t-Test on Means

In this section, we refer to t-tests which are used to compare independent sample means. H0 implies that the two means in the population are equal: H0: mu1 - mu2 = 0 For matched-pairs t-tests, use the "Other t-Tests" option. Chose "one-tailed" or "two-tailed," depending on your hypothesis. We have four examples on this page: Two group t-test, equal group sizes, equal sigma Unequal group sizes Unequal sigma

Two Group t-Test, Equal Group Sizes, Equal Sigma

You have 2 populations A and B which you want to compare with respect to x. Assume that the random variable x is normally distributed with a standard deviation of s (sigma) in both populations. Assume further that the population means of x are muA and muB in population A and B, respectively. Thus,

H0:	muA - muB = 0
H1:	muA - muB = c, c <> 0.

Which total sample size do you need such that the probability of obtaining a t statistic equal to or larger than a critical value is alpha = .05 under H0 and 1-beta = .9 under H1? Assume that the difference in means between the groups postulated by your H1 is equal to one half of the standard deviation, thus d = 0.5 (e.g., muA = 10, muB = 12, sigma = 4).

Select:	Type of Power Analysis:	A priori
	Type of Test:	t-Test (means), two-tailed
		Accuracy mode calculation

Input:	Alpha:	.05
	Power (1-beta):	.9
	Effect size "d":	0.5	(To calculate the effect size from muA, muB, and sigma, simply click "Calc d", insert the means and the standard deviation, and click "Calc & Copy".)

Result:	Total sample size:	172
	Actual power:	0.9032
	Critical t:	t(170) = 1.9740
	Delta:	3.2787

Assume further that you do not have enough money to pay 172 subjects. However, 140 would seam feasible. Which critical t would still result in a "fair" test of your H1? We use a compromise power analysis to compute an optimum critical value for the test statistic which satisfies the ratio q := beta/alpha. This optimum critical value can be regarded as a rational compromise between the demands for a low a-risk and a large power level, given a fixed sample size.

Select:	Type of Power Analysis:	Compromise
	Type of Test:	t-Test (means), two-tailed
		Accuracy mode calculation

Input:	n1: n2:	70 70
	Effect size "d":	0.5
	Beta/alpha ratio:	2	(That is, we are willing to commit a beta error twice as large as our alpha error.)

Result:	alpha:	0.0670
	Power (1-beta):	0.8661
	Critical t:	t(138) = 1.8465
	Delta:	2.9580

Two Group t-Test, Unequal Group Sizes, Equal Sigma

We have done a study in which, for some reasons, the group sizes are not equal. In Group A we have 24 subjects, in Group B we have 33. What is the power of the t-Test comparing the means of both groups, and how much power have we lost due to the unequal group sizes?

Select:	Type of Power Analysis:	Post-hoc
	Type of Test:	t-Test (means), one-tailed (This time assume that we know the direction of the difference between the groups.)
		Accuracy mode calculation

Input:	Alpha:	.05
	Effect size "d":	0.8	(We expect "large" effects according to the effect size conventions of Cohen, 1977.)
	n1: n2:	24 33

Result:	Power (1-beta):	0.9032
	Critical t:	t(55) = 1.6730
	Delta:	2.9821

Two Group t-Test, Equal Group Sizes, Unequal Sigma

What do you do if sigmaA <> sigmaB? This is not normally a problem because the t-test is known to be quite robust, at least as long as the groups sizes are equal. Cohen (1977, p. 44) suggests to adjust sigma to sigma':

             ______________________
            /                      \
sigma'=    /  sigmaA2  + sigmaB2  
          /  _______________________
         /             
       \/              2
			

Simply use sigma' instead of sigma to calculate the effect size using the "Calc 'd'" option, then proceed as in the examples given above. A word of caution is in order, however: The power values computed by G*Power will only be approximations in this case. Computer simulation results on the appropriateness of this approximation are not yet available.

Please report suggestions for improvements to Axel Buchner, Franz Faul, or Edgar Erdfelder.