Heiman G. Basic Statistics for the Behavioral Sciences

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228 CHAPTER 10 / Introduction to Hypothesis Testing

decision (with ). But, as in the lower row of the table, sometimes is really

false: Then if we retain , we make a Type II error (with ), and if we reject ,

we avoid a Type II error and make the correct decision (with ).

In any experiment, the results of your inferential procedure will place you in one of

the columns of Table 10.1. If you reject , then either you’ve made a Type I error, or

you’ve made the correct decision and avoided a Type II error. If you retain , then

either you’ve made a Type II error or you’ve made the correct decision and avoided a

Type I error.

The most serious error is a Type I, concluding that an independent variable works

when really it does not. For example, concluding that new drugs, surgical techniques,

or engineering procedures work when they really do not can cause untold damage. For

this reason, researchers always use a small to minimize the likelihood of these errors.

On the other hand, a Type II error is not as harmful because it is merely failing to iden-

tify an independent variable that works. We have faith that future research will eventu-

ally discover the variable. However, we still prefer to avoid Type II errors, and for that

we need power.

Power

Of the various outcomes back in Table 10.1, the goal of research is to reject when it

is false: We conclude that the pill works, and the truth is that the pill does work. Not

only have we avoided any errors, but we have learned about a relationship in nature.

This ability has a special name: Power is the probability that we will reject when it

is false, correctly concluding that the sample data represent a relationship. In other

words, power is the probability of not making a Type II error, so power equals

Power is important because, after all, why bother to conduct a study if we’re unlikely

to reject the null hypothesis even when there is a relationship present? Therefore,

power is a concern anytime we do not reject because we wonder, “Did we just miss

a relationship?” For example, previously, when we did not find a significant effect of

the pill, maybe the problem was that we lacked power: Maybe we were unlikely to

reject even if the pill really worked.

To avoid this doubt, we strive to maximize the power of a study (maximizing the size

of ). Then we’ll have confidence in our decision if we do ultimately retain null.

Essentially, the idea is to do everything we can to ensure that in case we end up in the

Type II situation where there is a relationship in nature, we—and our statistics—will

not miss the relationship. If we still end up retaining , we know that it’s not for lack

of trying. We’re confident that if the relationship was there, we would have found it,

so it must be that the relationship is not there. Therefore, we are confident in the deci-

sion to retain , and, in statistical lingo, we say that we’re confident we have avoided

a Type II error.

REMEMBER We seek to maximize power so that, if we retain , we are con-

fident we are not making a Type II error.

The time to build in power is when we design a study. We’re talking about being in

the Type II situation here, so it’s a given that the relationship exists in nature. We can’t

do anything to ensure that we’re in this situation (that’s up to nature), but assuming we

are, then the goal is to have significant results. Therefore, we increase power by

increasing the likelihood that our results will be significant. Results are significant if

is larger than , so anything that increases the size of the obtained value relative

to the critical value increases power.

crit

obt

1 2 ␤

1 2 ␤.

␣

p 5 1 2 ␤

p 5 ␤H

p 5 1 2 ␣

Essentially, the purpose of inferential statistics is to minimie the probability of making

Type I and Type II errors. If we had not performed the z-test in our initial IQ pill study,

we might have made a Type I error: We might have erroneously concluded that the pill

raises IQ to around 105 when, in fact, we were being misled by sampling error. We

would have no idea if this had occurred, nor even the chances that it had occurred. After

finding a significant result, however, we are confident that we did not make a Type I

error because the probability of doing so is less than . Likewise, if the results were

not significant, through power we minimize the probability of a Type II error, so we’d

be confident that we did not miss a pill that actually works.

.05

PUTTING IT

ALL TOGETHER

Errors in Statistical Decision Making 229

We influence power first through the statistics we use. It is better to design a study so

that you can use parametric procedures because parametric procedures are more power-

ful than nonparametric ones: Analyzing data using a parametric test is more likely to

produce significant results than analyzing the same data using a nonparametric test.

Then, in case we’re in the situation where is false, we won’t miss the relationship.

