The two-sample t test covered in this chapter is the independent-samples t test. The independent-samples t test is used when seeing if there is a difference between the means of two independent populations. For our example, let’s imagine following Dr. Villanova as he replicates part of a classic experiment about factors that influence how well one remembers information (Craik & Tulving, 1975).

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Dr. Villanova’s participants were 38 introductory psychology students, randomly assigned to two groups. He ended up with 18 in the control group (n₁) and 20 in the experimental group (n₂). The lowercase n is a new abbreviation that will be used to indicate the sample size for a specific group. Subscripts are used to distinguish the two groups. Here, one might say n₁ =18 and n₂ = 20 or n_Control = 18 and n_Experimental = 20. An uppercase N will still be used to indicate the total sample size. For this experiment, N = n₁ + n₂ = 18 + 20 = 38.

Each participant was tested individually. All participants were shown 20 words (e.g., giraffe, DOG, mirror), one at a time, and asked a question about each word. The control group was asked whether the word appeared in capital letters or not. The experimental group was asked whether the word would make sense in this sentence, “The passenger carried a __________ onto the bus.” The first question doesn’t require much thought, so the control group was called the “shallow” processing group. Answering the second question required more mental effort, so the experimental group was called the “deep” processing group. After all 20 words had been presented, the participants were asked to write down as many words as they could remember. This recall task was unexpected and the number of words recalled was the dependent variable. The shallow processing group recalled a mean of 3.50 words (s = 1.54 words), and the deep processing group a mean of 8.30 words (s = 2.74 words). Here is Dr. Villanova’s research question: Does deep processing lead to better recall than shallow processing?

Box-and-whisker plots are an excellent way to provide a visual comparison of two or more groups. Box-and-whisker plots are data-rich, providing information about central tendency and variability. In Figure 8.2, the number of words recalled is the dependent variable on the Y-axis and the two groups, shallow and deep, are on the X-axis. Both groups are in the same graph, making it easy to compare them on three things, the median, the interquartile range, and the range:

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It certainly appears as if deeper processing leads to better memory, and this was the result that Craik and Tulving found back in 1975. But, in order to see if he replicated their results, Dr. Villanova will need to complete a hypothesis test to see if the difference is a statistically significant one.

Step 1 Pick a Test. The first step in completing a hypothesis test is choosing the correct test. Here, Dr. Villanova is comparing the mean of one population to the mean of another population, so he will use a two-sample t test. The two sample sizes are different (n_Shallow = 18 and n_Deep = 20), which means they are independent samples (see Table 8.1). In addition, the cases in one sample aren’t paired with the cases in the other sample, another indicator of independent samples. As the samples are independent, his planned test is the independent-samples t test.

Step 2 Check the Assumptions. Several of the assumptions for the independent-samples t test, listed in Table 8.2, are familiar from the single-sample t test, but one is new.

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The first assumption, random samples, is one seen before. For the depth of processing study, the participants are intro psych students, specifically ones who volunteered to participate in this study. Dr. Villanova may have used random assignment to place the participants into a control group and an experimental group, but he doesn’t have a random sample of participants. Thus, this assumption was violated and this will have an impact on interpreting the results. However, such an assumption is robust, so he can proceed with the t test.

With participants being randomly assigned to groups, each case participating by him- or herself, and no one participating twice, the second assumption, that each participant’s responding was not influenced by any other participant, was not violated. Dr. Villanova was willing to assume that the third assumption, normality, is not violated. He’s willing to make this assumption because memory is a cognitive ability and most psychological variables like this are considered to be normally distributed. In addition, this assumption is robust as long as N > 30.¹

The final assumption is new, homogeneity of variance. This fourth assumption says that the amounts of variability in the two populations should be about equal. Dr. Villanova will assess this by comparing the two sample standard deviations. If the larger standard deviation is not more than twice the smaller standard deviation, then the amounts of variability are considered about equal (Bartz, 1999). For the depth of processing study, the larger standard deviation is 2.74, and the smaller standard deviation is 1.54. The larger standard deviation is not more than twice the smaller standard deviation, so the fourth assumption has not been violated. Dr. Villanova can proceed with the planned independent-samples t test.

