Heiman G. Basic Statistics for the Behavioral Sciences

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

For Practice

In a study, is that . The data are 6, 7, 9, 8,

and 8.

1. To compute the , what two descriptive statistics

are computed first?

2. What do you compute next?

3. Compute the .t

obt

␮ 5 10H

Answers

1. and

obt

5 17.6 2 102>.51 524.71

5 7.6, s

5 1.30, N 5 5; s

5 11.3>5 5 .51;

Lower means

X X X X X X X X X X X X

Higher means

–3 –2 –1

0+3+2+1

Values of t

FIGURE 11.1

Example of a t-distribution of random sample means

The t-Distribution and Degrees of Freedom

In our housekeeping study, the sample mean produced a of on the sampling

distribution of means in which . The question is, “Is this significant?”

To answer this, the final step is to compare to , and for that we examine the

t-distribution.

Think of the t-distribution in the following way. One last time we hire our very bored

statistician. She infinitely draws samples having our from the raw score population

described by . For each sample she computes , but she also computes and, ulti-

mately, . Then she plots the usual sampling distribution—a frequency distribution of

the sample means—but also labels the axis using each . Thus, the t-distribution is

the distribution of all possible values of t computed for random sample means selected

from the raw score population described by . For our example, the t-distribution

essentially shows all sample means—and their corresponding values of —that occur

when men and women belong to the same population of housekeeping scores.

You can envision the t-distribution as in Figure 11.1. As with z-scores, increasing

positive values of are located farther to the right of , and increasing negative val-

ues of are located farther to the left of . If places our mean close to the center

of the distribution, then this mean is frequent and thus likely when is true. (So in

our example, our sample of men is likely to be representing the population where is

75, the same population as for women.) But, if places our sample mean far into a

tail of the sampling distribution, then this mean is infrequent and thus unlikely when

is true. (Our sample is unlikely to represent the population where is 75, so it is

unlikely that men and women have the same population of scores.) To determine if our

mean is far enough into a tail, we find and create the region of rejection.t

crit

␮H

obt

␮

obt

␮t

obt

␮t

obt

crit

obt

␮ 5 75

23.61t

obt

Performing the One-Sample t-Test 239

X X X X X X X X X X X X XX X

Critical value = +2.5

for Distribution A

Distribution A

Distribution B

–4 –3 –2 –1

0+2+1 +3 +4

Means

Values of t

Critical value = –2.5

for Distribution A

FIGURE 11.2

Comparison of two t-distributions based on different sample Ns

But we have one important novelty here: There are actually many versions of the

t-distribution, each having a slightly different shape. The shape of a particular distribu-

tion depends on the sample size that is used when creating it. If the statistician uses

small samples, the t-distribution will be only a rough approximation to the normal

curve. This is because small samples will often contain large sampling error, so often

each estimate of the population variability will be very different from the next and

from the true population variability. This inconsistency produces a t-distribution that is

only approximately normal. However, large samples are more representative of the

population, so each estimate of the population variability will be very close to the true

population variability. As we saw when computing the z-test, using the true population

variability produces a sampling distribution that forms a normal curve. In-

between, as sample size increases, each t-distribution will be a successively closer

approximation to the normal curve.

However, in this context, the size of a sample is determined by the quantity ,

what we call the degrees of freedom, or . Because we compute the estimated popula-

tion standard deviation using , it is our that determines how close we are to

the true population variability, and thus it is the that determines the shape of the

t-distribution. The larger the , the closer the t-distribution comes to forming a normal

curve. However, a tremendously large sample is not required to produce a perfect nor-

mal t-distribution. When is greater than 120, the t-distribution is virtually identical

to the standard normal curve. But when is between 1 and 120 (which is often the case

in research), a differently shaped t-distribution will occur for each .

The fact that t-distributions are differently shaped is important for one reason: Our

region of rejection should contain precisely that portion of the area under the curve

defined by our . If , then we want to mark off precisely 5% of the area under

the curve. On distributions that are shaped differently, we mark off that 5% at different

locations. Because the location of the region of rejection is marked off by the critical

value, with differently shaped t-distributions we will have different critical values. For

example, Figure 11.2 shows two t-distributions. Notice the size of the (blue) region of

rejection in a tail of Distribution A. Say that this corresponds to the extreme 5% of

Distribution A and is beyond the of . However, if we also use as

on Distribution B, the region of rejection is larger, containing more than 5% of the

crit

;2.5;2.5t

crit

␣ 5 .05␣

dfN 2 1

N 2 1

1σ

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

distribution. Conversely, the marking off 5% of Distribution B will mark off less

than 5% of Distribution A. (The same problem exists for a one-tailed test.)

