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 t-test with the following formula:

\[t =\frac{\bar{X}-\mu }{s / \sqrt{n}}\]

This corresponds to a right-tailed t-test. The t-statistics 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 right-tailed 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 p-value approach. The right-tailed p-value for this test is calculated as

\[p=\Pr \left( {{t}_{14}}>6.3376 \right)=0.000\]

Considering that the p-value 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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