Step 3 List the Hypotheses. Nondirectional, or two-tailed, hypotheses are more common than directional, one-tailed hypotheses, so let’s mention those first. Nondirectional hypotheses are used when there is no prediction about which group will have a higher score than the other. In this situation, the null hypothesis will be a negative statement about the two populations represented by the two samples. It will say that no difference exists between the mean of one population, μ₁, and the mean of the other population, μ₂.

¹ There are objective ways to assess the normality assumption, but they are beyond the scope of an introductory statistics book like this one. Those who go on to take more classes in statistics will learn these techniques and will be freed from having to assume that variables are normally distributed.

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This doesn’t mean that the two sample means, M₁ and M₂, will be exactly the same. But, the difference between the sample means should be small enough that it can be explained by sampling error if the two samples were drawn from the same population.

The alternative hypothesis will state that the two population means are different. When using a two-tailed test, the alternative hypothesis will just say that the two population means are different from each other; it won’t state the direction of the difference. The implication of the alternative hypothesis is that the two sample means will be different enough that sampling error is not a likely explanation for the difference.

Written mathematically and using the names of the conditions, the hypotheses are

H₀: μ_Shallow = μ_Deep

H₁: μ_Shallow ≠ μ_Deep

Hypotheses like these will always be the null and alternative hypotheses used for a two-tailed independent-samples t test.

Dr. Villanova, however, has directional hypotheses and is doing a one-tailed test. Why is he doing a one-tailed test? Because the original study by Craik and Tulving in 1975 found that deep processing worked better than shallow processing, and Dr. Villanova expects that to be the case in his replication. He has made a prediction about the direction of the results in advance of collecting data, so he can do a one-tailed test.

One-tailed tests require a little more thought on the experimenter’s part in formulating the hypotheses. With one-tailed tests, it is easier to state the alternative hypothesis first. Dr. Villanova’s theory is that deep processing leads to better recall and the alternative hypothesis should reflect this: H₁: μ_Deep > μ_Shallow.

Once the alternative hypothesis is stated, the null hypothesis is formed by making sure that the two hypotheses are all-inclusive and mutually exclusive. If the alternative hypothesis says that deep processing is better than shallow, then the null hypothesis has to say that shallow processing is as good as, or better than, deep processing. The null hypothesis for the one-tailed independent-samples t test would be H₀: μ_Deep ≤ μ_Shallow.

Dr. Villanova, then, would state his hypotheses as

H₀: μ_Deep ≤ μ_Shallow

H₁: μ_Deep ≤ μ_Shallow

Step 4 Set the Decision Rule. The critical value of t, t_cv, is the border between the rare and common zone for the sampling distribution of t. t_cv is used to set the decision rule and decide whether to reject the null hypothesis for an independent-samples t test.

To find the critical value of t, Dr. Villanova will use Appendix Table 3. To use this table of critical values of t, he needs to know three things: (1) whether he is doing a one-tailed test or a two-tailed test, (2) what alpha level he wants to use, and (3) how many degrees of freedom the test has.

Dr. Villanova predicted in advance of collecting the data that the deep processing participants would do better than the shallow processing participants. And, because he had directional hypotheses, he has a one-tailed test. So, he will be looking at one-tailed values of t_cv.

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By convention, most researchers are willing to have a 5% chance of making a Type I error and set alpha at .05. (Type I error occurs when one erroneously rejects the null hypothesis.) As Dr. Villanova is replicating previous work, he wants to make sure that if he rejects the null hypothesis, it should have been rejected. He wants to have only a 1% chance of Type I error, so he’s setting α = .01.

Finally, the degrees of freedom need to be determined. The formula for calculating degrees of freedom (df) for an independent-samples t test is given in Equation 8.1.

For the depth of processing study, Dr. Villanova calculates df = 38 – 2 = 36. Looking in Appendix Table 3 at the row where df = 36 and under the column where α = .01, one-tailed, we find t_cv = 2.434.

The t distribution is symmetric, so this value could be either –2.434 or +2.434. As Dr. Villanova is doing a one-tailed test, he needs to decide whether his critical value of t will be a positive or a negative number. To do so requires knowing: (1) which mean will be subtracted from which in the numerator of Equation 8.1 when it is used to find the t value; and (2) what the sign of the difference should be. The expected sign of the difference is the sign to be associated with the critical value of t for a one-tailed test.

