Independent Samples t-test: Purpose, Hypotheses, and Assumptions

Chapter 8. Testing for Differences in Mean and Proportion: Independent Samples t-test

Independent Samples t-test: Purpose, Hypotheses, and Assumptions

In this chapter, we will consider research designs in which a continuous variable is measured once on two separate simple random samples. The measurements of the first and second sample will be denoted by #X_1# and #X_2#, respectively.

To conduct inferences about the difference between the means of two independent populations, an independent samples #t#-test should be used.
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Independent Samples t-test: Hypotheses

The independent samples #\boldsymbol{t}#-test is used to test hypotheses about the difference between two population means #\mu_1 - \mu_2#.

Specifically, the test is used to determine whether or not it is plausible that #\mu_1-\mu_2# differs from some value #\Delta#. In most situations #\Delta=0#, so we will only present this specific setting.

The hypotheses of an independent samples #t#-test are:

Two-tailed#^1#	Left-tailed	Right-tailed
#H_0: \mu_1 - \mu_2 = 0# #H_a: \mu_1 - \mu_2 \neq 0#	#H_0:\mu_1 - \mu_2 \geq 0# #H_a: \mu_1 - \mu_2 \lt 0#	#H_0: \mu_1 - \mu_2 \leq 0# #H_a: \mu_1 - \mu_2 \gt 0#

Note that #\mu_1 - \mu_2 = 0# means the same thing as #\mu_1 = \mu_2#.

Assumptions of the Independent Samples t-test

The following assumptions are required to hold in order for an Independent Samples #t#-test to produce valid results:

Random sampling is used to draw the samples.
Independence of observations, meaning:
- No individual can be part of both samples.
- No individual in either sample can influence individuals in the same sample.
- No individual in either sample can influence individuals in the other sample.
The sampling distribution of the difference between the two sample means is approximately normal. This condition of normality is met under the following circumstances:
- If either of the samples is small #(n_1 \lt 30 \text{ or } n_2 \lt 30)#, it is required that the measured variable is normally-distributed on each population:
  \[X_1 \sim N(\mu_1, \sigma_1) \phantom{000000} X_2 \sim N(\mu_2, \sigma_2)\]
- If both samples are sufficiently large #(n_1 \geq 30# and #n_2 \geq 30)#, the Central Limit Theorem can be invoked and this requirement is not needed.