Introduction to the t-Tests
Chapter Objectives
The reader will be able to:
Understand the concept of t test as a test of the difference between two sample means to determine statistical difference n Understand the factors affecting the probability level that the null
hypothesis is true n Understand the difference among independent samples, paired samples, and one sample t tests Researchers frequently need to determine the statistical significance of the difference between two sample means. Consider Example 1, which illustrates the need for making such a comparison. This chapter deals with how to compare two sample means for statistical significance. About a hundred years ago, a statistician named William Gosset developed the t test for exactly the situation described in Example 1 (i.e., to test the difference between two sample means to determine whether there is a significant difference between them or statistical significance).1 As a test of the null hypothesis, the t test yields a probabil- ity that a given null hypothesis is correct. As indicated in Chapter 20, when there is a low probability that it is correct—say, as low as .05 (5%) or less—researchers usually reject the null hypothesis. The computational procedures for conducting t tests are beyond the scope of this book. 2 However, the following material describes what makes the t test work. In other words, what leads the t test to yield a low probability that the null hypothesis is correct for a given pair of means? Here are the three basic factors, which interact with each other in determining the probability level:
Exercise for Chapter 21
Factual Questions
1. Example 1 mentions how many possible explanations for the 3-point difference?
2. What is the name of the hypothesis which states that the observed difference is due to sampling errors created by random sampling?
3. Which of the following statements is true (circle one)?
A. The t test is used to test the difference between two sample means to determine statistical significance.
B. The t test is used to test the difference between two population means to determine statistical significance.
4. If a t test yields a low probability, such as p < .05, what decision is usually made about the null hypothesis?
5. The larger the sample, the (circle one)
A. more likely the null hypothesis will be rejected.
B. less likely the null hypothesis will be rejected.
6. The smaller the observed difference between two means, the (circle one)
A. more likely the null hypothesis will be rejected.
B. less likely the null hypothesis will be rejected.
7. If there is no variation among members of a population, is it possible to have sampling errors when sampling from the population?
8. If participants are first paired before being randomly assigned to experimental and control groups, are the resulting data “independent” or “dependent”?
9. Which type of data tends to have less sampling error (circle one)?
A. Independent
B. Dependent
Notes
1. As indicated in Chapter 20, when a result is statistically significant, the null hypothesis is rejected.
2. See Appendix A for the computational procedures for t tests for reference.
3. In the types of studies we are considering, researchers do not know the variation in population.
However, the t test uses the standard deviations of the samples to estimate the variation of the population. In other words, the standard deviations of the samples provide the t test with an indication of the amount of variation in the populations from which the samples were drawn. Pyrczak, Fred, and Deborah M. Oh.
5. The formulas for the standard error of the difference between means are different for Independent Samples t Test and Dependent t Test. These along with the formula for the Standard error of mean are listed in Appendix A. = 0
