The purpose of the *t* -test is to make
inferences about single means, or inferences about two means or variances,
where sample sizes are small and/or the population distribution is
unknown. While not always used in practice—since the one-way Analysis of
Variance (ANOVA) is mathematically equivalent to the
*t*-test, and since most researchers attempt to gather
a reasonable number of samples to avoid Type II errors—understanding the
logic and outcomes of the *t* -test (and its
distribution) will make it much easier for you to follow ANOVA and more
sophisticated analytical techniques, especially where your sampling is
necessarily limited.

In Chapter 7, you
learned how to use the normal (or *Gaussian*)
distribution, which is a continuous probability distribution, to assist
in making inferences about a population. Recall that the known
mathematical properties of the distribution can be used to determine
probabilities of characteristics occurring within the population, even
when the population mean is unknown. Thus, hypothesis testing can be
carried out using limited sampling, and correct inferences drawn, if the
population is normally distributed. In many natural systems, populations
are normally distributed, but sometimes they are not, and thus, the
normal distribution cannot be used as a model.

However, if you have gathered enough samples, it may still be possible to use the properties of the normal distribution, since the sampling distribution of averages ...

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