Heiman G. Basic Statistics for the Behavioral Sciences

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

Test scores

Age

Sample

scatterplots

FIGURE 11.5

Scatterplot of a popula-

tion for which 0, as

described by H

Any r is a result of

sampling error when

selecting a sample from

this scatterplot.

␳ 5

2. The and scores each represent a normal distribution. Further, they represent a

bivariate normal distribution. This means that the scores at each form a nor-

mal distribution and the scores at each form a normal distribution. (If is

larger than 25, however, violating this assumption is of little consequence.)

3. The null hypothesis is that in the population there is zero correlation. This is the

most common approach and the one that we’ll use. (However, you can also test

the that your sample represents a nonzero . Consult an advanced statistics

book for the details.)

Our housekeeping and age scores meet these assumptions, so we set at and test

. First, we create the statistical hypotheses. You can perform either a one- or a two-

tailed test. Use a two-tailed test if you do not predict the direction of the relationship.

For example, let’s say that we are unsure whether men produce higher or lower scores

as they age. This is a two-tailed test because we’re predicting either a positive or a neg-

ative correlation. For our alternative hypothesis, if the correlation in the population is

either positive or negative, then does not equal zero. Therefore we have

On the other hand, the null hypothesis is always that the predicted relationship does

not exist, so here it says that the correlation in the population is zero. Thus,

This implies that if does not equal zero, it’s because of sampling error. You can under-

stand this by looking at Figure 11.5. It shows the scatterplot in the population that

says we would find: There is no relationship here, so equals 0. Recall, however, that a

slanting elliptical scatterplot reflects an that is not equal to zero. Thus, implies

that, by chance, we selected an elliptical sample scatterplot from this population plot.

Therefore, it says, although age and housekeeping scores are not really related, the

scores in our sample happen to pair up so that it looks like they’re related. Conversely,

implies that the population’s scatterplot would not look like Figure 11.5, but rather

it would be similar to our sample’s scatterplot.

␳

: ␳ 5 0

: ␳ ? 0

␳

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␳H

NYX

Significance Tests for Correlation Coefficients 249

–1.0 ...

–r

... +1.0

Values of r

–r –r –r –r –r –r –r +r +r +r +r +r +r +r +r

FIGURE 11.6

Sampling distribution of r

It is an approximately normal

distribution, with values of r

plotted along the X axis.

The Sampling Distribution of r We test by determining the likelihood of

obtaining our sample from the population where is zero. To do so, we envision the

sampling distribution of .

The bored statistician quit! By now, however, you could create the sampling distri-

bution yourself: Using the same as in our study, you would select an infinite number

of samples of pairs from the population where (as if you pulled each sam-

ple from the scatterplot in Figure 11.5). Each time, you would compute . If you then

plotted the frequency of the various values of , you would have the sampling distribu-

tion of . The sampling distribution of r is a frequency distribution showing all possi-

ble values of that occur by chance when samples are drawn from a population in

which is zero. Such a distribution is shown in Figure 11.6.

The only novelty here is that along the axis are now different values of . When

, the most frequent sample is also 0, so the mean of the sampling distribution—

the average —is 0. Because of sampling error, however, sometimes we’ll obtain a pos-

itive and sometimes a negative . Most often the will be relatively small and close

to 0. But, less frequently, we’ll obtain a larger that falls into a tail of the distribution.

Thus, the larger the (whether positive or negative), the less likely it is to occur when

the sample actually represents a population in which

To test , we determine where our lies on this distribution. To do so, we could per-

form a variation of the t-test, but luckily that is not necessary. Instead, directly com-

municates its location on the sampling distribution. The mean of the sampling

distribution is always zero, so, for example, our of is a distance of below the

mean. Therefore, we test simply by examining our obtained , which is . To

determine whether is in the region of rejection, we compare it to .

As with the t-distribution, the shape of the sampling distribution of is slightly dif-

ferent for each , so there is a different value of for each . But, here’s a new one:

With the Pearson correlation coefficient, the degrees of freedom equals , where

is the number of pairs of scores in the sample.

