In the formula t m1 m2 sdifference sdifference is

270

✪ The Distribution of Differences Between Means 271

✪ Hypothesis Testing with a t Test for Independent Means 278

✪ Assumptions of the t Test for Independent Means 286

✪ Effect Size and Power for the t Test for Independent Means 288

✪ Review and Comparison of the Three Kinds of t Tests 290

✪ Controversy: The Problem of Too Many t Tests 291

In the previous chapter, you learned how to use the t test for dependent means tocompare two sets of scores from a single group of people (such as the same menmeasured on communication quality before and after premarital counseling). In this chapter, you learn how to compare two sets of scores, one from each of

two entirely separate groups of people. This is a very common situation in psychol- ogy research. For example, a study may compare the scores from individuals in an experimental group and individuals in a control group (or from a group of men and a group of women). This is a t test situation because you don’t know the population variances (so they must be estimated). The scores of the two groups are indepen- dent of each other; so the test you learn in this chapter is called a t test for inde- pendent means.

✪ The t Test for Independent Means in Research Articles 292

✪ Advanced Topic: Power for the t Test for Independent Means When Sample Sizes Are Not Equal 293

✪ Summary 294

✪ Key Terms 295

✪ Example Worked-Out Problems 295

✪ Practice Problems 298

✪ Using SPSS 305

✪ Chapter Notes 309

The t Test for Independent Means

Chapter Outline

CHAPTER 8

t test for independent means hypothesis-testing procedure in which there are two separate groups of people tested and in which the popula- tion variance is not known.

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The t Test for Independent Means 271

Let’s consider an example. A team of researchers is interested in the effect on physical health of writing about thoughts and feelings associated with traumatic life events. This kind of writing is called expressive writing. Suppose the researchers recruit undergraduate students to take part in a study and randomly assign them to be in an expressive writing group or a control group. Students in the expressive writ- ing group are instructed to write four 20-minute essays over four consecutive days about their most traumatic life experiences. Students in the control group write four 20-minute essays over four consecutive days describing their plans for that day. One month later, the researchers ask the students to rate their overall level of physi- cal health (on a scale from to ). Since the expressive writing and the control group contain different students, a t test for inde- pendent means is the appropriate test of the effect of expressive writing on physical health. We will return to this example later in the chapter. But first, you will learn about the logic of the t test for independent means, which involves learning about a new kind of distribution (called the distribution of differences between means).

The Distribution of Differences Between Means In the previous chapter, you learned the logic and figuring for the t test for dependent means. In that chapter, the same group of people each had two scores; that is, you had a pair of scores for each person. This allowed you to figure a difference score for each person. You then carried out the hypothesis-testing procedure using these dif- ference scores. The comparison distribution you used for this hypothesis testing was a distribution of means of difference scores.

In the situation you face in this chapter, the scores in one group are for different people than the scores in the other group. So you don’t have any pairs of scores, as you did when the same group of people each had two scores. Thus, it wouldn’t make sense to create difference scores, and you can’t use difference scores for the hypothesis- testing procedure in this chapter. Instead, when the scores in one group are for differ- ent people than the scores in the other group, what you can compare is the mean of one group to the mean of the other group.

So the t test for independent means focuses on the difference between the means of the two groups. The hypothesis-testing procedure, however, for the most part works just like the hypothesis-testing procedures you have already learned. Since the focus is now on the difference between means, the comparison distribution is a distribution of differences between means.

A distribution of differences between means is, in a sense, two steps removed from the populations of individuals: First, there is a distribution of means from each population of individuals; second, there is a distribution of differences between pairs of means, one of each pair from each of these distributions of means.

Think of this distribution of differences between means as being built up as follows: (a) randomly select one mean from the distribution of means for the first group’s population, (b) randomly select one mean from the distribution of means for the second group’s population, and (c) subtract. (That is, take the mean from the first distri- bution of means and subtract the mean from the second distribution of means.) This gives a difference score between the two selected means. Then repeat the process. This creates a second difference score, a difference between the two newly selected means. Repeating this process a large number of times creates a distribution of differences be- tween means. You would never actually create a distribution of differences between means using this lengthy method. But it shows clearly what makes up the distribution.

100 = perfect health0 = very poor health

distribution of differences between means distribution of differences between means of pairs of samples such that, for each pair of means, one is from one population and the other is from a second population; the comparison distribution in a t test for independent means.

T I P F O R S U C C E S S The comparison distributions for the t test for dependent means and the t test for independent means have similar names: a distribution of means of difference scores, and a distribution of differences be- tween means, respectively. Thus, it can be easy to confuse these com- parison distributions. To remember which is which, think of the logic of each t test. The t test for depen- dent means involves difference scores. So, its comparison distrib- ution is a distribution of means of difference scores. The t test for independent means involves differences between means. Thus, its comparison distribution is a dis- tribution of differences between means.

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272 Chapter 8

Distributions of means

Populations

Samples

Distribution of differences between

means

Figure 8–1 Diagram of the logic of a distribution of differences between means.

