Heiman G. Basic Statistics for the Behavioral Sciences

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258 CHAPTER 11 / Performing the One-Sample t-Test and Testing Correlation Coefficients

computations should he perform to describe the relationship in these data?

(d) Identify three problems with his study. (e) Why will correcting these problems

improve his study?

17. A scientist suspects that as a person’s stress level changes, so does the amount of

his or her impulse buying. With , his is . (a) What are and ?

(b) With , what is ? (c) Report these results using the correct format.

(d) What conclusions should he draw? (e) What other calculations should be per-

formed to describe the relationship in these data?

18. Foofy computes the correlation between an individual’s physical strength and his

or her college grade point average. Using a computer, the correlation for a sample

of 2000 people is , . She claims this is a useful tool

for predicting which college applicants are likely to succeed academically. Do you

agree or disagree? Why?

19. We study the influence of wearing uniforms in middle school on attitudes toward

achieving good grades. On a national survey, the average attitude score for

students who do not wear uniforms is . A sample of students who wear

uniforms has scores of 8, 12, 10, 9, 6, and 7. Perform all parts of the t-test and

draw the appropriate conclusions.

20. A newspaper article claims that the academic rank of a college is negatively

related to the rank of its football team. From a sample of 28 colleges, you obtain a

correlation coefficient of . (a) Which type of correlation coefficient did you

compute? (b) What are and ? (c) With , what is the critical value?

(d) What should you conclude about the accuracy of the newspaper’s claim?

(e) In predicting a particular school’s academic ranking in your sample, how

important is it that you look at the school’s football ranking?

21. (a) How would you report your results if , , and is

significant? (b) How would you report your results if , , and

is not significant?

22. While reading a published research report, you encounter the following statements.

For each, identify the , the procedure performed and the outcome, the relationship,

and the type of error possibly being made. (a) “When we examined the perceptual

skills data, the mean of 55 for the sample of adolescents differed significantly from

the population mean of 70 for adults, .” (b) “The correlation

between personality type and emotionality, however, was not significantly different

from zero, with , .”

23. You wish to compute the 95% confidence interval for a sample with a of 80.

Using interpolation, determine the that you should use.

24. In a two-tailed test, is 35. (a) Is the of significant? (b) Is the of

significant?

INTEGRATION QUESTIONS

25. (a) Why must a relationship be significant to be important? (b) Why can a

relationship be significant and still be unimportant? (Chs. 8, 10, 11)

26. (a) What is the difference between the purpose of descriptive and inferential

statistics? (b) When should you should use a parametric versus a nonparametric

inferential procedure? (Chs. 2, 10)

27. What is the design of the study when we compute the z-test and t-test versus when

we compute a correlation coefficient? (Chs. 2, 7, 10, 11)

14.0

obt

12.019t

obt

crit

p 7 .05r125251.42

t14525 3.76, p 6 .01

obt

521.72

N 5 6␣ 5 .05

obt

516.72N 5 43␣ 5 .05

␣ 5 .05H

2.32

␮ 5 12

p 5 6 .0001r11998251.08

crit

␣ 5 .05

1.38rN 5 72

Integration Questions 259

28. (a) What does a correlation coefficient tell you? (b) When do you compute ?

29. (a) Study A reports a result with . Study B reports results with .

What is the difference between the results of A and B in terms of (a) how signifi-

cant they are? (b) Their critical values? (c) Their region of rejection? (d) The

probability of a statistical error? (Chs. 9, 10, 11)

30. (a) What do we mean by the restriction of range? (b) Why is it a problem for the

size of correlation coefficient? (c) Why is it a problem for the power of a correla-

tion coefficient? (Chs. 7, 11)

31. For the following, specify which descriptive and inferential procedures should be

performed, explain what is being compared, and identify the key to answering the

researcher’s question. (a) A researcher measures a group of participants using stan-

dard tests of “social nervousness” and “introversion” to determine if introversion is

a good predictor of nervousness. (Scores are interval scores.) (b) The average

worker at a calculator plant can assemble 106 calculators in his or her first hour of

work. During their final hour, a sample of workers produced an average of only

97.4 calculators . Should we conclude that performance decreases dur-

ing the final hour for all workers? (c) For 20 years the basketball coach recorded

his team’s performance when making free throws, with and .