Dr. Villanova has decided that he’ll subtract the shallow processing group’s mean from the deep processing group’s mean. That is, it will be M_Deep – M_Shallow. If his theory is correct, the deep processing group will remember more words than the shallow group, and the difference will have a positive sign. This means that the critical value of t will be positive: 2.434. This value separates the rare zone of the sampling distribution of t from the common zone. The decision rule is shown in Figure 8.3, with the rare and common zones labeled. Here is how Dr. Villanova states the decision rule for his one-tailed test:

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How would things be different if Dr. Villanova were doing a two-tailed test and had set alpha at .05? The degrees of freedom would still be 36, but t_cv would be 2.028 and the decision rule would be:

This critical value of t, 2.028, is closer to zero, the midpoint of the t distribution, than is the critical value of t that Dr. Villanova is actually using, 2.434. This means it would be easier to reject the null hypothesis if Dr. Villanova were using the two-tailed test with α = .05 than with the one-tailed test set at .01 that he is actually using because the rare zone would be larger. He wanted to reduce the likelihood of Type I error, so he has achieved his goal. Table 8.3 summarizes the decision rules for one-tailed and two-tailed two-sample t tests.

Step 5 Calculate the Test Statistic. It is now time for Dr. Villanova to calculate the test statistic, t. In the last chapter, it was pointed out that the formula for a single-sample t test, , was similar to the z score formula, . Though it won’t look like it by the time we see it, the independent-samples t test formula is also like a z score formula. The numerator is a deviation score, the deviation of the difference between the two sample means from the difference between the two population means: (M₁ – M₂) – (μ₁ – μ₂). Because hypothesis testing proceeds under the assumption that the null hypothesis is true, if we assume μ₁ = μ₂, we can simplify the numerator to M₁ – M₂.

The denominator is a standard deviation, the standard deviation of the sampling distribution of the difference scores. Standard deviations of sampling distributions are called standard errors and this one, called the standard error of the difference, is abbreviated . Calculating the standard error of the difference proceeds in two steps. First we need to calculate variance for the two samples combined, or pooled, and then we need to adjust this pooled variance to turn it into a standard error. Equation 8.2 gives the formula for calculating the pooled variance.

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This equation says the pooled variance is calculated by multiplying each sample variance by 1 less than the number of cases that are in its sample and adding together these products. That sum is then divided by 2 less than the total number of cases, which is the same as the degrees of freedom.

Let’s follow Dr. Villanova as he plugs in the values from his depth of processing study into Equation 8.2. Remember, the shallow processing group had 18 cases, with a standard deviation of 1.54; the sample size and standard deviation for the deep processing group were 20 and 2.74, respectively. The equation calls for variances, not standard deviations, but a variance is simply a squared standard deviation, so we are OK. And the equation calls for the degrees of freedom, which we calculated in the previous step as N – 2 = 38 – 2 = 36:

The pooled variance is 5.08. We can now use the pooled variance to find the standard error of the mean by using Equation 8.3.

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This formula says that to find the standard error of the difference, one finds the quotient of the total sample size divided by the two individual sample sizes multiplied together. This quotient is multiplied by the pooled variance. Finally, in a step that is often overlooked, the square root of the product is found. In essence, this is similar to what was done in calculating the standard error of the mean, where the standard deviation was divided by the square root of N.

Let’s do the math for the shallow vs. deep processing study:

At this point, the hardest part of calculating an independent-samples t test is over and Dr. Villanova knows that the standard error of the mean is 0.73. All that is left to do is use the formula in Equation 8.4 to find t.

Equation 8.4 says that an independent-samples t value is calculated by dividing the numerator, the difference between the two sample means, by the denominator, the standard error of the difference. Dr. Villanova, because he is doing a one-tailed test, has already decided that he will be subtracting the shallow processing group’s mean, 3.50, from the deep processing group’s mean, 8.30:

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The test statistic t that Dr. Villanova calculated for his independent-samples t test is 6.58 and Step 5 of the hypothesis test is over. We’ll follow Dr. Villanova as he covers Step 6, interpretation, after working through the first five steps with another example.