu_0$.

We define the test statistic as before, i.e., we define $W$ as \begin{align}%\label{} W(X_1,X_2, \cdots,X_n)=\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}, \end{align} if $\sigma=\sqrt{\textrm{Var}(X_i)}$ is known, and as \begin{align}%\label{} W(X_1,X_2, \cdots,X_n)=\frac{\overline{X}-\mu_0}{S / \sqrt{n}}, \end{align} if $\sigma$ is unknown. If $H_0$ is true (i.e., $\mu \leq \mu_0$), we expect that $\overline{X}$ (and thus $W$) to be relatively small, while if $H_1$ is true, we expect $\overline{X}$ (and thus $W$) to be larger. This suggests the following test: Choose a threshold $c$. If $W \leq c$, accept $H_0$, and if $W>c$, accept $H_1$. To choose $c$, note that \begin{align} P(\textrm{type I error}) &= P(\textrm{Reject }H_0 \; | \; H_0) \\ &= P(W > c \; | \; \mu \leq \mu_0)\\ &\leq P(W > c \; | \; \mu=\mu_0). \end{align} Note that the last inequality resulted because if we make $\mu$ larger, the probability of $W>c$ can only increase. In other words, we assumed the worst case scenario, i.e, $\mu=\mu_0$ for the probability of error. Thus, we can choose $c$ such that $P(W > c \; | \; \mu=\mu_0)=\alpha$. By doing this procedure, we obtain the acceptance regions reflected in Table 8.3.

Case	Test Statistic	Acceptance Region
$X_i \sim N(\mu, \sigma^2)$, $\sigma$ known	$W=\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}$	$W \leq z_{\alpha}$
$n$ large, $X_i$ non-normal	$W=\frac{\overline{X}-\mu_0}{S / \sqrt{n}}$	$W \leq z_{\alpha}$
$X_i \sim N(\mu, \sigma^2)$, $\sigma$ unknown	$W=\frac{\overline{X}-\mu_0}{S / \sqrt{n}}$	$W \leq t_{\alpha,n-1}$

Note that the tests mentioned in Table 8.3 remain valid if we replace the null hypothesis by $\mu=\mu_0$. The reason for this is that in choosing the threshold $c$, we assumed the worst case scenario, i.e, $\mu=\mu_0$. Finally, if we need to decide between

$\quad$ $H_0$: $\mu \geq \mu_0$,

$\quad$ $H_1$: $\mu \lt \mu_0$,

we can again repeat the above analysis and we obtain the acceptance regions reflected in Table 8.4.

Case	Test Statistic	Acceptance Region
$X_i \sim N(\mu, \sigma^2)$, $\sigma$ known	$W=\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}$	$W \geq -z_{\alpha}$
$n$ large, $X_i$ non-normal	$W=\frac{\overline{X}-\mu_0}{S / \sqrt{n}}$	$W \geq -z_{\alpha}$
$X_i \sim N(\mu, \sigma^2)$, $\sigma$ unknown	$W=\frac{\overline{X}-\mu_0}{S / \sqrt{n}}$	$W \geq -t_{\alpha,n-1}$