Also, when we can predict the direction of the relationship, using a one-tailed test is

more powerful than a two-tailed test. This is because the for a one-tailed test

(1.645) is smaller than the for a two-tailed test (1.96). All other things being equal,

a is more likely to be beyond , so it’s more likely to be significant.

In later chapters, you’ll see additional ways to maximize power. Do not think that we

are somehow “rigging” the decisions here. We are simply protecting ourselves against

errors. Setting at or less protects us if we end up in the situation where is true

(limiting Type I errors). Maximizing power protects us if we end up in the situation

where is false (limiting Type II errors). Together, these strategies minimize our

errors, regardless of whether or not there is really a relationship.

REMEMBER When discussing power, it is a given that is false. Power is

increased by increasing the size of the obtained value relative to the critical

value so that the results are more likely to be significant.

.05␣

1.645z

obt

crit

■

A Type I error is rejecting a true . A Type II error

is retaining a false .

■

Power is the probability of not making a Type II

error.

MORE EXAMPLES

When is true, there is no relationship: If the data

cause us to reject , we make a Type I error. To

decrease the likelihood of this, we keep alpha small. If

the data cause us to retain , we avoid this error.

When is false, there is a relationship: If the data

cause us to retain , we make a Type II error. If the

data cause us to reject , we avoid this error. To

increase the likelihood of this, we increase power.

For Practice

1. Claiming that an independent variable works

although in nature it does not is a ____ error.

2. Failing to conclude that an independent variable

works although in nature it does is a ____ error.

3. If we reject , we cannot make a ____ error.

4. If we retain , we cannot make a ____ error.

5. To be confident in a decision to retain , our

power should be ____.

Answers

1. Type I

2. Type II

3. Type II

4. Type I

5. high

␮

A QUICK REVIEW

230 CHAPTER 10 / Introduction to Hypothesis Testing

All parametric and nonparametric inferential procedures follow the logic described

here: is the hypothesis that says your data represent the populations you would find

if the predicted relationship does not exist; says that your data represent the

predicted relationship. You then compute something like a z-score for your data on

the sampling distribution when is true. If the z-score is larger than the critical value,

the results are unlikely to represent the populations described by , so we reject

and accept . The results are called significant, meaning essentially that they are

“believable“: The relationship depicted in the sample data can be believed as existing

in nature rather than being a chance pattern resulting from sampling error. That’s it!

That’s inferential statistics (well, not quite).

CHAPTER SUMMARY

1. Inferential statistics are procedures for deciding whether sample data represent a

particular relationship in the population.

2. Parametric inferential procedures require assumptions about the raw score

populations being represented. They are performed when we compute the mean.

3. Nonparametric inferential procedures do not require stringent assumptions about

the populations being represented. They are performed when we compute the

median or mode.

4. The alternative hypothesis is the statistical hypothesis that describes the

population being represented if the predicted relationship exists.

5. The null hypothesis is the statistical hypothesis that describes the population

being represented if the predicted relationship does not exist.

6. A two-tailed test is used when we do not predict the direction in which the

dependent scores will change. A one-tailed test is used when the direction of the

relationship is predicted.

7. The z-test is the parametric procedure used in a one-sample experiment if

(a) the population contains normally distributed interval or ratio scores and

(b) the standard deviation of the population is known.

8. If lies beyond , then the corresponding sample mean is unlikely to occur

when sampling from the population described by . Therefore, we reject and

accept . This is a significant result and is evidence of the predicted relationship

in the population.

9. If does not lie beyond , then the corresponding sample mean is likely

to occur when sampling the population described by . Therefore, we retain .

This is a nonsignificant result and is not evidence for or against the predicted

relationship.

10. A Type I error occurs when a true is rejected. Its theoretical probability equals .

If a result is significant, the probability of a Type I error is . The theoretical

probability of avoiding a Type I error when retaining is .

11. A Type II error occurs when a false is retained. Its theoretical probability is .

The theoretical probability of avoiding a Type II error when rejecting is .