This issue is important because is not only the size of the region of rejection, but it is

also the probability of a Type I error. Unless we use the appropriate , the actual proba-

bility of a Type I error will not equal our and that’s not supposed to happen! Thus, there

is only one version of the t-distribution to use when testing a particular : the one that

the bored statistician would create by using the same as in our sample. Therefore, we

are no longer automatically using the critical values of 1.96 or 1.645. Instead, when your

is between 1 and 120, use the to first identify the appropriate sampling distribution

for your study. The on that distribution will accurately mark off the region of rejec-

tion so that the probability of a Type I error equals your . Thus, in the housekeeping

study with an of 9, we will use the from the t-distribution for . In a different

study, however, where might be 25, we would use the different from the t-distribu-

tion for . And so on.

REMEMBER The appropriate for the one-sample t-test comes from the

t-distribution that has equal to , where is the number of scores in

the sample.

Using the t-Tables

We obtain the different values of from Table 2 in Appendix C, entitled “Critical

Values of t.” In these “t-tables,” you’ll find separate tables for two-tailed and one-tailed

tests. Table 11.2 contains a portion of the two-tailed table.

To find the appropriate , first locate the appropriate column for your (either

or ). Then find the value of in the row at the for your sample. For example, in

the housekeeping study, is 9, so is . For a two-tailed test with

and , is 2.306.

Here’s another example: In a different study, is 61. Therefore, the

Look in Table 2 of Appendix C to find the two-tailed with . It is 2.000. The

one-tailed here is 1.671.

The table contains no positive or negative signs. In a two-tailed test, you add the

“ ,” and, in a one-tailed test, you supply the appropriate “ ” or “ .” Also, the table

uses the symbol for infinity for greater than 120. With this , using the esti-

mated population standard deviation is virtually the same as using the true population

standard deviation. Therefore, the t-distribution matches the standard normal curve,

and the critical values are those of the z-test.

Interpreting the t-Test

Once you’ve calculated and identified , you can make a

decision about your results. In our housekeeping study, we must

decide whether or not the men’s mean of 65.67 represents the same

population of scores that women have. Our is , and is

, producing the sampling distribution in Figure 11.3.

Remember, this can be interpreted as showing the frequency of all

means that occur by chance when is true. Essentially, here the

distribution shows all sample means that occur when the men’s

population of scores is the same as the women’s population, with a

of 75. Our lies beyond , so the results are significant: Ourt

crit

obt

␮

;2.306

crit

23.61t

obt

crit

obt

dfdf1q 2

21;

crit

␣ 5 .05t

crit

df 5 N 2 1 5 60.N

crit

df 5 8

␣ 5 .05N 2 1 5 8dfN

dft

crit

.01

.05␣t

crit

NN 2 1df

crit

df 5 24

crit

df 5 8t

crit

␣

crit

dfdf

obt

␣

crit

␣

crit

Alpha Level

df a .05 a .01

1 12.706 63.657

2 4.303 9.925

3 3.182 5.841

4 2.776 4.604

5 2.571 4.032

6 2.447 3.707

7 2.365 3.499

8 2.306 3.355

TABLE 11.2

A Portion of the t-Tables

Performing the One-Sample t-Test 241

obt

= –3.61

X X X X X X X X X X X XXX

–3 –2 –1

0+3+2+1

crit

= +2.306t

crit

= –2.306

X = 65.67

Means

Values of t

FIGURE 11.3

Two-tailed t-distribution

for df 8 when H

true and 75␮ 5

sample mean is so unlikely to occur if it had been representing the population in which

is 75, that we reject the that it represents this population.

