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Math

Unless otherwise stated, the material in this guide is from The Learning Centre at Centennial College.  Content has been adapted for the NWP Learning Commons in March 2022. This work is licensed under a Creative Commons BY 4.0 International License.

Hypothesis Testing Process

There are many different test statistics to choose from. It depends on what parameter you are testing (e.g., \(\mu,\sigma,p\)), what variables are given (is \(\sigma\) known?), and the distribution of the population (e.g., normally distributed). The following are some test statistics you will encounter for hypothesis testing with one sample.

Parameter Sampling Distribution Requirements Test Statistic
Proportion p Normal (z) \(np\geq 5\) and \(nq \geq 5\) \[z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}}\]
Mean \(\mu\) t

\(\sigma\) is not known and normally distributed population

or

\(\sigma\) not known and \(n\leq 30\)

\[t=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}\]
Mean \(\mu\) Normal (z)

\(\sigma\) is known and normally distributed population

or

\(\sigma\) known or \(n>30\)

\[z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\]
Standard deviation \(\sigma\) \(\chi^2\) Strict requirement: normally distributed population \[\chi^2=\frac{(n-1)s^2}{\sigma^2}\]

Example 1: 93 student course evaluations report an average rating of 3.91 with standard deviation 0.53. What test statistic should be used to test the hypothesis that the population student course evaluations has a mean equal to 4.00?

Solution: The given values are \(\bar{x}=3.91\), \(s=0.53\) and \(\mu=4.00\). \(\sigma\) is not known and \(n>30\) so the t-statistic should be used.

\begin{align} t&=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}\\ &=\frac{3.91-4.00}{\frac{0.53}{\sqrt{93}}}\\&=-1.63760\end{align}

Example 2: A study of of 19,136 people found that 29.2% of the people sleep-walked. Would a reporter be justified in stating that "fewer than 30% of adults have sleep-walked"? What test statistic should be used?

Solution: The given value is in proportions with \(\hat{p}=0.292\), \(p=0.30\), which means \(q=1-p=0.7\).

First, we have to meet the requirements \(np\geq 5\) and \(nq \geq 5\) to apply the test statistic. 

\(np=(19,136)(0.3)=5740.8\geq 5\) and \(nq=(19,136)(0.7)=13,395.2\geq 5\)

With the conditions satisfied, we can calculate the test statistic for proportions. 

\begin{align} z&=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}}\\ &=\frac{0.292-0.30}{\sqrt{\frac{(0.3)(0.7)}{19,136}}} \\ &=-2.41494\end{align}

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