3.4 Summary

Hypothesis tests are a set of techniques to allow us to test characteristics of a population using data from a sample of this population.

Every hypothesis test involves the calculation of a test statistic.

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  This test statistic is compared to a sampling distribution, derived under the assumption that the null hypothesis is true.

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  This is done by either finding a critical value or a $p$-value.

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  If the test statistic is sufficiently extreme relative to the sampling distribution, we conclude that the null hypothesis cannot be true.

The one-sample $t$-test allows us to test whether or not the population mean $\mu$ is equal to a pre-specified value. The test statistic

is compared to the $t_{n-1}$-distribution.

Two sample $t$-tests allow us to compare the means of two populations. We first need to determine whether the data are

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  Paired (matched by a secondary variable)

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  Unpaired.

For unpaired data, first calculate the pooled sample variance,

and then the test statistic

For paired data, first calculate the differences and then compare the mean of these differences to zero using the test statistic,