Heiman G. Basic Statistics for the Behavioral Sciences

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218 CHAPTER 10 / Introduction to Hypothesis Testing

not work as predicted. Therefore, we will reject that our pill does not work. If this makes

your head spin, it may be because the logic actually involves a “double negative.” When

our sample falls in the region of rejection, we say “no” to the that says we’re repre-

senting the population with . But says this as a way of saying there is “no

relationship” involving our pill. By rejecting ,we are saying no to “no relationship.”

This is actually saying yes, there is a relationship involving our pill, which is what

says. Therefore, by rejecting and accepting , we also accept the corresponding

experimental hypothesis that the independent variable does work as predicted. Here, it

appears that we have demonstrated a relationship in nature such that the pill would

change the population of IQ scores. In fact, we can be more specific: A sample mean of

105 is most likely to represent the population where is 105. Thus, without the pill, the

population is 100, but with the pill, we expect that scores would increase to a of

around 105.

INTERPRETING SIGNIFICANT RESULTS

The shorthand way to communicate that we have rejected and accepted is to

say that the results are significant. (Statistical hypothesis testing is sometimes called

“significance testing.”) Significant does not mean important or impressive. Signifi-

cant indicates that our results are unlikely to occur if the predicted relationship does

not exist in the population. Therefore, we imply that the relationship found in the

experiment is “believable,” representing a “real” relationship found in nature, and

that it was not produced by sampling error from the situation in which the relation-

ship does not exist.

REMEMBER The term significant means that we have rejected the null

hypothesis and believe that the data reflect a relationship found in nature.

Notice that your decision is simply either yes, reject , or no, do not reject . All

z-scores in the region of rejection are treated the same, so one cannot be “more sig-

nificant” than another. Likewise, there is no such thing as “very significant” or “highly

significant.” (That’s like saying “very yes” or “highly yes.”) If is beyond ,

regardless of how far it is beyond, the results are simply significant, period!

Although we accept that a relationship exists, we have three very important restric-

tions on how far we can go when interpreting significant results in any experiment.

First, we did not prove that is false. With our pill, the only thing we have

“proven” is that a sample mean of 105 is unlikely to come from a population where

. However, the sampling distribution shows that means of 105 do occur once

in a while when we are representing this population. Maybe our sample was one of

them. Maybe the pill did not work, and our sample was very unrepresentative of this.

Second, we did not prove it was our independent variable that caused the scores to

change. Although our pill might have caused the higher IQ scores, some other, hid-

den variable also might have produced them. Maybe our participants cheated on the

IQ test, or there was something in the air that made them smarter, or there were

sunspots, or who-knows-what! If we’ve performed a good experiment and can elimi-

nate such factors, then we can argue that it is our independent variable that changed

the scores.

Finally, the represented by our sample may not equal our . Even assuming that

our pill does increase IQ, the population is probably not exactly 105. Our sample may

reflect (you guessed it) sampling error! That is, the sample may accurately reflect that

␮

␮ 5 100

crit

obt

␮␮

␮

␮ 5 100

Interpreting Nonsignificant Results 219

the pill increases IQ, but it may not perfectly represent how much the pill increases

scores. Therefore, if we gave the pill to the population, we might find a of 104, or

106, or any other value. However, a sample mean of 105 is most likely when the popu-

lation is 105, so we would conclude that the resulting from our pill is probably

around 105.

Bearing these qualifications in mind, we interpret the of 105 the way we wanted

to several pages back: Apparently, the pill increases IQ scores by about 5 points. But

now, because the results are significant, we are confident that we are not being misled

by sampling error. Therefore, we are more confident that we have discovered a rela-

tionship in nature. (But stay tuned, we could be wrong.) At this point, we return to

being behavioral researchers and interpret the results “psychologically”: We describe

how the ingredients in the pill affect intelligence, what brain mechanisms are

involved, and so on.

