Statistics Tutorials: How to Solve Hypothesis Testing Problems
One common type of problem you will find in Basic Statistics homework is the type of problem that involves using sample data to test a hypothesis.
A hypothesis is a statement about a population parameter. This is, it is a claim that we make about a certain population parameter, such as the population mean, or the population standard deviation.
For example, an engineer from a car manufacturer may claim that the population mean gas mileage of a new car model is 25 mpg. That would be an hypothesis. Or for example, a political polls researcher may claim that the voting share of certain candidate is 53%. That would be another hypothesis, about the true proportion of voters who support that certain candidate.
Consider the following example: A psychologist claims that the mean IQ scores of statistics instructors is greater than 100. She collects sample data from 15 statistics instructors and she finds that \(\bar{X}=118\) and s = 11. The sample data appear to come from a normally distributed population with unknown \(\mu\) and \(\sigma\).
Let us solve this problem:
Notice that we want to test the following null and alternative hypotheses
\[\begin{align}{{H}_{0}}:\mu {\le} {100}\, \\ {{H}_{A}}:\mu {>} {100} \\ \end{align}\]
Considering that the population standard deviation \(\sigma\) is not provided, we have to use a ttest with the following formula:
\[t =\frac{\bar{X}\mu }{s / \sqrt{n}}\]
This corresponds to a righttailed ttest. The tstatistics is given by the following formula:
\[t=\frac{\bar{X}\mu }{s /\sqrt{n}}=\frac{{118}100}{11/\sqrt{15}}={6.3376}\]
The critical value for \(\alpha = 0.05\) and for \(df = n 1 = 15 1 = 14\) degrees of freedom for this righttailed test is \(t_{c} = 1.761\). The rejection region is given by
\[R = \left\{ t:\,\,\,t>{ 1.761 } \right\}\]
Since \(t = 6.3376 {>} t_c = 1.761\), then we reject the null hypothesis H_{0}.
Alternatively, we can use the pvalue approach. The righttailed pvalue for this test is calculated as
\[p=\Pr \left( {{t}_{14}}>6.3376 \right)=0.000\]
Considering that the pvalue is such that \(p = 0.000 {<} 0.05\), we reject the null hypothesis H_{0}.
Hence, we have enough evidence to support the claim that the mean IQ scores of statistics instructors is greater than 100.

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