The Logic Figure 8–1 shows the entire logical construction for a distribution of differences between means. At the top are the two population distributions. We do not know the characteristics of these population distributions, but we do know that if the null hy- pothesis is true, the two population means are the same. That is, the null hypothesis is that . We also can estimate the variance of these populations based on the sample information (these estimated variances will be and ).

Below each population distribution is the distribution of means for that popula- tion. Using the estimated population variance and knowing the size of each sample, you can figure the variance of each distribution of means in the usual way. (It is the estimated variance of its parent population divided by the size of the sample from that population that is being studied.)

Below these two distributions of means, and built from them, is the crucial distribution of differences between means. This distribution’s variance is ultimately based on estimated population variances. Thus, we can think of it as a t distribution. The goal of a t test for independent means is to decide whether the difference be- tween the means of your two actual samples is a more extreme difference than the cutoff difference on this distribution of differences between means. The two actual samples are shown (as histograms) at the bottom.

Remember, this whole procedure is really a kind of complicated castle in the air. It exists only in our minds to help us make decisions based on the results of an actual ex- periment. The only concrete reality in all of this is the actual scores in the two samples. You estimate the population variances from these sample scores. The variances of the two distributions of means are based entirely on these estimated population variances (and the sample sizes). And, as you will see shortly, the characteristics of the distribu- tion of differences between means are based on these two distributions of means.

Still, the procedure is a powerful one. It has the power of mathematics and logic behind it. It helps you develop general knowledge based on the specifics of a particu- lar study.

With this overview of the basic logic, we now turn to six key details: (1) the mean of the distribution of differences between means, (2) the estimated population variance, (3) the variance of the two distributions of means, (4) the variance and standard deviation of the distribution of differences between means, (5) the shape of the distribution of differences between means, and (6) the t score for the difference between the two means being compared.

S22S 2 1

�1 = �2

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The t Test for Independent Means 273

Mean of the Distribution of Differences Between Means In a t test for independent means, you are considering two populations: for example, one population from which an experimental group is taken and one population from which a control group is taken. In practice, you don’t know the mean of either popu- lation. You do know that if the null hypothesis is true, these two populations have equal means. Also, if these two populations have equal means, the two distributions of means have equal means. (This is because each distribution of means has the same mean as its parent population of individuals.) Finally, if you take random samples from two distributions with equal means, the differences between the means of these random samples, in the long run, balance out to 0. The result of all this is the follow- ing: whatever the specifics of the study, you know that, if the null hypothesis is true, the distribution of differences between means has a mean of 0.

Estimating the Population Variance In Chapter 7, you learned to estimate the population variance from the scores in your sample. It is the sum of squared deviation scores divided by the degrees of freedom (the number in the sample minus 1). To do a t test for independent means, it has to be reasonable to assume that the populations the two samples come from have the same variance (which, in statistical terms, is called homogeneity of variance). (If the null hypothesis is true, they also have the same mean. However, whether or not the null hypothesis is true, you must be able to assume that the two populations have the same variance.) Therefore, when you estimate the variance from the scores in either sample, you are getting two separate estimates of what should be the same number. In practice, the two estimates will almost never be exactly identical. Since they are both supposed to be estimating the same thing, the best solution is to average the two estimates to get the best single overall estimate. This is called the pooled estimate of the population variance ( ).

In making this average, however, you also have to take into account that the two samples may not be the same size. If one sample is larger than the other, the estimate it provides is likely to be more accurate (because it is based on more information). If both samples are exactly the same size, you could just take an ordinary average of the two estimates. On the other hand, when they are not the same size, you need to make some adjustment in the averaging to give more weight to the larger sample. That is, you need a weighted average, an average weighted by the amount of infor- mation each sample provides.

Also, to be precise, the amount of information each sample provides is not its number of scores, but its degrees of freedom (its number of scores minus 1). Thus, your weighted average needs to be based on the degrees of freedom each sample provides. To find the weighted average, you figure out what proportion of the total degrees of freedom each sample contributes and multiply that proportion by the pop- ulation variance estimate from that sample. Finally, you add up the two results, and that is your weighted, pooled estimate. In terms of a formula,

(8–1)

In this formula, is the pooled estimate of the population variance. is the degrees of freedom in the sample from Population 1, and is the degrees of freedom in the sample from Population 2. (Remember, each sample’s df is its number of scores minus 1.) is the total degrees of freedom . is theS2 1(dfTotal = df1 + df2)dfTotal

df2

df1S 2 Pooled

S2Pooled = df1

dfTotal (S21) +

df2 dfTotal

(S22)

S2Pooled

pooled estimate of the population variance ( ) in a t test for inde- pendent means, weighted average of the estimates of the population variance from two samples (each estimate weighted by the proportion consisting of its sample’s degrees of freedom divided by the total degrees of freedom for both samples).