INTERPRETING NONSIGNIFICANT RESULTS

Let’s say that the IQ pill had instead produced a sample mean of 99. Now the z-score

for the sample is

As in Figure 10.6, a of is not beyond the of , so the sample is not in

the region of rejection. This indicates that we will frequently obtain a sample mean of

99 when sampling the population where . Therefore, the null hypothesis is rea-

sonable: our sample is likely to be a poor representation of the population where is

100. So we will not reject . Sampling error from this population can explain our

results just fine, thank you, so we will not reject this explanation. In such situations, we

say that we have “failed to reject ” or that we “retain .”

When we retain , we also retain the experimental hypothesis that our independent

variable does not work as predicted. We’ve found that our sample mean was likely to

occur if we were representing the situation where the pill is not present. Therefore, it

makes no sense to conclude that the pill works if our results were likely to occur without

the pill. Likewise, we never conclude that an independent variable works if the results

were likely to be due to sampling error from the situation where it does not work.

␮

␮ 5 100

;1.96z

crit

2.40z

obt

X 2 ␮

99 2 100

2.5

52.40

␮␮

␮

crit

= –1.96 z

crit

= +1.96

obt

= –.40

(X = 99)

X X X X X

100

X X X X X

–3 –2 –1

0+3+2+1

Sample means

z-scores

FIGURE 10.6

Sampling distribution

of IQ means

The sample mean of 99 has

a of 40. 2z

obt

220 CHAPTER 10 / Introduction to Hypothesis Testing

The shorthand way to communicate all of this is to say that the results are not signif-

icant or that they are nonsignificant. (Don’t say insignificant.) Nonsignificant indicates

that the results are likely to reflect chance, sampling error, without there being a rela-

tionship in nature.

REMEMBER Nonsignificant indicates that we have failed to reject

because the results are not in the region of rejection and are thus likely to

occur when there is not the predicted relationship in nature.

When we retain , we also retain the experimental hypothesis that the independent

variable does not work as predicted. However, we have not proven that is true, so

we have not proven that our independent variable does not work. We have simply failed

to find convincing evidence that it does work. The only thing we’re sure of is that sam-

pling error could have produced our data. Therefore, we still have two hypotheses that

are both viable: (1) , that the data do not really represent a relationship, and (2) ,

that the data do represent a relationship. Thus, maybe in fact the pill does not work. Or

maybe the pill does work, but our sample poorly represents this. We simply don’t know

whether the pill works or not.

Thus, with nonsignificant results, you should not say anything about whether the

independent variable influences behavior or not, and do not even begin to interpret

the results “psychologically.” All that you can say is that you failed to demonstrate that

the predicted relationship exists

REMEMBER Nonsignificant results provide no convincing evidence—one

way or the other—as to whether a relationship exists in nature.

For this reason, you cannot design a study to show that a relationship does not exist.

For example, you could not set out to show that the IQ pill does not work. At best,

you’ll end up retaining both and , and at worst, you’ll end up rejecting , show-

ing that it does work.

SUMMARY OF THE z-TEST

Altogether, the preceding discussion can be summarized as the following four steps.

For a one-sample experiment that meets the assumptions of the z-test:

1. Determine the experimental hypotheses and create the statistical hypothesis: Pre-

dict the relationship the study will or will not demonstrate. Then describes the

that the represents if the predicted relationship does not exist. describes

the that the represents if the relationship does exist.

2. Compute , compute , and then compute . In the formula for , the value of

is the of the sampling distribution, which is also the of the raw score popula-

tion that says is being represented.

3. Set up the sampling distribution: Select , locate the region of rejection, and

determine the critical value.

4. Compare to : If lies beyond , then reject , accept , and the

results are “significant.” Then interpret the relationship. If does not lie beyond

, do not reject and the results are “nonsignificant.” Do not draw any conclu-

sions about the relationship.

crit

obt

crit

obt

crit

obt

␣

␮␮

␮zz

obt

X␮

The One-Tailed Test 221

THE ONE-TAILED TEST

Recall that a one-tailed test is used when we predict the direction in which scores will

change. The statistical hypotheses and sampling distribution are different in a one-

tailed test.