We interpret significant results using the same rules as discussed in the previous

chapter. Thus, although we consider whether we’ve made a Type I error, with a sample

mean of 65.67, our best estimate is that the for men is around 65.67. Because we

expect a different population for women located at 75, we conclude that the results

demonstrate a relationship in the population between gender and test scores. Then we

return to being a researcher and interpret the relationship in psychological or sociologi-

cal terms: What do the scores and relationship indicate about the underlying behaviors

and their causes? Are men really more ignorant about housekeeping than women, and

if so, why? Do men merely pretend to be ignorant, and if so, why? And so on.

Conversely, if was not beyond (for example, if ), it would not be

significant. Then we would have no evidence for a relationship between gender and test

scores, one way or the other. We would, however, consider whether we had sufficient

power so that we were not making a Type II error.

The One-Tailed t-Test

As usual, we perform one-tailed tests when we predict the direction of the difference

between our conditions. Thus, if we had predicted that men score higher than women

would be that the sample represents a population with greater than 75

. would be that is less than or equal to 75 . We then

examine the sampling distribution that occurs when (as we did in the two-

tailed test). We find the one-tailed from the t-tables for our and . To decide in

which tail of the sampling distribution to put the region of rejection, we determine

what’s needed to support . Here, for the sample to represent a population of higher

scores, the must be greater than 75 and be significant. As shown in the left-hand

graph in Figure 11.4, such means are in the upper tail, which is where we place the

region of rejection, and is positive.

However, predicting that men score lower than women would produce the sampling

distribution on the right in Figure 11.4. Now is that is less than 75, and is that

is greater than or equal to 75. Because we seek a that is significant and lower than

75, the region of rejection is in the lower tail, and is negative.

In either case, calculate using the previous formulas. If the absolute value of

is larger than and has the same sign, then the is unlikely to be representing a

described by . Therefore, reject , accept , and the results are significant.H

␮Xt

crit

obt

crit

X␮

␮H

crit

␣dft

crit

␮ 5 75

: ␮ # 752␮H

: ␮ 7 752

␮H

obt

511.32t

crit

obt

␮

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

X X X XX X X X X

.05

–t –t –t –t

crit

+t +t +t

: μ ≤ 75

: μ > 75

X X X XX X X X X

.05

–t –t –t –t

⫺t

crit

+t +t +t

: μ ≥ 75

: μ < 75

FIGURE 11.4

sampling distributions of t for one-tailed tests

■

Perform the one-sample t-test when is

unknown.

MORE EXAMPLES

In a study, is 40. We predict our condition will

change scores relative to this . . This is a

two-tailed test, so ; . Then

compute .

The scores produce and .s

5 196X 5 46

obt

: ␮ ? 40H

: ␮ 5 40

N 5 25␮

␮

Find : With and , .

The lies beyond the . Conclusion: The inde-

pendent variable significantly increases scores from a

of 40 to a around 46.␮␮

crit

obt

crit

5 ;2.064df 5 24␣ 5 .05t

crit

obt

X 2 ␮

46 2 40

2.80

512.14

196

5 2.80

A QUICK REVIEW

Some Help When Using the t-Tables

In the t-tables, you will not find a critical value for every between 1 and 120. When

the of your sample does not appear in the table, you can take one of two approaches.

First, remember that all you need to know is whether is beyond , but you do

not need to know how far beyond it is. Often you can determine this by examining the

critical values given in the table for the immediately above and below your . For

example, say that we perform a one-tailed t-test at with 49 . The t-tables give

a of 1.684 for 40 and a of 1.671 for 60 . Arrange the values like this:

60 49 40

1.671 ? 1.684

nonsignificant less than 1.671 greater than 1.684 significant

Because 49 lies between 40 and 60 , our unknown is a number larger than

1.671 but smaller than 1.684. Therefore, any that is beyond 1.684 is already beyond

the smaller unknown , and so it is significant. On the other hand, any that is not

beyond 1.671 will not be beyond the larger unknown , and so it is not significant.

The second approach is used when falls between the two critical values given in

the tables. Then you must compute the exact by performing “linear interpolation,”

as described in Appendix A.2.

crit

obt

crit

obt

crit

obt

crit

dfdfdf

crit

dft

crit

dft

crit

df␣ 5 .05

dfdf

crit

obt

(continued)

Estimating by Computing a Confidence Interval 243␮

ESTIMATING ␮ BY COMPUTING A CONFIDENCE INTERVAL

As you’ve seen, after rejecting , we estimate the population that the sample mean

represents. There are two ways to estimate .

