There are two types of t-tests.  A one sample t-test compares the mean of a sample to a known or expected number.  A two sample t-test compares the means of two samples.  The discussion of the One Sample t-test begins below.  Follow this link to go to a discussion of a two sample t-test.

One Sample t-test

A one-sample t-test is used to compare the mean of a sample to a known, or expected, number. This, however, is not simply a comparison of whether or not the two are different numerically, which is almost always true.  Instead, it is a comparison to determine whether or not a statistically significant difference exists between the two means.  Take, for example, the following situation.

Everyone has heard that the average person sleeps 8 hours per day. However, a researcher believes that her college's students sleep less. She asks a group of 12 randomly selected college students at her institution how much sleep they obtain on an average night. After obtaining their answers,  the researcher calculates the mean of the sample which consists of 12 data points. Is the difference between this sample mean and the hypothesized mean, which is 8 hours, statistically significant or is the difference just due to expected random variation within the population.   If the difference is too large to be attributed to random variation, then it is statistically significant. 

 To compare the mean of the data set to the hypothesized mean, the researcher would have to do a one-sample t-test. The statistical hypotheses are:

Null Hypothesis
HO : μpopulation = μexpected

Alternate Hypothesis
HA : μpopulation < μexpected 


HA : μpopulation≠ μexpected 


After the researcher conducts the one-sample t-test, she will compare the results of the t-test to determine whether or not a significant difference exists. She has two choices:

  1. Reject the null hypothesis and accept the alternate hypothesis, which means that the two means are different significantly for reasons other than that of normal random variation. This occurs when the  t-statistic > t-critical; or when p<alpha.
  1. Retain the null hypothesis, which means there is not sufficient evidence to conclude that the two means are significantly different. The observed difference is small enough that it could be due to random variation within the population. This occurs when the t-statistic < t-critical; or when p>alpha

In the previous menu, the Acid Rain activity and the Speed of Light activity use one-sample t-tests to analyze the data set. 

Copyright 2002 Central Virginia Governor's School for Science and Technology Lynchburg, VA