REMEMBER For the Pearson , the degrees of freedom equals , where

is the number of pairs of scores.

Table 3 in Appendix C gives the critical values of the Pearson correlation coefficient.

Use these “r-tables” in the same way that you’ve used the t-tables: Find for either a

one- or a two-tailed test at the appropriate and . For the housekeeping correlation,

was 25, so , and, for a two-tailed test with , is . We set up;.396r

crit

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

–1.0 ...

0.0

... +1.0

Values of r +r+r+r+r+r+r

–r –r –r –r –r –r

obt

= –.45

–r

crit

= –.396

crit

= +.396

FIGURE 11.7

sampling distribution

of r

For the two-tailed test, there

is a region of rejection for

positive values of and for

negative values of .r

obt

the sampling distribution in Figure 11.7. Our of is beyond the of ,

so it is in the region of rejection. As usual, this means that the results are significant:

This is so unlikely to occur if we had been representing the population where is 0,

that we reject the that we were representing this population. We conclude that the

is “significantly different from 0.”

Interpreting r The rules for interpreting a significant result here are the same as

before. Remember that rejecting does not prove anything. In particular, this was a

correlational study, so we have not proven that changes in age cause test scores to

change. In fact, we have not even proven that the relationship exists because we may

have made a Type I error. Here, a Type I error is rejecting the that there is zero cor-

relation in the population, when in fact there is zero correlation in the population. As

usual, though, with , the probability that we have made a Type I error is slightly

less than .

Report the Pearson correlation coefficient using the same format as with previous

statistics. Our of was significant with 23 , so in a published report we’d say

that , . As usual, our is in parentheses and because

and the coefficient is significant, we indicate that the probability is less than that

we’ve made a Type I error.

Because the sample is , our best estimate is that, in this population, equals

. However, recognizing that the sample may contain sampling error, we expect

that is probably around . (We could more precisely describe by computing a

confidence interval. However, this is computed using a very different procedure from

the one discussed previously.)

In Chapter 8, you saw that we further describe a relationship by computing the

regression equation and the proportion of variance accounted for. However, do this only

when is significant! Only then are we confident that we’re describing a “real” rela-

tionship. Thus, for the housekeeping study, we would now compute the linear regres-

sion equation for predicting test scores if we know a man’s age. We would also

compute , which is or . Recall, this is the proportion of variance in

scores that is accounted for by the relationship with . Here, an of tells us that

we are 20% more accurate when we use the relationship with age to predict housekeep-

ing scores than when we do not use the relationship.

Remember that it is and not “significance” that determines how important a

relationship is. Significant indicates only that the sample relationship is unlikely to

be a fluke of chance. The indicates the importance of a relationship because it indi-

cates the extent to which knowing participants’ scores improves our accuracy in

predicting and understanding differences in their scores. This allows us to understandY

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Significance Tests for Correlation Coefficients 251

–1.0 ...

... +1.0

+r+r +r +r

crit

Predicting positive

correlation

: ρ ≤ 0

: ρ > 0

–r –r –r –r

–1.0 ...

... +1.0

+r+r +r +r

–r

crit

Predicting negative

correlation

: ρ ≥ 0

: ρ < 0

–r –r –r –r

FIGURE 11.8

sampling distribution of r where 0 for one-tailed test␳ 5

differences in their behavior, and so the larger the , the greater is the importance of

the relationship.

Thus, a relationship must be significant to be even potentially important (because it

must first be believable). But, a significant relationship is not necessarily important.

For example, an of might be significant, but it is not statistically important:

is only , so this relationship accounts for only 1% of the variance and so is

virtually useless in explaining differences in scores. (We’re only 1% better off with it

than without it.) Thus, although this relationship is unlikely to occur through sampling

error, it is also an unimportant relationship.

After describing the relationship, as usual the final step is to interpret it in terms of

behaviors. For example, perhaps our correlation coefficient reflects socialization

processes, with older men scoring lower on the housekeeping test because they come

from generations in which wives typically did the housekeeping, while men were the

“breadwinners.”