12. Power is the probability of rejecting a false , and it equals .1 – ␤H

1 – ␤H

␤H

1 – ␣H

p 6 ␣

␣H

crit

obt

crit

obt

1σ

␮s

Application Questions 231

KEY TERMS

alpha 216

alternative hypothesis 211

beta 227

experimental hypotheses 209

inferential statistics 208

nonparametric statistics 209

nonsignificant 220

null hypothesis 212

␤z

obt

crit

␣H

?#$67

one-tailed test 210

parametric statistics 209

power 228

significant 218

statistical hypotheses 211

two-tailed test 210

Type I error 224

Type II error 226

-test 215z

REVIEW QUESTIONS

(Answers for odd-numbered questions are in Appendix D.)

1. Why does the possibility of sampling error present a problem to researchers when

inferring a relationship in the population?

2. What are inferential statistics used for?

3. What does stand for, and what two things does it determine?

4. (a) What are the two major categories of inferential procedures? (b) What charac-

teristics of your data determine which you should use? (c) What happens if you

seriously violate the assumptions of a procedure? (d) What is a statistical reason

to design a study so you can use parametric procedures?

5. What are experimental hypotheses?

6. (a) What does communicate? (b) What does communicate?

7. (a) When do you use a one-tailed test? (b) When do you use a two-tailed test?

8. (a) What does “significant” convey about the results of an experiment? (b) Why is

obtaining significant results a goal of behavioral research? (c) Why is declaring

the results significant not the final step in a study?

9. (a) What is power? (b) Why do researchers want to maximize power? (c) What

result makes us worry whether we have sufficient power? (d) Why is a one-tailed

test more powerful than a two-tailed test?

10. (a) What are the advantage and disadvantage of two-tailed tests? (b) What are the

advantage and disadvantage of one-tailed tests?

␣

APPLICATION QUESTIONS

11. Describe the experimental hypotheses and the independent and dependent variables

when we study: (a) whether the amount of pizza consumed by college students dur-

ing finals week increases relative to the rest of the semester, (b) whether breathing

exercises alter blood pressure, (c) whether sensitivity to pain is affected by increased

hormone levels, and (d) whether frequency of day-dreaming decreases as a function

of more light in the room.

12. For each study in question 11, indicate whether a one- or a two-tailed test should

be used and state the and . Assume that when the amount of the

independent variable is zero.

13. Listening to music while taking a test may be relaxing or distracting. We test 49

participants while listening to music, and they produce an . The meanX

5 54.36

␮ 5 50H

232 CHAPTER 10 / Introduction to Hypothesis Testing

of the population taking this test without music is 50 . (a) Is this a one-

tailed or two-tailed test? Why? (b) What are our and ? (c) Compute .

(d) With , what is ? (e) Do we have evidence of a relationship in the

population? If so, describe the relationship.

14. We ask whether attending a private school leads to higher or lower performance

on a test of social skills. A sample of 100 students from a private school produces

a mean of 71.30 on the test, and the national mean for students from public

schools is 75.62 . (a) Should we use a one-tailed or a two-tailed test?

Why? (b) What are and ? (c) Compute . (d) With , what is ?

(e) What should we conclude about this relationship?

15. (a) In question 13, what is the probability that we made a Type I error? What

would be the error in terms of the independent and dependent variables? (b) What

is the probability that we made a Type II error? What would be the error in terms

of the independent and dependent variables?

16. (a) In question 14, what is the probability that we made a Type I error? What

would be the error in terms of the independent and dependent variables? (b) What

is the probability that we made a Type II error? What would be the error in terms

of the independent and dependent variables?

17. Foofy claims that a one-tailed test is cheating because we use a smaller , and

therefore it is easier to reject than with a two-tailed test. If the independent

variable doesn’t work, she claims, we are more likely to make a Type I error.

Why is she correct or incorrect?

18. Poindexter claims that the real cheating occurs when we increase power by

increasing the likelihood that results will be significant. He reasons that if we are

more likely to reject , then we are more likely to do so when is true. There-

fore, we are more likely to make a Type I error. Why is he correct or incorrect?

