t-Tests

Abstract

The t-test is used for the mean of a normal distribution with estimated standard deviation s, or for comparing the means of two normal distributions.

We will look at several datasets, graphically and numerically, and focus on two types of questions.

Testing:   We have null (H ₀) and alternative (H ₁) hypotheses about the true value μ of the mean of the population from which the data was drawn. We wish to test whether there is enough evidence to reject the null hypothesis. The possible answers to the test are “reject the null hypothesis” or “do not reject the null hypothesis.” The possible answers do not contain any numbers.

The null hypothesis is a statement about the world. It might be a true statement. It might be a false statement. The phrase “reject the null hypothesis” means the evidence from the data suggests that the null hypothesis is a false statement.

Estimation:   We have some data, and we wish to estimate the location of the population mean μ. We estimate the location with a confidence interval with a specified confidence level. Frequently, the level is 95%. The answer is an interval, a set of two numbers. The interval is written in the form

$$95\%{\rm CI}(\mu)=(L,U)$$

((6.1))

where the numbers L and U stand for “lower bound” and “upper bound,” respectively. The interval is written as a set of parentheses with the smaller number on the left, the larger number on the right, and a comma separating them.