The first way is point estimation, in which we describe a point on the variable at

which the is expected to fall. Earlier we estimated that the of the population of men

is located on the variable of housekeeping scores at the point identified as 65.67. How-

ever, if we actually tested the entire population, would probably not be exactly 65.67.

The problem with point estimation is that it is extremely vulnerable to sampling error.

Our sample probably does not perfectly represent the population of men, so we can say

only that the is around 65.67.

The other, better way to estimate is to include the possibility of sampling error and

perform interval estimation. With interval estimation, we specify a range of values

within which we expect the population parameter to fall. You often encounter such

intervals in real life, although they are usually phrased in terms of “plus or minus”

some amount (called the margin of error). For example, the evening news may report

that a sample survey showed that 45% of the voters support the president, with a mar-

gin of error of plus or minus 3%. This means that the pollsters expect that, if they asked

the entire population, the result would be within of 45%: They believe that the

true portion of the population that supports the president is inside the interval that is

between 42% and 48%.

We will perform interval estimation in a similar way by creating a confidence inter-

val. Confidence intervals can be used to describe various population parameters, but

the most common is for a single . The confidence interval for a single describes a

range of values of , one of which our sample mean is likely to represent. Thus, when

we say that our sample of men represents a around 65.67, a confidence interval is the

way to define around. To do so, we’ll identify those values of above and below 65.67

that the sample mean is likely to represent, as shown here:

values of , one of which is likely to be

represented by our sample mean

The is the lowest value of that our sample mean is likely to represent, and

is the highest value of that the mean is likely to represent. When we compute these

two values, we will have the confidence interval.

␮

high

␮␮

low

␮

low

. . . ␮ ␮ ␮ ␮ 65.67 ␮ ␮ ␮ ␮ . . . ␮

high

␮

␮␮

;3%

␮

␮␮

␮

␮H

For Practice

We test if artificial sunlight during the winter months

lowers one’s depression. Without the light, a depres-

sion test has . With the light, our sample scored

4, 5, 6, 7, and 8.

1. What are the hypotheses?

2. Compute .

3. What is ?

4. What is the conclusion?

crit

obt

␮ 5 8

Answers

1. To “lower” is a one-tailed test: ; .

3. With and .

4. is beyond . Conclusion: Artificial sunlight signif-

icantly lowers depression scores from a of 8 to a

around 6.

␮␮

crit

obt

df 5 4, t

crit

522.132␣ 5 .05

obt

5 16 2 82>.707 522.83

5 6; s

5 2.5; s

5 12.5>5 5 .707;

: ␮ $ 8H

: ␮ 6 8

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

When is a sample mean likely to represent a particular ? It depends on sampling

error. For example, intuitively we know that sampling error is unlikely to produce a

sample mean of 65.67 if is, say, 500. In other words, 65.67 is significantly different

from 500. But sampling error is likely to produce a sample mean of 65.67 if, for exam-

ple, is 65 or 66. In other words, 65.67 is not significantly different from these .

Thus, a sample mean is likely to represent any that the mean is not significantly dif-

ferent from. The logic behind a confidence interval is to compute the highest and low-

est values of that are not significantly different from the sample mean. All

between these two values are also not significantly different from the sample mean, so

the mean is likely to represent one of them.

REMEMBER A confidence interval describes the highest and lowest values

of that are not significantly different from our sample mean, and so the

mean is likely to represent one of them.

We usually compute a confidence interval only after finding a significant . This is

because we must be sure that our sample is not representing the described in

before we estimate any other that it might represent.

Computing the Confidence Interval

The t-test forms the basis for the confidence interval, and here’s what’s behind the for-

mula for it. We seek the highest and lowest values of that are not significantly differ-

ent from the sample mean. The most that can differ from a sample mean and still not

be significant is when equals . We can state this using the formula for the t-test:

To find the largest and smallest values of that do not differ significantly from our

sample mean, we determine the values of that we can put into this formula along with

our and . Because we are describing the above and below the sample mean, we

use the two-tailed value of . Then by rearranging the above formula, we create the

formula for finding the value of to put in the t-test so that the answer equals .