19. Bubbles reads that in study A the . She also reads that in

study B the . (a) She concludes that the results of study

B are way beyond the critical value used in study A, falling into a region of rejec-

tion containing only .0001 of the sampling distribution. Why is she correct or

incorrect? (b) She concludes that the results of study B are more significant than

those of study A, both because the is so much larger and because is so much

smaller. Why is she correct or incorrect? (c) In terms of their conclusions, what is

the difference between the two studies?

20. Researcher A finds a significant relationship between increasing stress level and

ability to concentrate. Researcher B replicates this study but finds a nonsignificant

relationship. Identify the statistical error that each researcher may have made.

21. A report indicates that brand X toothpaste significantly reduced tooth decay rela-

tive to other brands, with . (a) What does “significant” indicate about the

researcher’s decision about brand X? (b) What makes you suspicious of the claim

that brand X works better than other brands?

22. We ask if the attitudes toward fuel costs of 100 owners of hybrid electric cars

are different from those on a national survey of owners of non-hybrid cars

. Higher scores indicate a more positive attitude. (a) Is this a one-

or two-tailed test? (b) In words what is and ? (c) Perform the z-test. (d) What

do you conclude about attitudes here? (e) Report your results in the correct format.

23. We ask if visual memory ability for a sample of 25 art majors is better

than that of engineers who, on a nationwide test, scored and .

Higher scores indicate a better memory. (a) Is this a one- or two-tailed test? (b) In

words what is and ? (c) Perform the z-test. (d) What do you conclude about

memory ability here? (e) Report your results in the correct format.

5 14␮ 5 45

1X 5 492

1␮ 5 65, σ

5 242

1X 5 762

p 6 .44

␣z

obt

5114.21, p 6 .0001

obt

511.97, p 6 .05

crit

␣ 5 .05z

obt

1σ

5 28.02

crit

␣ 5 .05

obt

1σ

5 122

Integration Questions 233

INTEGRATION QUESTIONS

24. We measure the self-esteem scores of a sample of statistics students, reasoning

that this course may lower their self-esteem relative to that of the typical college

student . We obtain these scores:

44 55 39 17 27 38 36 24 36

(a) Summarize your sample data. (b) Is this a one-tailed or two-tailed test? Why?

What should we conclude about the relationship here? (Chs. 4, 5, 10)

25. (a) What is the difference between the independent variable and the dependent

variable in an experiment? (b) When the assumptions of a procedure require nor-

mally distributed interval/ratio scores, are we referring to scores on the independ-

ent or dependent variable? (c) What distinguishes an interval and ratio variable

from nominal or ordinal variables? (d) What distinguishes a skewed versus a

normal distribution? (Chs. 2, 3, 4, 10)

26. For the following, identify the independent variable and the dependent variable

and explain why we should use a parametric or nonparametric procedure?

(a) When ranking the intelligence of a group of people given a smart pill.

(b) When comparing the median income for a group of college professors to that

of the national population of all incomes. (c) When comparing the mean reading

speed for a sample of hearing-impaired children to the average reading speed in

the population of hearing children. (d) When measuring interval scores from a

personality test given to a group of emotionally troubled people and comparing

them to the population for emotionally healthy people. (Chs. 2, 4, 10)

27. We have a of 40 under the condition of people tested in the morning versus a

of 60 for people tested in the evening. Assuming they accurately represent their

populations, how do you envision this relationship in the population? (Chs. 4, 10)

28. (a) What does a sampling distribution of means show? (b) A mean having a z

beyond is where? (c) How often do means in the region of rejection occur

when dealing with a particular raw score population? (d) What does this tell you

about your mean? (Chs. 6, 9, 10)

29. (a) Why do researchers want to discover relationships? (b) What is the difference

between a real relationship and one produced by sampling error? (c) What does a

relationship produced by sampling error tell us about nature? (Chs. 2, 10)

30. (a) Why can no statistical result prove that changing the independent variable

causes the dependent scores to change? (b) What one thing does a significant

result prove? (Chs. 2, 10)

;1.96

␮

crit

␣ 5 .05z

obt

1␮ 5 55 and σ

5 11.352

■ ■ ■ SUMMARY OF

FORMULAS

To perform the z-test,

where is the mean of the population described by .H

␮

obt

X – ␮

GETTING STARTED

To understand this chapter, recall the following:

■

From Chapter 5, that is the estimated population standard deviation, that

is the estimated population variance, and that both involve degrees of

freedom, or , which equals .