The One-Tailed Test for Increasing Scores

Say that we had developed a “smart” pill, so the experimental hypotheses are (1) the pill

makes people smarter by increasing IQ scores, or (2) the pill does not make people

smarter. For the statistical hypotheses, start with the alternative hypothesis: People with-

out the pill produce , so if the pill makes them smarter, their will be greater

than 100. Therefore, our alternative hypothesis is that our sample represents this popula-

tion, so . On the other hand, if the pill does not work as predicted, either it

will leave IQ scores unchanged or it will decrease them (making people dumber). Then

will either equal 100 or be less than 100. Therefore, our null hypothesis is that our

sample represents one of these populations, so .

We again test , and we do so by testing whether the sample represents the raw

score population in which equals 100. This is because first the pill must make the

sample smarter, producing a above 100, or we have no evidence that the “smart” pillX

␮

: ␮ # 100

␮

: ␮ 7 100

␮␮ 5 100

■

If lies beyond , reject , the results are

significant, and conclude there is evidence for the

predicted relationship. Otherwise, the results are

not significant, and we make no conclusion about

the relationship.

MORE EXAMPLES

We test a new technique for teaching reading. Without

it, the on a reading test is 220, with . An

of 25 participants has . Then:

1. With a two-tailed test, .

2. Compute

3. With , is , and the sampling dis-

tribution is like Figure 10.6.

4. The of is beyond the of ,

so the results are significant: the data reflect a

relationship, with the of the population using

the technique at around , while for those

not using it at .

Above a different mean produced . This

is not beyond the so the results are not signifi-z

crit

obt

Z 521.83

␮ 5 220

211.55

␮

21 .96z

crit

22 .817z

obt

;1 .96z

crit

␣ 5 .05

22.817

obt

5 1X 2 ␮2>σ

5 1211.55 2 2202>3 5

obt

: σ

5 σ

>1N 5 15>125 5 3;

: ␮ ? 220H

: ␮ 5 220;

X 5 211.55

Nσ

5 15␮

crit

obt

cant: Make no conclusion about the influence of the

technique on reading.

For Practice

We test whether a sample of 36 successful dieters are

more or less satisfied with their appearance than in the

population of nondieters, where .

1. What are and ?

2. The for dieters is 45. Compute .

3. Set up the sampling distribution.

4. What should we conclude?

Answers

3. With the sampling distribution has a region

of rejection in each tail, with (as in

Figure 10.6).

4. The of is beyond of , so the results

are significant: The population of dieters are more satis-

fied (at a around 45) than the population of nondieters

(at ).␮ 5 40

␮

;1.96z

crit

12.50z

obt

crit

5 ;1.96

␣ 5 .05

5 12>136 5 2; z

obt

5 145 2 402>2 512 .50

: ␮ 5 40; H

: ␮ ? 40

obt

␮ 5 40 1σ

5 122

A QUICK REVIEW

222 CHAPTER 10 / Introduction to Hypothesis Testing

X X X X X

100

X X X X X

–3 –2 –1

0+3+2+1

Sample means

z-scores

crit

= +1.645

obt

= +2.63

Region of rejection

equals 5%

FIGURE 10.7

Sampling distribution of

IQ means for a one-tailed

test of whether scores

increase

The region of rejection is

entirely in the upper tail.

works. If we then conclude that the population is above 100, then it is automatically

above any value less than 100.

REMEMBER A one-tailed null hypothesis always includes a population with

equal to some value. Test by testing whether the sample data represent

that population.

Thus, as in Figure 10.7, the sampling distribution again shows the means that occur

when sampling the population where is 100. We again set , but because we

have a one-tailed test, the region of rejection is in one tail of the sampling distribution.