A sports psychologist trained 20 players on this year’s team to visualize each shot

beforehand. They shot an average of 77.6 for the year. Is visualization a way to

improve the performance of all players? (Chs. 4, 5, 7, 8, 10, 11)

5 14.2␮ 5 71.1

5 17.32

p 6 .001p 5 .031

■ ■ ■ SUMMARY OF

FORMULAS

1. To perform the one-sample t-test,

Values of are found in Table 2 of Appendix

C, for .df 5 N 2 1

crit

obt

X 2 ␮

©X

1©X2

N 2 1

2. The formula for a confidence interval for a single

where is the two-tailed value for .

3. Critical values of the Pearson are found in

Table 3 of Appendix C, using ,

where is the number of pairs.

4. Critical values of the Spearman are found in

Table 4 of Appendix C, using , the number of

pairs.

df 5 N 2 1

df 5 N 2 1t

crit

212t

crit

21 X # ␮ # 1s

211t

crit

21 X

␮

GETTING STARTED

To understand this chapter, recall the following:

■

From Chapter 2, what the terms condition, independent variable, and

dependent variable refer to.

■

From Chapter 4, how to graph the results of experiments.

■

From Chapter 8, how to interpret the proportion of variance

accounted for.

■

From Chapter 11, what a confidence interval indicates and what you’ve

learned about inferential statistics.

Your goals in this chapter are to learn

■

The logic of a two-sample experiment.

■

The difference between independent samples and related samples.

■

How to perform the independent-samples t-test and related-samples t-test.

■

How to compute a confidence interval for the difference between two ms

and for the of difference scores.

■

How is used to describe effect size in a two-sample experiment.r



The Two-Sample t-TEST12

260

This chapter presents the two-sample t-test, which is the major parametric procedure

used when an experiment involves two samples. As the name implies, this test is simi-

lar to the one-sample t-test you saw in Chapter 11, except that a two-sample design

requires that we use slightly different formulas. This chapter discusses (1) one version

of this t-test, called the independent-samples t-test, and its confidence interval; (2) the

other version of this t-test, called the related-samples t-test, and its confidence interval;

and (3) procedures for summarizing the results of any two-sample experiment.

NEW STATISTICAL NOTATION

So far, has stood for the number of scores in a sample. Actually, indicates the total

number of scores in the study, but with only one condition, was also the number of scores

in the sample. However, now we will be discuss experiments with two conditions, so the

lowercase with a subscript will stand for the number of scores in each sample. Thus,

is the number of scores in condition 1, and is the number of scores in condition 2. is

the total number of scores in the experiment, so adding the together equals .

REMEMBER stands for the total number of scores in an experiment;

stands for the number of scores in a condition.n

Nns

Understanding the Two-Sample Experiment 261

Low scores

Dependent scores

High scores

Condition 1

XXXXXXXXXXXXXXXXX

Condition 2

⇒ μ

FIGURE 12.1

Relationship in the population in a two-sample experiment

As the conditions change, scores in the population tend to change.

WHY IS IT IMPORTANT TO KNOW ABOUT THE TWO-SAMPLE t-TEST?

The one-sample experiments discussed in previous chapters are not often found in real

research, because they require that we know under one condition of the independent

variable. Usually, however, researchers explore new behaviors and variables, so they

do not know any beforehand. Instead, the usual approach is to measure a sample of

participants under each condition of the independent variable and to use the sample

mean to estimate the corresponding population that would be found. Often we test

only two conditions, and then our inferential procedures involve two-sample t-tests.

Thus, it is important for you to know about these procedures because they apply to a

more realistic and common way of conducting experiments that you’ll often encounter.

Further, by understanding studies with two conditions, you will understand the more

complicated designs and analyses that we’ll discuss in the remaining chapters and that

also are common in the literature.

UNDERSTANDING THE TWO-SAMPLE EXPERIMENT

In a two-sample experiment, we measure participants’ scores under two conditions of

the independent variable. Condition 1 produces that represents , the we would

find if we tested everyone in the population under condition 1. Condition 2 produces

that represents , the we would find if we tested everyone in the population

under condition 2. A possible outcome from such an experiment is shown in Figure

12.1. If each sample mean represents a different population and for each condition,

then the experiment has demonstrated a relationship in nature.