Of course, if does not lie beyond , then you would retain and conclude that

the correlation is not significant. In this case, make no claims about the relationship that

may or may not exist, and do not compute the regression equation or .

One-Tailed Tests of r If we had predicted only a positive correlation or only a neg-

ative correlation, then we would have performed a one-tailed test. When we predict a

positive relationship, we are predicting a positive (a number greater than 0) so our

alternative hypothesis is . The says that we’re wrong (that is 0 or less

than 0), so . On the other hand, when we predict a negative relationship, we

are predicting a negative (a number less than 0) so we have . The says

we’re wrong (that is 0 or greater than 0) so .

We test each by again testing whether the sample represents a population in which

there is zero relationship—so again we examine the sampling distribution for .

From the -tables in Appendix C, find the one-tailed critical value for and . Then set

up one of the sampling distributions shown in Figure 11.8. When predicting a positive

correlation, use the left-hand distribution: is significant if it is positive and falls

beyond the positive . When predicting a negative correlation, use the right-hand

distribution: is significant if it is negative and falls beyond the negative .r

crit

obt

crit

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

–1.0 ...

–r

... +1.0

–r

Critical r

= +.600 Obtained r

= +.85

FIGURE 11.9

One-tailed H

sampling

distribution of values

of r

when H

is ␳

5 0

Testing the Spearman

We also use the above logic when testing the Spearman correlation coefficient .

Recall that describes the linear relationship in a sample when and are both

ordinal (ranked) scores. Again our ultimate goal is to use the sample coefficient to

estimate the correlation coefficient we would see if we could measure everyone in the

population.

The symbol for the Spearman correlation coefficient in the population is ␳

However, before we can use to estimate , we must first deal with the usual prob-

lem: That’s right, maybe our merely reflects sampling error. Perhaps if we measured

the population, we’d find that is 0. Therefore, before we can conclude that the corre-

lation reflects a relationship in nature, we must perform hypothesis testing. So, we

1. Set alpha: how about ?

2. Consider the assumptions of the test: The requires a random sample of pairs of

ranked (ordinal) scores. (Note: Because of the data involved and the lack of para-

metric assumptions, is technically a nonparametric procedure.)

3. Create the statistical hypotheses: You can test the one- or two-tailed hypotheses

that we saw previously with , except now use the symbol .

The only new aspect of testing is the sampling distribution. The sampling distri-

bution of is a frequency distribution showing all possible values of that occur

when samples are drawn from a population in which is zero. This creates a new fam-

ily of sampling distributions and a different table of critical values. Table 4 in Appen-

dix C, entitled “Critical Values of the Spearman Rank-Order Correlation Coefficient,”

contains the critical values for one- and two-tailed tests of . Obtain critical values as

in previous tables, except here use , not degrees of freedom. (Note the instructions at

the top of this table for when your is not listed.)

REMEMBER The critical value of is obtained using , the number of pairs

of scores in the sample.

Here’s an example. In Chapter 7, we correlated the aggressiveness rankings given to

nine children by two observers and found that . We had assumed that the

observers’ rankings would agree, predicting a positive correlation. Therefore, we have

a one-tailed test with the hypotheses and . From Table 4 in

Appendix C, with and , the one-tailed critical value is . This pro-

duces the sampling distribution shown in Figure 11.9.

1.600N 5 9␣ 5 .05

: ␳

7 0H

: ␳

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51.85

␳

.05

␳

YXr

Maximizing the Power of Statistical Tests 253

■

Always perform hypothesis testing on a correlation

coefficient so that we are confident it is not due to

sampling error.

MORE EXAMPLES

We compute an . We predicted

some kind of relationship, so ; .

With and , the two-tailed

. The is beyond , so it is signifi-

cant: We expect the population correlation coefficient

to be around .

We compute an , using . With

, the two-tailed critical value is . The

is not in the region of rejection and is not significant.

