Statistics in a Nutshell

Chapter 8. Thet-Test

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.

The t Distribution

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 ...