■

From Chapter 7, the uses and interpretation of and .

■

From Chapter 10, the basics of significance testing, including one- and

two-tailed tests, and , Type I and Type II errors, and power.

Your goals in this chapter are to learn

■

When and how to perform the t-test.

■

How the t-distribution and degrees of freedom are used.

■

What is meant by the confidence interval for m and how it is computed.

■

How to perform significance testing of and .

■

How to increase the power of a study.

N 2 1df

Performing the One-Sample

t-Test and Testing Correlation

Coefficients

234

The logic of hypothesis testing discussed in the previous chapter is common to all infer-

ential statistical procedures. Your goal now is to learn how slightly different procedures

are applied to different research designs. This chapter begins the process by introducing

the t-test, which is very similar to the z-test. The chapter presents (1) when and how to

perform the t-test, (2) how to use a similar procedure to test correlation coefficients, and

(3) a new procedure—called the confidence interval—that is used to estimate .

WHY IS IT IMPORTANT TO KNOW ABOUT t-TESTS?

The t-test is important because, like the z-test, the t-test is used for significance testing

in a one-sample experiment. In fact, the t-test and the “t-distribution” are used more

often in behavioral research. That’s because with the z-test we must know the standard

deviation of the raw score population . However, usually researchers do not know

such things because they’re exploring uncharted areas of behavior. Instead, we usually

estimate by using the sample data to compute the unbiased, estimated population

standard deviation . Then we compute something like a z-score for our sample

mean. However, because we are estimating, we are computing . The one-sample t-test

is the parametric inferential procedure for a one-sample experiment when the standard

deviation of the raw score population must be estimated.

REMEMBER Use the z-test when is known; use the t-test when is not

known.

1σ

␮

Performing the One-Sample t-Test 235

Further, t-tests are important because they form the basis for several other procedures

that we’ll see in this and later chapters.

PERFORMING THE ONE-SAMPLE t-TEST

The one-sample t-test is applied when we have a one-sample experiment. Here’s an

example: Say that one of those “home-and-gardening/good-housekeeper” magazines

describes a test of housekeeping abilities. The magazine is targeted at women, and it

reports that the national average score for women is 75 (so their is 75), but it does

not report the standard deviation. Our question is, “How do men score on this test?” To

answer this, we’ll give the test to a random sample of men and use their to estimate

the for the population of all men. Then we can compare the for men to the of 75

for women. If we can conclude that men produce one population of scores located at

one , but women produce a different population of scores at a different , then we’ve

found a relationship in which, as gender changes, test scores change.

As usual, we first set up the statistical test.

1. The statistical hypotheses: Say that we’re being open minded and look for

any kind of difference, so we have a two-tailed test. If men are different from

women, then the for men will not equal the for women of 75, so is

. If men are not different, then their will equal that of women, so

is .

2. Alpha: We select alpha; sounds good.

3. Check the assumptions: The one-sample t-test is appropriate if we can assume the

following about the dependent variable:

a. We have one random sample of interval or ratio scores.

b. The raw score population forms a normal distribution.

c. The standard deviation of the raw score population is estimated by

computing .

Based on similar research that we’ve read, our test scores meet these assumptions, so

we proceed. For simplicity, we test nine men. (For power, you should never collect so

few scores.) Say that the sample produces a . Based on this, we might con-

clude that the population of men has a of 65.67, while women have a of 75. On the

other hand, maybe we are being misled by sampling error: Maybe by chance we

selected some exceptionally sloppy men for our sample, but men in the population are

not different from women, and so our sample actually poorly represents that the male

population also has a .