;.409␣ 5 .05

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crit

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crit

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df 5 42 2 2 5 40␣ 5 .05

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r 51.32 1N 5 422

For Practice

We predict a negative relationship and obtain

1. What are and ?

2. With and , what is ?

3. What is the conclusion about ?

4. What is the conclusion about the relationship in

the population and ?

Answers

3. Not significant

4. Make no conclusion and do not compute .r

df 5 8, r

crit

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: ␳ $ 0; H

: ␳ 6 0

obt

crit

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52.44

A QUICK REVIEW

Because an of is beyond the critical value of , we reject . Thus, our

is significantly different from zero, and we estimate that in the population of such

rankings is around . In a publication, this would be reported as ,

. Note that the of the sample is given in parentheses. We would also compute

the squared to determine the proportion of variance accounted for. Then interpret

these results as we have done previously. (With different predictions, we might have

performed the other one-tailed test or a two-tailed test.)

Summary of Testing a Correlation Coefficient

All of the preceding boil down to the following steps:

1. Check the assumptions of or .

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

3. Compute the correlation coefficient.

4. Obtain the critical value from Appendix C: The critical value for is in Table 3,

using . The critical value for is in Table 4, using .

5. Compare the obtained to the critical value: If the obtained coefficient is beyond

the critical value, the results are significant. If the coefficient is not beyond the

critical value, the results are not significant.

6. For significant results, compute the proportion of variance accounted for by

squaring the obtained coefficient. For , compute the linear regression equation.r

df 5 Nr

df 5 N 2 2

Np 6 .05

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␳

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MAXIMIZING THE POWER OF STATISTICAL TESTS

The t-test is our first design in which we—the researcher—have some control over all

components of our data, because we are not stuck with a given . Therefore, it is

appropriate to revisit the topic of power, so that you can understand how researchers

use this control to increase the power of a study. Recall that power is the probability of

not committing a Type II error. We’re talking about those times when really is false,H

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

and the error would be to retain . Instead, we should reject , correctly concluding

that the predicted relationship exists in nature. Essentially, power is the probability that

we will not miss a relationship that really exists in nature. We maximize power by

doing everything we can to reject so that we don’t miss the relationship. If we still

end up retaining , we can be confident that we did not do so incorrectly and miss a

relationship that exists, but rather that the relationship does not exist.

We maximize power by the way that we design an experiment or correlational

study.

We’re talking about those times when we should reject the null hypothesis, so

maximizing power boils down to maximizing the probability that our results will be

significant. This translates into designing the study to maximize the size of our

obtained statistic relative to the critical value, so that the obtained will be significant.

For the one-sample t-test, three aspects of the design produce a relatively larger

and thus increase power. (These also apply to other types of experiments that we will

discuss.) Look at the formulas:

First, larger differences produced by changing the independent variable increase

power. In the housekeeping study, the greater the difference between the sample mean

for men and the for women, the greater the power. Logically, the greater the differ-

ence between men and women, the less likely we are to miss that a difference exists.

Statistically, in the formula this translates to a larger difference between and that

produces a larger numerator, which results in a larger that is more likely to be sig-

nificant. Therefore, when designing any experiment, the rule is to select conditions that

are substantially different from one another, so that we produce a big difference in

dependent scores between the conditions.

Second, smaller variability in the raw scores increases power. Recall that variability

refers to the differences among the scores. Logically, smaller variability indicates more

consistent behavior and a more consistent, stronger relationship. This makes a clearer

pattern that we are less likely to miss. Statistically, in the formula, smaller variability pro-

duces a smaller estimated variance , which produces a smaller standard error .

Then in the t-test, dividing by a smaller denominator produces a larger . We will see

smaller variability in scores the more that all participants experience the study in the same

way. Therefore, the rule is to conduct any study in a consistent way that minimizes the

variability of scores within each condition.

Third, a larger increases power. Logically, a larger provides a more accurate

representation of the population, so we are less likely to make any type of error. Statis-

tically, dividing by a larger produces a smaller , which results in a larger .

