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### Assumptions Underlying the t Test

#### Assumptions Underlying the Independent-Samples t Test

#### Assumptions Underlying the Paired-Samples t Test

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Level of measurement. The criterion variable should be assessed on an interval-or ratio-level of measurement. The predictor variable should be a nominal-level variable that includes just two categories (i.e., two groups).

Independent observations. A given observation should not be dependent on any other observation in either group. In an experiment, you normally achieve this by drawing a sample and randomly assigning each participant to only one of the two treatment conditions. This assumption would be violated if a given participant contributed scores on the criterion variable under both treatment conditions.

The independence assumption is also violated when one participant’s behavior influences another’s behavior within the same condition. For example, if participants are given experimental instructions in groups of five and are allowed to interact in the course of providing scores on the criterion variable, it is likely that their scores will not be independent. That is, each participant score is likely to be affected by the others in that group. In these situations, scores from participants constituting a given group of five should be averaged and these average scores should constitute the unit of analysis. None of the tests discussed in this text are robust against violations of the independence assumption.

Random sampling. Scores on the criterion variable should represent a random sample drawn from the populations of interest.

Normal distributions. The distribution of observed values for all continuous variables should approximate normal distributions.

Homogeneity of variance. To use the equal-variances t test, you should draw the samples from populations with equal variances on the criterion. If the null hypothesis of equal population variances is rejected, you should use the unequal-variances t test.

Level of measurement. The criterion variable should be assessed on an interval-or ratio-level of measurement. The predictor variable should be a nominal-level variable that includes just two categories.

Paired observations. A given observation appearing in one condition must be paired in some meaningful way with a corresponding observation appearing in the other condition. You can accomplish this by having each participant contribute one score under Condition 1 and a separate score under Condition 2. Observations could also be paired by using a matching procedure.

Independent observations. A given participant’s score in one condition should not be affected by any other participant’s score in either condition. It is, of course, acceptable for a given participant’s score in one condition to be dependent upon his or her own score in the other. This is another way of stating that it is acceptable for participants’ scores in Condition 1 to be correlated with their scores in Condition 2.

Random sampling. Participants contributing data should represent a random sample drawn from the populations of interest.

Normal distribution for difference scores. Differences in paired scores should be normally distributed. These difference scores are normally created by beginning with a given participant’s score on the dependent variable obtained under one treatment condition and subtracting from it that participant’s score on the dependent variable obtained under the other treatment condition. It is not necessary that the individual dependent variables be normally distributed so long as difference scores are normally distributed.

Homogeneity of variance. The populations represented by the two conditions should have equal variances on the criterion.