We also rearrange this formula to find the value of to put in so that the answer equals

. Our sample mean represents a between these two , so we combine these

rearranged formulas to produce:

␮s␮1t

crit

␮

crit

␮

crit

␮s

␮

crit

X 2 ␮

crit

obt

␮

obt

␮

␮s␮

␮

␮s␮

␮

The symbol stands for the unknown value represented by the sample mean. The

and are computed from your data. Find the two-tailed value of in the t-tables at

your for , where is the sample .NNdf 5 N 2 1␣

crit

X␮

The formula for the confidence interval for a single is

212t

crit

21 X # ␮ # 1s

211t

crit

21 X

␮

Estimating by Computing a Confidence Interval 245␮

REMEMBER

Use the two-tailed critical value when computing a confidence

interval, even if you performed a one-tailed t-test.

For our housekeeping study, the and . The two-tailed for

and is . Filling in the formula, we have

After multiplying 2.582 times and times , we have

The formula at this point tells us that our mean represents a of 65.67, plus or minus

5.954.

After adding to 65.67, we have

This is the finished confidence interval. Returning to our previous diagram, we replace

the symbols and with the numbers 59.72 and 71.62, respectively.

values of , one of which is likely to be

represented by our sample mean

As shown, our sample mean probably represents a around 65.67, meaning that is

greater than or equal to 59.72, but less than or equal to 71.62.

Because we created this interval using the for an of , there is a 5% chance

that we are in error and the being represented by our mean is outside of this interval.

On the other hand, there is a 95% chance that the being represented

is within this interval. Therefore, we have created what is called the 95% confidence

interval: We are 95% confident that the interval between 59.72 and 71.62 contains the

represented by our sample mean.

For greater confidence, we could have used the for . Then we would cre-

ate the 99% confidence interval of . Notice, however, that greater

confidence comes at the cost of less precision: This interval spans a wider range of val-

ues than did the 95% interval, so we have less precisely identified the value of . Usu-

ally, researchers compromise between precision and confidence by creating the 95%

confidence interval.

Thus, we conclude our one-sample t-test by saying, with 95% confidence, that our

sample of men represents a between 59.72 and 71.62. Because the center of the inter-

val is at 65.67, we still communicate that is around 65.67, but we have much more

information than if we merely said that is somewhere around 65.67. In fact, there-

fore, you should compute a confidence interval anytime you are describing the repre-

sented by the mean of a condition in any significant experiment.

␮

57.01 # ␮ # 74.33

␣ 5 .01t

crit

␮

␮311 2 ␣2110024

␮

.05␣t

crit

␮␮

␮

59.72 . . . ␮ ␮ ␮ ␮ 65.67 ␮ ␮ ␮ ␮ . . . 71.62

␮

high

␮

low

59.72 # ␮ # 71.62

;5.954

␮

25.954 1 65.67 # ␮ #15.954 1 65.67

12.30622.306

12.5822122.30621 65.67 # ␮ # 12.5822112.30621 65.67

;2.306␣ 5 .05df 5 8

crit

5 2.582X 5 65.67

You can also compute a confidence interval when performing the z-test. Use the formula above, except use

the critical values of z. If , then . If , then .z

crit

5 ;2.575␣ 5 .01z

crit

5 ;1.96␣ 5 .05

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

■

A confidence interval for provides a range of ,

any one of which our is likely to represent.

MORE EXAMPLES

A is significant with , , and,

. To compute the 95% confidence interval,

, so the two-tailed . Then,

For Practice

1. What does this 95% confidence indicate:

?15 # ␮ # 20

40.16 # ␮ # 59.84

129.83721 50 # ␮ # 119.83721 50

14.72122.09321 50 # ␮ # 14.72112.09321 50

212t

crit

21 X # ␮ # 1s

211t

crit

21 X

crit

5 ;2.093df 5 19

5 4.7

N 5 20X 5 50t

obt

␮s␮

2. With , you perform a one-tailed test

What is for computing the

confidence interval?

3. The is significant when , , and

. Compute the 95% confidence interval.

Answers

1. We are 95% confident that our represents a between

15 and 20.

2. With , the two-tailed

28.07 # ␮ # 41.93

13.332122.08021 35 # ␮ # 13.332112.08021 35 5

crit

5 ;2.080df 5 21

␮X

N 5 22

5 3.33X 5 35t

obt

crit

1␣ 5 .05.2

N 5 22

A QUICK REVIEW

Summary of the One-Sample t-Test

All of the preceding boils down to the following steps for the t-test.