However, there is the usual problem of sampling error. Even though we may have

different sample means, the relationship may not exist in the population. Instead, if we

tested the population, we might find the same population of scores under both condi-

tions. In Figure 12.1 we might find only the lower or upper distribution, or we might

find one in-between. Then there would be only one value of : Call it or , it

wouldn’t matter because it’s the same Therefore, before we make any conclusions

about the experiment, we must determine whether the difference between the sample

means reflects sampling error.

.







s

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262 CHAPTER 12 / The Two-Sample t-Test

The parametric statistical procedure for determining whether the results of a two-

sample experiment are significant is the two-sample t-test. However, we have two differ-

ent ways to create the samples, so we have two different versions of the t-test: One is

called the independent-samples t-test and the other is called the related-samples t-test.

REMEMBER The two ways to calculate the two-sample t-test are the

independent-samples t-test or the related-samples t-test.

THE INDEPENDENT-SAMPLES t-TEST

The independent-samples t-test is the parametric procedure for testing two sample

means from independent samples. Two samples are independent when we randomly

select participants for a sample, without regard to who else has been selected for either

sample. Then the scores are independent events, which, as in Chapter 9, means that the

probability of a particular score occurring in one sample is not influenced by the scores

that occur in the other sample. You can recognize independent samples by the absence

of anything fancy when selecting participants, such as creating pairs of participants or

repeatedly testing the same participants in both conditions.

Here is a study that calls for the independent-samples t-test. People who witness a

crime or other event may recall the event differently when they are hypnotized. We’ll

select two samples of participants who watch a videotape of a supposed robbery. Later,

one group will be hypnotized and then answer 30 questions about the event. The other

group will answer the questions without being hypnotized. Thus, the conditions of

the independent variable are the presence or absence of hypnosis, and the dependent

variable is the amount of information correctly recalled. This design is shown in

Table 12.1. After replacing the with the actual recall scores, we will compute the

mean of each condition (each column). If the means differ, we’ll have evidence of a

relationship where, as amount of hypnosis changes, recall scores also change.

First, as always, we check that the study meets the assumptions of the statistical test.

In addition to requiring independent samples, this t-test also requires

1. The dependent scores measure an interval or ratio variable.

2. The populations of raw scores form at least roughly normal distributions.

3. And here’s a new one: The populations have homogeneous variance. Homogene-

ity of variance means that the variances of the populations being represented are

equal. That is, we assume that if we computed for each population, we would

have the same answer each time.

4. It is not required that each condition have the same , but the should not be mas-

sively unequal—a difference in the neighborhood of 10 to 20 is best. (The more the

differ from each other, the more important it is to have homogeneity of variance.)

You’ll know if you meet these assumptions by seeing how the vari-

ables are treated in previously published research related to your study.

Statistical Hypotheses for the

Independent-Samples t-Test

As usual, we may have a one- or a two-tailed test. For now, say that

we don’t predict whether hypnosis will increase or decrease recall

scores so we have a two-tailed test.

nsn

No Hypnosis Hypnosis

Recall XX

Scores XX

➝

TABLE 12.1

Diagram of Hypnosis

Study using an

Independent-Samples

Design

The independent variable

is amount of hypnosis,

and the dependent

variable is recall.

The Independent-Samples t-Test 263

First, the alternative hypothesis: A relationship exists if one population mean is

larger or smaller than the other , producing two distributions, similar to that back in

Figure 12.1. In other words, should not equal . We could state this as

, but there is a better way. If the two are not equal, then their difference does not equal

zero. Thus, the two-tailed alternative hypothesis for our study is

implies that the means from our two conditions each represent a different popula-

tion of recall scores having a different , so a relationship is present.

Of course, there’s also our old nemesis, the null hypothesis. Perhaps there is no rela-

tionship, so if we tested everyone under the two conditions, we would find the same

population and . In other words, equals . We could state this as ,

but, again, there is a better way. If the two are equal, then their difference is zero.

Thus, our two-tailed null hypothesis is

implies that both samples represent the same population of scores, having the same

, so a relationship is not present. If our sample means differ, it’s because of sampling

error in representing that one .

Notice that these hypotheses do not contain a specific value of . Therefore, these

are the two-tailed hypotheses for any independent-samples t-test, when you are testing

an that says there is zero difference between the populations. This is the most com-

mon approach and the one that we’ll use. (Consult an advanced statistics book to test

for nonzero differences.)

As usual, we test the null hypothesis, and to do that we examine the sampling

distribution.