To test this null hypothesis, we use the logic that we’ve used previously: says that

the men’s mean represents a population where is 75, so we will create a sampling dis-

tribution showing the means that occur by chance when representing this population.

Then using the formula for the t-test, we will compute , which will locate our sam-

ple mean on this sampling distribution in the same way that z-scores did. The larger the

absolute value of , the farther our sample mean is into the tail of the sampling distri-

bution. Therefore, we will compare to the critical value, called . If is beyond

, our sample mean lies in the region of rejection, so we’ll reject that the sample

poorly represents the population where is 75.

The only novelty here is that is calculated differently than and comes

from the t-distribution.

crit

obt

␮

crit

obt

crit

obt

␮

␮ 5 75

␮␮

X 5 65.67

.05

␮ 5 75

␮␮ ? 75

␮␮

␮␮␮

␮

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

Computing

The computation of consists of three steps that parallel the three steps in the z-test.

After computing our sample mean, the first step in the z-test was to determine the stan-

dard deviation of the raw score population. For the t-test, we could compute the

estimated standard deviation , but to make your computations simpler, we’ll use the

estimated population variance . Recall that the formula for the estimated popula-

tion variance is

The second step of the z-test was to compute the standard error of the mean ,

which is like the “standard deviation” of the sampling distribution. However, because

now we are estimating the population variability, we compute the estimated standard

error of the mean, which is an estimate of the “standard deviation” of the sampling

distribution of means. The symbol for the estimated standard error of the mean is .

(The lowercase s stands for an estimate of the population, and the subscript indicates

that it is for a population of means.)

Previously we computed by dividing by . We could use a similar formula

here so that equals divided by . However, recall that to get any standard devi-

ation we first compute the variance and then find its square root. Thus, buried in the

above is the extra step of finding the square root of the variance that we then divide by

. Instead, by using the variance, we only need to take the square root once.1N

1Ns

1Nσ

1σ

©X

1©X2

N 2 1

1σ

obt

This says to divide the estimated population variance by the of our sample and then

find the square root.

The third step in the z-test was to compute using the formula

This says to find the difference between our sample mean and the in and then divide

by the standard error. Likewise, the final step in computing is to use this formula:t

obt

␮

obt

X 2 ␮

obt

The formula for the estimated standard error of the

mean is

The formula for the one-sample t-test is

obt

X 2 ␮

Performing the One-Sample t-Test 237

Here, the is our sample mean, is the mean of the sampling distribution (which

equals the value of described in the null hypothesis), and is the estimated standard

error of the mean. The is like a z-score, however, indicating how far our sample

mean is from the of the sampling distribution, when measured in estimated standard

error units.

For our housekeeping study, say that we obtained the data in Table 11.1. First, com-

pute . Substituting the data into the formula gives

Thus, the estimated variance of the population of housekeeping scores is 60.

Next, compute the estimated standard error of the mean. With and ,

Finally, compute . Our sample mean is 65.67, the that says we’re represent-

ing is 75, and the estimated standard error of the mean is 2.582. Therefore,

Our is .23.61t

obt

X 2 ␮

65.67 2 75

2.582

29.33

2.582

523.61

␮t

obt

60.00

5 16.667 5 2.582

N 5 9s

5 60.00

©X

1©X2

N 2 1

39,289 2

349,281

9 2 1

5 60.00

␮

obt

␮

␮X

Subject Grades (X) X

1 50 2,500

2 75 5,625

3 65 4,225

4 72 5,184

5 68 4,624

6 65 4,225

7 73 5,329

8 59 3,481

9 64 4,096

X 5 65.67

1©X2

5 349,281

©X

TABLE 11.1

Test Scores of Nine Men

■

Perform the one-sample t-test in a one-sample

experiment when you do not know the population

standard deviation.

MORE EXAMPLES

In a study, is that . To compute , say that

, , and .N 5 36s

5 25X 5 62

obt

␮ 5 60H

obt

X 2 ␮

62 2 60

.833

512.40

5 1.694 5 .833

A QUICK REVIEW

(continued)