1. Check that the experiment meets the assumptions.

2. Create the hypotheses: Create either the two-tailed or one-tailed and .

3. Compute : From the sample data, compute , compute , and then

compute .

4. Find the appropriate : Use the that equal .

5. If is beyond : Reject , the results are significant, and so interpret the

relationship “psychologically.”

6. If is not beyond : the results are not significant. Draw no conclusion.

7. Compute the confidence interval: For significant results, use the two-tailed

to describe the represented by your .

STATISTICS IN PUBLISHED RESEARCH: REPORTING THE t-TEST

Report the results of a t-test in the same way that you reported the z-test, but also include

the . For our housekeeping study, we had 8 , the was , and it was signifi-

cant. We always include the descriptive statistics too, so in a report you might read: “the

national average for women is 75, although this sample of men scored lower

. This difference was significant, with ”

First, note the in parentheses. Also note that the results are significant because the

probability is less than our alpha of that we are making a Type I error. (Had these

results not been significant, we’d report )

In the t-tables, when is , the is , so our of would also

be significant if we had used the level. Therefore, instead of saying , we

would provide more information by reporting that , because this communicatesp 6 .01

p 6 .05.01

23.47t

obt

;3.355t

crit

.01␣

t182523.47, p 7 .05.

.05

t182523.47, p 6 .05.1M 5 65.67, SD 5 7.752

23.47t

obt

dfdf

X␮

crit

obt

crit

obt

N 2 1dft

crit

obt

Significance Tests for Correlation Coefficients 247

that the probability of a Type I error is not in the neighborhood of . For

this reason, researchers usually report the smallest values of alpha at which a result is

significant.

Usually, confidence intervals are reported in sentence form (and not symbols), but

we always indicate the confidence level used. Thus, in a report you might see: “The

95% confidence interval for this mean was between 59.72 and 71.62.”

SIGNIFICANCE TESTS FOR CORRELATION COEFFICIENTS

It’s time to shift mental gears and consider that other type of one-sample study—a cor-

relational study. Here’s a new example: We examine the relationship between a man’s

age and his housekeeping score in a correlational design. We measure the test scores

and the ages of a sample of 25 men and determine that the Pearson correlation coeffi-

cient is appropriate. Say that, using the formula from Chapter 7, we compute an

, indicating that the older a man is, the lower his housekeeping score is.

Although this correlation coefficient describes the relationship in the sample, ultimately,

we want to describe the relationship in the population. That is, we seek the correlation

coefficient that would be produced if we could measure everyone’s and scores in the

population. Of course, we cannot do that, so instead we use the sample coefficient to esti-

mate the correlation that we’d expect to find if we could measure the entire population.

Recall that symbols for population parameters involve the Greek alphabet, so we

need a new symbol:

The symbol for the Pearson correlation coefficient in the population is the

Greek letter called “rho,” which looks like this: .

We interpret in the same way as : It is a number between 0 and , indicating

either a positive or a negative linear relationship in the population. The larger the

absolute value of , the stronger the relationship: The more that one value of is asso-

ciated with each , the more closely the scatterplot for the population hugs the regres-

sion line, and the better we can predict unknown scores by using scores.

Thus, using the sample in our study of , we might estimate that would

equal if we measured the ages and housekeeping scores of the entire population

of men.

But hold on, there’s a problem here: That’s right, sampling error. The problem of

sampling error applies to all statistics. Here the idea is that, because of the luck of the

draw of who was selected for the sample, their scores happened to produce this corre-

lation. But, in the population—in nature—there is no relationship, and so is really

zero. So, here we go again! For any correlation coefficient you compute, you must

determine whether it is significant.

REMEMBER Never accept that a sample correlation coefficient reflects a real

relationship in nature unless it is significant.

Testing the Pearson r

As usual, the first step is to make sure that a study meets the assumptions of the statis-

tical procedure. The Pearson correlation coefficient has three assumptions:

1. We have a random sample of pairs of and scores, and each variable is an inter-

val or ratio variable.

␳

2.45

␳2.45r

Y␳

;1.0r␳

r 52.45

.04, .03, or .02