You can identify which tail by identifying the result you must see to claim that your

independent variable works as predicted (to support ). For us to believe that the smart

pill works, we must conclude that the is significantly larger than 100. On the sam-

pling distribution, the means that are significantly larger than 100 are in the region of

rejection in the upper tail of the sampling distribution. Therefore, the entire region is in

the upper tail of the distribution. Then, as in the previous chapter, the region of rejec-

tion is 5% of the curve, so is .

Say that after testing the pill we find . The sampling distribution

is still based on the IQ population with and , so .

Then . As in Figure 10.7, this is beyond , so

it is in the region of rejection. Therefore, the sample mean is unlikely to represent the

population having . If the sample is unlikely to represent the population where

is 100, it is even less likely to represent a population where is below 100. Therefore,

we reject the null hypothesis that , and accept the alternative hypothesis that

. We conclude that the pill produces a significant increase in IQ scores and esti-

mate that would equal about (keeping in mind all of the cautions and qualifica-

tions for interpreting significant results that we discussed previously).

Notice that a one-tailed is significant only if it lies beyond and has the same

sign. Thus, if had not been in our region of rejection, we would retain and have

no evidence whether the pill works or not. This would be the case even if we

had obtained very low scores producing a very large negative z-score. We have no

region of rejection in the lower tail for this study and, no, you cannot move the region

of rejection to make the results significant. Remember, which tail you use is determined

by your experimental hypothesis. After years of developing a “smart pill,” it would

make no sense to suddenly say, “Whoops, I meant to call it a dumb pill.” Likewise, you

obt

crit

obt

106.58␮

␮ 7 100

␮ # 100

␮␮

␮ 5 100

crit

obt

5 1106.58 2 1002>2.5 512.63

5 15>136 5 2.5σ

5 15␮ 5 100

X 5 106.581N 5 362

11.645z

crit

␣ 5 .05␮

␮

The One-Tailed Test 223

crit

= –1.645

Region of rejection

equals 5%

X X X X X

100

X X X X X

–3 –2 –1

0+3+2+1

Sample means

z-scores

FIGURE 10.8

Sampling distribution of

IQ means for a one-tailed

test of whether scores

decrease

The region of rejection is

entirely in the lower tail.

cannot switch from a one-tailed to a two-tailed test. Therefore, use a one-tailed test

only when confident of the direction in which the dependent scores will change. When

in doubt, use a two-tailed test.

The One-Tailed Test for Decreasing Scores

Say that we had created a pill to lower IQ scores. If the pill works, then would be less

than 100, so . But, if the pill does not work, it would produce the same

scores as no pill (with ), or it would make people smarter (with ). So

We again test using the previous sampling distribution. Now, however, to conclude

that the pill lowers IQ, our sample mean must be significantly less than 100. Therefore,

the region of rejection is in the lower tail of the distribution, as in Figure 10.8.

With , is now minus . If the sample produces a negative beyond

(for example, ), then we reject the that the sample mean repre-

sents a equal to or greater than 100 and accept the that the sample represents a

less than 100. However, if does not fall in the region of rejection (for example, if

), we do not reject , and we have no evidence as to whether the pill

works or not.

obt

521.25

obt

␮

obt

521.6921.645

obt

1.645z

crit

␣ 5 .05

: ␮ $ 100

␮ 7 100␮ 5 100

: ␮ 6 100

␮

■

Perform a one-tailed test when predicting the direc-

tion the scores will change.

■

When predicting that will be higher than , the

region of rejection is in the upper tail of the

sampling distribution. When predicting that will

be lower than , the region of rejection is in the

lower tail.

MORE EXAMPLES

We predict that learning statistics will increase a stu-

dent’s IQ. Those not learning statistics have

and . For 25 statistics students, .X

5 108.6σ

5 15

␮ 5 100

␮

␮X

1. With a one-tailed test, .

2. Compute

3. With , is ⫹1.645. The sampling distri-

bution is as in Figure 10.7.

4. The of is beyond , so the results are

significant: Learning statistics gives a around

, whereas people not learning statistics have

Say that a different mean produced .