The Sampling Distribution for the

Independent-Samples t-Test

To understand the sampling distribution here, say that we find a mean recall score of 20

in the no-hypnosis condition and a mean of 23 in the hypnosis condition. We can sum-

marize these results by looking at the difference between the means: Changing from no

hypnosis to hypnosis results in a difference in mean recall of 3 points. We always test

by finding the probability of obtaining our results when there is not a relationship,

so here we will determine the probability of obtaining a difference of 3 between two

when they both actually represent the same .

REMEMBER The independent-samples t-test determines the probability of

obtaining our difference between when is true.

You can think of the sampling distribution as follows. Using the same as in our

study, we select two random samples from one raw score population. (Just like says

we did in our study.) We compute the two sample means and subtract one from the other.

The result is the difference between the means, which we symbolize by . We do

this an infinite number of times and plot a frequency distribution of these differences,

producing the sampling distribution of differences between means. This is the distri-

bution of all possible differences between two means when they are drawn from one raw

score population. You can envision this sampling distribution as shown in Figure 12.2.

On the axis, each score is the difference between two randomly selected sample

means. The axis is labeled twice, first using the symbols of and, beneath them,X

2 X



: 

2 

5 0

s

: 

5 



: 

2 

? 0

s

: 

? 



1

264 CHAPTER 12 / The Two-Sample t-Test

– X

Differences:

Values of t:

. . .. . .

–1 +1 +2 +3 +4 +5 +6 +7 +8–2–3–4–5–6–7–8 0

FIGURE 12.2

Sampling distribution of differences between means when H

: 



Each symbolizes the difference between two sample means. Larger differences are less likely when is true.H

2 X

actual differences we might find. The mean of the sampling distribution is zero because,

most often, both sample means will equal the of the population of raw scores, so their

difference will be zero. However, sometimes is larger, so the difference will be

a positive or negative amount. Small negative or positive differences will occur relatively

frequently, but larger differences occur less frequently. The larger the difference between

the means, the farther into the tail of the distribution it lies.

To test , we determine where our difference between means lies on this sampling dis-

tribution. To do so, we compute a new version of but it provides information similar to

previous t-tests: A difference of zero between , located at the of the distribu-

tion, produces a of zero. A positive difference produces a positive and a negative

difference produces a negative . Larger differences between the means are further into a

tail of the distribution and have a larger . Therefore, if the difference between our sam-

ple means produces a close to the center of the distribution, then our difference occurs

frequently when is true: In our example, our two samples are likely to represent the

same population of recall scores. But, if places our difference beyond , far into a

tail of the sampling distribution, then this difference is unlikely when is true: Our two

samples are unlikely to represent the same population of recall scores.

So now we compute .

Computing the Independent-Samples t-Test

In the previous chapter, you computed by computing and then performing three

steps: (1) estimating the variance of the raw score population, (2) computing the esti-

mated standard error of the sampling distribution, and (3) computing . For the two-

sample t-test, after computing and , you perform three similar steps.

Estimating the Population Variance First, calculate for each condition, using

the formula

Each time, use the from only one condition, and is the number of scores in that

condition.

nXs

©X

1©X2

n 2 1

obt

crit

obt

X

and X

obt

or X



The Independent-Samples t-Test 265

This will give us and , and each is an estimate of the population variance. How-

ever, each may contain sampling error. (Because of this, if does not equal , we

have not necessarily violated the assumption of homogeneity of variance.) To obtain

the best estimate, we compute a “weighted average” of the two variances. The

“weight” we give to each variance is based on the number of participants in a sample,

using each sample’s This weighted average is called the pooled variance, and its

symbol is .s

pool

df.

This says to multiply the from each sample times for that sample. Then add

the results together and divide by the sum of .

For example, say that the hypnosis study produced the results shown in Table 12.2.

Let’s label the hypnosis condition as condition 1, so it produces , and . The

no-hypnosis condition produces , , and . Filling in the above formula, we have

In the numerator, 16 times 9 is 144, and 14 times 7.5 is 105. In the denominator, 16 plus

14 is 30, so

Thus, we estimate that the variance of the population of recall scores represented by

our samples is 8.30.

Computing the Standard Error of the Difference The next step is to use

to compute the standard error of the sampling distribution. It is called the standard error

of the difference. The standard error of the difference is the estimated “standard

deviation” of the sampling distribution of differences between the means. The symbol

for the standard error of the difference is . (The subscript indicates that we are

dealing with differences between pairs of means.)