S2Pooled

weighted average average in which the scores being averaged do not have equal influence on the total, as in figur- ing the pooled variance estimate in a t test for independent means.

The pooled estimate of the population variance is the de- grees of freedom in the first sample divided by the total degrees of freedom (from both samples), multiplied by the population estimate based on the first sample, plus the degrees of freedom in the second sample divided by the total degrees of freedom mul- tiplied by the population variance estimate based on the second sample.

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274 Chapter 8

estimate of the population variance based on the scores in Population 1’s sample; is the estimate based on the scores in Population 2’s sample.

Consider a study in which the population variance estimate based on an experi- mental group of 11 participants is 60, and the population variance estimate based on a control group of 31 participants is 80. The estimate from the experimental group is based on 10 degrees of freedom (11 participants minus 1), and the estimate from the control group is based on 30 degrees of freedom (31 minus 1). The total information on which the estimate is based is the total degrees of freedom—in this example, 40 (that is, ). Thus, the experimental group provides one-quarter of the infor- mation ( ), and the control group provides three-quarters of the informa- tion ( ).

You then multiply the estimate from the experimental group by , making 15 (that is, ), and you multiply the estimate from the control group by

, making 60 (that is, ). Adding the two gives an overall estimate of 15 plus 60, which is 75. Using the formula,

Notice that this procedure does not give the same result as ordinary averaging (without weighting).

Ordinary averaging would give an estimate of 70 (that is, ). Your weighted, pooled estimate of the population variance of 75 is closer to the esti- mate based on the control group alone than to the estimate based on the experimen- tal group alone. This is as it should be, because the control group estimate in this example was based on more information.

Figuring the Variance of Each of the Two Distributions of Means The pooled estimate of the population variance is the best estimate for both popula- tions. (Remember, to do a t test for independent means, you have to be able to as- sume that the two populations have the same variance.) However, even though the two populations have the same variance, if the samples are not the same size, the dis- tributions of means taken from them do not have the same variance. That is because the variance of a distribution of means is the population variance divided by the sam- ple size. In terms of formulas,

(8–2)S2M1 = S2Pooled

N1

360 + 804>2 = 70

= 1

4 (60) +

3

4 (80) = 15 + 60 = 75.

S2Pooled = df1

dfTotal (S21) +

df2 dfTotal

(S22) = 10

40 (60) +

30

40 (80)

80 * 3>4 = 603>4 60 * 1>4 = 15 1>430>40 = 3>4

10>40 = 1>410 + 30

T I P F O R S U C C E S S You know you have made a mis- take in figuring if it does not come out between the two esti- mates of the population variance. (You also know you have made a mistake if it does not come out closer to the estimate from the larger sample.)

S2Pooled

The variance of the distribu- tion of means for the first population (based on an estimated population vari- ance) is the pooled estimate of the population variance divided by the number of participants in the sample from the first population.

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The t Test for Independent Means 275

(8–3)

Consider again the study with 11 in the experimental group and 31 in the control group. We figured the pooled estimate of the population variance to be 75. For the experimental group, the variance of the distribution of means would be , which is 6.82. For the control group, the variance would be , which is 2.42. Using the formulas,

The Variance and Standard Deviation of the Distribution of Differences Between Means The variance of the distribution of differences between means is the variance of Population 1’s distribution of means plus the variance of Population 2’s distribution of means. (This is because, in a difference between two numbers, the variation in each contributes to the overall variation in their difference. It is like sub- tracting a moving number from a moving target.) Stated as a formula,

(8–4)

The standard deviation of the distribution of differences between means ( ) is the square root of the variance:

(8–5)

In the example we have been considering, the variance of the distribution of means for the experimental group was 6.82, and the variance of the distribution of means for the control group was 2.42; the variance of the distribution of the differ- ence between means is thus 6.82 plus 2.42, which is 9.24. This makes the standard deviation of this distribution the square root of 9.24, which is 3.04. In terms of the formulas,

Steps to Find the Standard Deviation of the Distribution of Differences Between Means

●A Figure the estimated population variances based on each sample. That is, figure one estimate for each population using the formula ).S2 = SS>(N - 1

SDifference = 2S2Difference = 29.24 = 3.04. S2Difference = S2M1 + S

2 M2 = 6.82 + 2.42 = 9.24

SDifference = 2S2Difference

SDifference

S2Difference = S2M1 + S 2 M2

(S2Difference)

S2M2 = S2Pooled

N2 =

75

31 = 2.42.

S2M1 = S2Pooled

N1 =

75

11 = 6.82

75>31 75>11

S2M2 = S2Pooled

N2

T I P F O R S U C C E S S Remember that when figuring esti- mated variances, you divide by the degrees of freedom. But when fig- uring the variance of a distribution of means, which does not involve any additional estimation, you di- vide by the actual number in the sample.

The variance of the distribution of differences between means is the variance of the distribution of means for the first population (based on an estimated population variance) plus the variance of the distribution of means for the second population (based on an estimated population variance).