This is not beyond , so it is not significant. We’d

have no evidence that learning statistics raises IQ.

crit

obt

511.47

␮ 5 100

108.6

␮

crit

12.87z

obt

crit

␣ 5 .05

obt

5 1X 2 ␮2>σ

5 1108.6 2 1002>3 512.87.

obt

: σ

5 σ

>1N 5 15>125 5 3;

: ␮ 7 100; H

: ␮ # 100

A QUICK REVIEW

continued

224 CHAPTER 10 / Introduction to Hypothesis Testing

STATISTICS IN PUBLISHED RESEARCH: REPORTING SIGNIFICANCE TESTS

Every study must indicate whether the results are significant or nonsignificant. With

our IQ pill, a report might say, “The pill produced a significant difference in IQ scores.”

This indicates that the difference between the sample mean and the without the pill

is too large to accept as being due to sampling error. Or a report might say that we

obtained a “significant ”: The is beyond the . Or we might observe a “signifi-

cant effect of the pill”: The change in IQ scores reflected by the sample mean is

unlikely to be caused by sampling error, so presumably it is the effect of—caused by—

changing the conditions of the independent variable.

Whether any result is significant depends on how we have defined unlikely. We do

not always use , so you must always report the used. The APA format for

reporting a result is to indicate the symbol for the statistic, the obtained value, and then

the alpha level. For example to report a significant of , we write: ,

. Notice that instead of using we use (for probability), and with significant

results, we say that is less than . (We’ll discuss the reason for this shortly.) For a

nonsignificant of say, , we report , . Notice, with nonsignifi-

cant results, is greater than .

ERRORS IN STATISTICAL DECISION MAKING

We have one other issue to consider, and it involves potential errors in our decisions:

Regardless of whether we conclude that the sample does or does not represent the pre-

dicted relationship, we may be wrong.

Type I Errors: Rejecting H

When H

Is True

Sometimes, the variables we investigate are not related in nature, so is really true.

When in this situation, if we obtain data that cause us to reject , then we make an

error. A Type I error is defined as rejecting when is true. In other words, we

conclude that the independent variable works when it really doesn’t.

Thus, when we rejected and claimed that the pill worked, it’s possible that it did

not work and we made a Type I error. How could this happen? Because our sample was

exactly what the sampling distribution indicated it was: a very unlikely and unrepre-

sentative sample from the population having a of 100. In fact, the sample so poorly

represented the situation where the pill did not work, we mistakenly thought that

the pill did work. In a Type I error, there is so much sampling error that we—and our

␮

.05p

p 7 .05z 52.402.40z

obt

.05p

p␣p 6 .05

z 512.0012.00z

obt

␣␣ 5 .05

crit

obt

␮

For Practice

You test the effectiveness of a new weight-loss diet.

1. Why is this a one-tailed test?

2. For the population of nondieters, . What

are and ?

3. In which tail is the region of rejection?

4. With , the for the sample of dieters is

. What do you conclude?21.86

obt

␣ 5 .05

␮ 5 155

Answers

1. Because a successful diet lowers weight scores

2. and

3. The left-hand tail

4. The is beyond of , so it is significant:

The for dieters will be less than the of 155 for

nondieters.

␮␮

21.645

crit

obt

: ␮ $ 155H

: ␮ 6 155

Errors in Statistical Decision Making 225

X X X X

100

X X X X X

Critical valueCritical value

Region of rejectionRegion of rejection

FIGURE 10.9

Sampling distribution of

sample means showing

that 5% of all sample

means fall into the

region of rejection

when H

is true

statistical procedures—are fooled into concluding that the predicted relationship exists

when it really does not.

Any time researchers discuss Type I errors, it is a given that is true. Think of it as

being in the “Type I situation” whenever you discuss the situation in which the pre-

dicted relationship does not exist. If you reject in this situation, then you’ve made a

Type I error. If you retain in this situation, then you’ve avoided a Type I error: By

not concluding that the pill works, you’ve made the correct decision because, in reality,

the pill doesn’t work.