In the previous chapter, we computed the standard error by dividing the variance by

and then taking the square root. However, instead of dividing by , we can multiplyNN

˛˛X

pool

144 1 105

249

5 8.30

pool

117 2 129.0 1 115 2 127.5

117 2 121 115 2 12

2 121 1n

2 12

n 2 1s

Condition 1: Condition 2:

Hypnosis No Hypnosis

Mean details recalled

Number of participants

Estimated variance s

5 7.5s

5 9.0

5 15n

5 17

5 20X

5 23

TABLE 12.2

Data from the Hypnosis

Study

The formula for the pooled variance is

pool

2 12s

1 1n

2 12s

2 121 1n

2 12

by . Then, for the two-sample t-test, we substitute the pooled variance and our two

, producing this formula:ns

1>N

266 CHAPTER 12 / The Two-Sample t-Test

To use this formula, first reduce the fractions and to decimals. Then add them

together and multiply the sum times . Then find the square root.

For the hypnosis study, is 8.30, is 17, and is 15. Thus:

First, is .059 and is .067. Their sum is .126. Then

Thus, is 1.023.

Computing In previous chapters we found how far the result of the study

was from the mean of the sampling distribution , measured in standard error

units. In general, this formula is

Now the “result of the study” is the difference between the two sample means, so in the

formula we will put in . Likewise, instead of one we have the difference

described by , so we put in Finally, we replace “standard error” with

All together we have

.

2 

X

2 X

obt

1result of the study22 1mean of H

sampling distribution2

standard error

12H

1X2t

obt

5 18.31.12625 11.046 5 1.023

1>151>17

8.3 a

pool

1>n

Here, and are our sample means, is computed as above, and the value of

is the difference specified by the null hypothesis. This value is always 0

(unless you are testing for a nonzero difference.)

For the hypnosis study, our sample means were 23 and 20, the difference between

and is 0, and is 1.023. Therefore,

obt

123 2 2022 0

1.023

113.022 0

1.023

13.0

1.023

512.93



2 

The formula for the standard error of the difference is

pool

The formula for the independent-samples t

obt

2 X

22 1

2 

Our is . Thus, the difference of between our sample means is located at

something like a z-score of on the sampling distribution of differences produced

when both samples represent the same population.

12.93.

13.012.93t

obt

To compute the independent-samples :

■

Compute , and ; , and .

■

Then compute the pooled variance .

■

Then compute the standard error of the difference

■

Then compute .

MORE EXAMPLES

An independent-samples study produced the follow-

ing data: , , , ,

, and .

34.5 a

b5 2.506

pool

110236 1 110233

10 1 10

5 34.5

pool

2 12s

1 1n

2 12s

2 121 1n

2 12

5 11s

5 33

5 21n

5 11s

5 36X

5 27

obt

pool

obt

For Practice

We find , , , ,

and .

1. Compute the pooled variance .

2. Compute the standard error of the difference

3. Compute .

Answers

3. t

obt

133 2 2722 0

1.18

515.08

14.5 a

b5 1.18

pool

120216 1 120213

20 1 20

5 14.5

obt

pool

5 21s

5 13

5 27n

5 21s

5 16X

5 33

512.394

obt

2 X

22 1␮

2 ␮

127 2 2122 0

2.506

A QUICK REVIEW

Interpreting the Independent-Samples t-Test

To determine if is significant, we compare it to , which is found in the t-tables

(Table 2 in Appendix C). As usual, we obtain using degrees of freedom, but with

two samples, the are computed differently: Now the degrees of freedom equals

REMEMBER Critical values of for the independent-samples t-test are found

for .

Another way of expressing this is .

For the hypnosis study, and , so .

With alpha at , the two-tailed is Figure 12.3 locates these values on the

sampling distribution of differences.

The sampling distribution shows the frequency of various differences between sam-

ple means that occur when the samples really represent no difference in the population.

Our says that the difference between our sample means is merely a poor representa-

tion of no difference. But, looking at the sampling distribution, we see that our differ-

ence of hardly ever occurs when the samples represent no difference. In fact our t

obt

;2.042.t

crit

.05

df 5 117 2 121 115 2 125 30n

5 15n

5 17

df 5 1n

1 n

22 2

df 5 1n

2 121 1n

2 12

2 121 1n

2 12

crit

obt

The Independent-Samples t-Test 267