t-test for One Group

So far we have looked at only counts or frequencies. These types of statistics are useful, but limited in what they can tell us. For example, suppose that a client believes that income is an important variable in determining who will buy her product. She believes that people who have incomes over $50,000 are her primary niche of focus. A sample of her product users was taken and found to have an average income of $48,400. Income is a continuous variable (ratio data). Counts mean nothing here. Is your client wrong?

Many statistical tests were named after the person that first proposed them. For example, Fisher developed the F-test. The t-test was developed in 1908 by W.S. Gossett and should be the g-test, but Gossett used the pen name “student” when reporting his statistic. “S” is already used for the standard deviation, so the test became known as the “t” test (the second letter in “student.”) Gossett’s interest in statistics originated in problems associated with the control of the quality of materials used to make beer.

Essentially the value of a t-test is simply the number of standard errors a sample mean is away from the hypothesized mean.

Where SE is the sampling error, or the standard deviation of a distribution of means:

SE = SD of the population divided by the square root of the sample size, or:

Since the standard deviation of the population is not known, it has to be estimated by the sample itself, hence the need for a t-test. Each t-test has a degree of freedom, it is simply n –1.

What to do:

1. Ask 20 students what their GPA is.

2. Input those numbers into SPSS under a variable called “gpa.”

3. Test the hypothesis that the average GPA = 3.00 for your sample.

Analyze

Compare mean

One-sample T Test

(put gpa in box)

(set “test value” to 3.00)

4. Cut and paste result to report

5. Explain the table you printed, and explain why you rejected or did not reject H₀.