We never know if we’re making a Type I error because only nature knows if the

variables are related. However, we do know that the theoretical probability of a Type I

error equals our . Here’s why. Assume that the IQ pill does not work, so we’re in the

Type I situation. Therefore, we can only obtain IQ scores from the population where

is 100. If we repeated this experiment many times, then the sampling distribution in

Figure 10.9 shows the different means we’d obtain over the long run. With a ,

the total region of rejection is 5% of the distribution, so sample means in the region of

rejection would occur 5% of the time in this situation. These means would cause us to

reject even though is true. Rejecting when it is true is a Type I error, so over

the long run, the relative frequency of Type I errors would be . Therefore, anytime

we reject , the theoretical probability that we’ve just made a Type I error is . (The

same is true in a one-tailed test.)

You either will or will not make the correct decision when is true, so the proba-

bility of avoiding a Type I error, is . This is because, if 5% of the time samples

are in the region of rejection when is true, then 95% of the time they are not in the

region of rejection when is true. Therefore, 95% of the time we will not obtain sam-

ple means that cause us to erroneously reject : Anytime you retain , the theoreti-

cal probability is that you’ve avoided a Type I error.

Although the theoretical probability of a Type I error equals , the actual probabil-

ity is slightly less than . This is because the region of rejection includes the critical

value. Yet to reject , the must be larger than . We cannot determine the pre-

cise area under the curve at , so we can’t remove it from our 5%. We can only say

that the region of rejection is slightly less than 5% of the curve. Therefore, the actual

probability of a Type I error is also slightly less than .

Thus, in our examples when we rejected , the probability that we made a

Type I error was slightly less than . That is why we report a significant result using

. This is code for “the probability of a Type I error is less than .” The reason

you must always report your alpha level is so that you indicate the probability of mak-

ing a Type I error.

.05p 6 .05

.05

␣

crit

obt

␣

.95

1 2 ␣

.05H

.05

␣ 5 .05

␮

␣

226 CHAPTER 10 / Introduction to Hypothesis Testing

On the other hand, we reported a nonsignificant result using . This commu-

nicates that we did not call this result significant because to do so would require a

region greater than 5% of the curve. But then the probability of a Type I error would be

greater than our of , and that’s unacceptable.

Typically, researchers do not use an larger than because then it is too likely that

they will make a Type I error. This may not sound like a big deal, but the next time you

fly in an airplane, consider that the designer’s belief that the wings will stay on may

actually be a Type I error: He’s been misled by sampling error into erroneously think-

ing the wings will stay on. A 5% chance of this is scary enough—we certainly don’t

want more than a 5% chance that the wings will fall off. In science, we are skeptical

and careful, so we want to be convinced that sampling error did not produce our results.

Having only a 5% chance that it did is reasonably convincing.

Type I errors are the reason a study must meet the assumptions of a statistical proce-

dure. If we violate the assumptions, then the true probability of a Type I error will be

larger than our (so it’s larger than we think it is). Thus, if we severely violate a pro-

cedure’s assumptions, we may think that is when in fact it is, say, ! But recall

that with parametric tests we can violate the assumptions somewhat. This is allowed

because the probability of a Type I error will still be close to (it will be only, say,

when we’ve set at ).

Sometimes making a Type I error is so dangerous that we want to reduce its proba-

bility even further. Then we usually set alpha at . For example, say that the smart pill

had some dangerous side effects. We would not want to needlessly expose the public to

such dangers, so would make us even less likely to conclude that the pill works

when it does not. When is , the region of rejection is the extreme 1% of the sam-

pling distribution, so the probability of making a Type I error is now .

However, we use the term significant in an all-or-nothing fashion: A result is not

“more” significant when than when . If lies in the region of rejec-

tion that was used to define significant, then the result is significant, period! The only

difference is that when the probability that we’ve made a Type I error is

smaller.

