Colorado State University — Pueblo; Fall 2016
Math 156, Introduction to Statistics
Summary of Hypothesis Testing

We shall do hypothesis testing in essentially the same way in all cases we can handle. This page outlines that common approach and emphasizes the features that all solutions to hypothesis testing must contain.

1. Formalize the problem: Starting with a description of the issue to be investigated, figure out — and clearly explain in your work — the following
   1. the population under investigation
   2. the variable of interest for that population
   3. the population parameter of interest about that RV [which can only be, for us in this class, either the population mean $\mu$ or population proportion $p$]
   4. the null hypothesis $H_0$ for this test, which is of the form $H_0:\ \ $param = param₀ where param is the parameter of interest and param₀ is a specific number
   5. the alternative hypothesis $H_a$ for this test, which is one of the following: $H_a:\ \ $param < param₀ $H_a:\ \ $param > param₀ $H_a:\ \ $param ≠ param₀ where param is that parameter of interest and param₀ is that same specific number as in $H_0$.
2. Collect sample data: Now, collect data for a SRS of size $n$. From this, compute the sample mean $\overline{X}$, if the parameter param of interest is $\mu$, or the sample proportion $\widehat{p}$, if param is $p$.
3. Compute the test statistic: Compute:
   1. if param is $\mu$ and the population standard deviation $\sigma$ is known, then the test statistic is $$z=\frac{\overline{X}-\mu_0}{\sigma/\sqrt{n}}$$ where $\mu_0$=param₀
   2. if param is $\mu$ and the $\sigma$ is unknown, then the test statistic is $$t=\frac{\overline{X}-\mu_0}{s/\sqrt{n}}$$ where $\mu_0$=param₀ and $s$ is the sample standard deviation
   3. if param is $p$ then the test statistic is $$z=\frac{\widehat{p}-p_0}{\sqrt{p_0(1-p_0)/n}}\ .$$ where $p_0$=param₀
4. Compute the $p$-value:
   1. if param is $\mu$ and the population standard deviation $\sigma$ is known, then
      1. if $H_a$ is param < param₀ the the $p$-value is $p=P(Z<z)$ where $Z$ is the standard Normal RV
      2. if $H_a$ is param > param₀ the the $p$-value is $p=P(Z>z)$ where $Z$ is the standard Normal RV
      3. if $H_a$ is param ≠ param₀ the the $p$-value is $p=2P(Z>|z|)$ where $Z$ is the standard Normal RV
   2. if param is $\mu$ and the population standard deviation is unknown, then you can use an online $t$-distribution calculator such as this one to help in the following:
      1. if $H_a$ is param < param₀ the the $p$-value is $p=P(T<t)$ where $T$ is Student's $t$-distribution with $df=n-1$ degrees of freedom.
      2. if $H_a$ is param > param₀ the the $p$-value is $p=P(T>t)$ where $T$ is Student's $t$-distribution with $df=n-1$ degrees of freedom.
      3. if $H_a$ is param ≠ param₀ the the $p$-value is $p=2P(T>|t|)$ where $T$ is Student's $t$-distribution with $df=n-1$ degrees of freedom.
   3. if param is the population proportion, then
      1. if $H_a$ is param < param₀ the the $p$-value is $p=P(Z<z)$ where $Z$ is the standard Normal RV
      2. if $H_a$ is param > param₀ the the $p$-value is $p=P(Z>z)$ where $Z$ is the standard Normal RV
      3. if $H_a$ is param ≠ param₀ the the $p$-value is $p=2P(Z>|z|)$ where $Z$ is the standard Normal RV
5. Draw the statistical conclusion: If the $p$-value is sufficiently small (as determined by the context), then you say We reject the null hypothesis; if the $p$-value is too large, you say We fail reject the null hypothesis. If a significance level $\alpha$ is given, then "sufficiently small" for the $p$-value means "less than $\alpha$."
6. Restate the conclusion in the original terms: Finish by restating what it means to reject $H_0$ or to fail to reject $H_0$, depending upon whichever was just chosen, in the terminology of the original problem statement. Typically, this looks like one of the following
   1. "There is strong statistical evidence to reject the null hypothesis [stated without statistical terminology or notation] in favor of the alternative hypothesis [stated without statistical terminology or notation] ($p=$value)." if the $p$-value $p$ is small.
   2. "There is insufficient statistical evidence to reject the null hypothesis [stated without statistical terminology or notation] in favor of the alternative hypothesis [stated without statistical terminology or notation] ($p=$value)." if the $p$-value $p$ is large.
   3. "We reject the null hypothesis [stated without statistical terminology or notation] in favor of the alternative hypothesis [stated without statistical terminology or notation] at significance level $\alpha$." if the $p$-value satisfies $p<\alpha$.
   4. "We fail to reject the null hypothesis [stated without statistical terminology or notation] in favor of the alternative hypothesis [stated without statistical terminology or notation] at significance level $\alpha$." if the $p$-value satisfies $p≥\alpha$.