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Different hypothesis tests make different assumptions about the distribution of the random variable being sampled in the data. These assumptions must be considered when choosing a test and when interpreting the results.

For example, the *z*-test (`ztest`) and the *t*-test
(`ttest`) both assume that the
data are independently sampled from a normal distribution. Statistics Toolbox™ functions
are available for testing this assumption, such as `chi2gof`, `jbtest`, `lillietest`, and `normplot`.

Both the *z*-test and the *t*-test
are relatively robust with respect to departures from this assumption,
so long as the sample size *n* is large enough. Both
tests compute a sample mean
,
which, by the Central Limit Theorem, has an approximately normal sampling
distribution with mean equal to the population mean *μ*,
regardless of the population distribution being sampled.

The difference between the *z*-test and the *t*-test
is in the assumption of the standard deviation *σ* of
the underlying normal distribution. A *z*-test assumes
that *σ* is known; a *t*-test
does not. As a result, a *t*-test must compute an
estimate *s* of the standard deviation from the sample.

Test statistics for the *z*-test and the *t*-test
are, respectively,

Under the null hypothesis that the population is distributed
with mean *μ*, the *z*-statistic
has a standard normal distribution, *N*(0,1). Under
the same null hypothesis, the *t*-statistic has Student's *t* distribution
with *n* – 1 degrees of freedom. For small
sample sizes, Student's *t* distribution is flatter
and wider than *N*(0,1), compensating for the decreased
confidence in the estimate *s*. As sample size increases,
however, Student's *t* distribution approaches the
standard normal distribution, and the two tests become essentially
equivalent.

Knowing the distribution of the test statistic under the null
hypothesis allows for accurate calculation of *p*-values.
Interpreting *p*-values in the context of the test
assumptions allows for critical analysis of test results.

Assumptions underlying Statistics Toolbox hypothesis tests are given in the reference pages for implementing functions.

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