Finally, computer programs such as SPSS compute the exact probability of a Type I

error. For example, we might see . This indicates that the lies in the extreme

2% of the sampling distribution, and thus the probability of a Type I error here is . If

our is , then this result is significant. However, we might see , which indi-

cates that to call this result significant we’d need a region of rejection that is the

extreme 7% of the sampling distribution. This implies an of , which is greater than

, and thus this result is not significant.

REMEMBER When is true: Rejecting is a Type I error, and its probabil-

ity is ; retaining is avoiding a Type I error, and its probability is .

Type II Errors: Retaining H

When H

Is False

It is also possible to make a totally different kind of error. Sometimes the variables we

investigate really are related in nature, and so really is false. When in this situation,

if we obtain data that cause us to retain , then we make a Type II error. A Type II

error is defined as retaining when is false (and is true). In other words, here

we fail to identify that the independent variable really does work.

Thus, when our IQ sample mean of 99 caused us to retain and not claim the pill

worked, it’s possible that the pill did work and we made a Type II error. How could this

happen? Because the sample mean of 99 was so close to 100 (the without the pill)␮

1 2 ␣H

␣

.05

.07␣

p 5 .07.05␣

.02

obt

p 5 .02

␣ 5 .01

obt

␣ 5 .05␣ 5 .01

p 6 .01

.01␣

␣ 5 .01

.01

.050␣

.051␣

.20.05␣

␣

.05␣

p 7 .05

Errors in Statistical Decision Making 227

that the difference could easily be explained as sampling error, so we weren’t

convinced the pill worked. Or perhaps the pill would actually increase IQ greatly,

say to a of 105, but we obtained an unrepresentative sample of this. Either way, in a

Type II error, the sample mean is so close to the described by that we—and

our statistics—are fooled into concluding that the predicted relationship does not exist

when it really does.

Anytime we discuss Type II errors, it’s a given that is false and is true. That is,

we’re in the “Type II situation,” in which the predicted relationship does exist. If we

retain in this situation, then we’ve made a Type II error. If we reject in this situa-

tion, then we’ve avoided a Type II error: We’ve made the correct decision because we

concluded that the pill works and it does work.

We never know when we make a Type II error, but we can determine its probability.

The computations of this are beyond the introductory level, but you should know that the

symbol for the theoretical probability of a Type II error is , the Greek letter beta. When-

ever you retain , is the probability that you’ve made a Type II error. On the other

hand, is the probability of avoiding a Type II error. Thus, anytime you reject ,

the probability is that you’ve made the correct decision and rejected a false .

REMEMBER When is false: Retaining is a Type II error, and its proba-

bility is ; rejecting is avoiding a Type II error, and its probability is .

Comparing Type I and Type II Errors

You probably think that Type I and Type II errors are two of the most confusing inven-

tions ever devised. So, first recognie that if there’s a possibility you’ve made one type

of error, then there is no chance that you’ve made the other type of error. Remember: In

the Type I situation, is really true (the variables are not related in nature). In the

Type II situation, is really false (the variables are related in nature). You can’t be in

both situations simultaneously. Second, if you don’t make one type of error, then you

are not automatically making the other error because you might be making a correct

decision. Therefore, look at it this way: The type of error you can potentially make is

determined by your situation—what nature “says” about whether there is a relation-

ship. Then, whether you actually make the error depends on whether you agree or dis-

agree with nature.

Thus, four outcomes are possible in any study. Look at Table 10.1. As in the upper

row of the table, sometimes is really true: Then if we reject , we make a Type I

error (with a ). If we retain , we avoid a Type I error and make the correctH

p 5 ␣

1 2 ␤H

␤

1 2 ␤

␤H

␤

␮

Our Decision

We Reject H

We Retain H

Type I situation: We make a We are correct,

is true Type I error a voiding a Type I error

(no relationship exists)

The truth

about H

Type II situation: We are correct, We make a Type II error

is false avoiding a Type

(a relationship exists) II error

1 p 5 1 – ␤2

p 5 ␤2

1p 5 1 – ␣21p 5 ␣ 2

TABLE 10.1

Possible Results

of Rejecting

or Retaining H