Also, a larger produces larger , which produces a smaller . Then our is more

likely to be significant. Therefore, the rule is to design any experiment with the largest

practical . However, this is for small samples. Generally, an of 30 per condition is

needed for minimal power, and increasing up to 121 adds substantially to it. How-

ever, an of, say, 500 is not substantially more powerful than an of, say, 450.

REMEMBER Increase power in an experiment by maximizing differences in

dependent scores between conditions, minimizing differences among scores

within conditions, and testing a larger .N

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More advanced textbooks contain procedures for determining the amount of power that is present in a given

study.

In one sense, I hope that you found this chapter rather boring—not because it is bor-

ing, but because, for each statistic, we performed virtually the same operations. In

testing any statistic, we ultimately do and say the same things. In all cases, if the

obtained statistic is out there far enough in the sampling distribution, it is too

unlikely for us to accept as representing the situation, so we reject . Any

implies that the sample does not represent the predicted relationship, so rejecting

increases our confidence that the data do represent the predicted relationship. We’re

especially confident because the probability is less than that we’ve made an error

in this decision. If we fail to reject , then hopefully we have sufficient power, so

that we’re unlikely to have made an error here, too. These are the fundamentals of all

inferential statistics.

Using the SPSS Appendix As described in Appendix B.5, SPSS will compute the

one-sample . It also indicates the smallest two-tailed region of rejection (and alpha

level) for which your is significant. Further, it computes the and for the sample

and it computes the 95% confidence interval.

As part of computing the Pearson or the Spearman , SPSS automatically per-

forms a one- or two-tailed significance test. This includes indicating the smallest alpha

level at which the coefficient is significant.

CHAPTER SUMMARY

1. The one-sample t-test is for testing a one-sample experiment when the standard

deviation of the raw score population is not known.

2. A t-distribution is a theoretical sampling distribution of all possible values

of t when a raw score population is infinitely sampled using a particular .

A t-distribution that more or less forms a perfect normal curve will occur

depending on the degrees of freedom of the samples used to create it.

3. In point estimation, a is assumed to be at a point on the variable equal

to . Because the sample probably contains sampling error, a point estimate

is likely to be incorrect. In interval estimation, a is assumed to lie within a

specified interval. Interval estimation is performed by computing a confidence

interval.

␮

1df2

obt

␣

PUTTING IT

ALL TOGETHER

Chapter Summary 255

Likewise, we maximize the power of a correlational study by maximizing the size of

the correlation coefficient relative to the critical value. Three aspects will increase the

power of a correlation coefficient. First, avoiding a restricted range increases power.

Recall from Chapter 7 that having a small range of scores on the or variable pro-

duces a coefficient that is smaller than it would be without a restricted range. There-

fore, always measure the full range of possible and scores. Second, minimizing

the variability of the scores at each increases power. Recall that the smaller the

variability in scores at each , the larger the correlation coefficient. Therefore,

always test participants in a consistent fashion to minimize the variability in scores

at each . Third, increasing increases power. With a larger , the are larger, so the

critical value is smaller, and thus a given coefficient is more likely to be significant.

REMEMBER Increase power in a correlation coefficient by avoiding a

restricted range, minimizing the variability in scores, and increasing .NY

dfNNX

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

4. The confidence interval for a single m describes a range of , one of which the

sample mean is likely to represent. The interval contains the highest and lowest

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

5. The symbol for the Pearson correlation coefficient in the population is (called

rho). The symbol for the Spearman correlation coefficient in the population is .

6. The sampling distribution of the Pearson is a frequency distribution showing all

possible values of that occur when samples are drawn from a population in

which is zero.

7. The sampling distribution of the Spearman is a frequency distribution showing

all possible values of that occur when samples are drawn from a population in

which is zero.

8. Only when a correlation coefficient is significant is it appropriate to compute the

linear regression equation and the proportion of variance accounted for.

9. Maximize the power of experiments by (a) creating large differences in scores

between the conditions of the independent variable, (b) minimizing the variability

of the scores within each condition, and (c) increasing the of small samples.

10. Maximize the power of a correlation coefficient by (a) avoiding a restricted range,

(b) minimizing the variability in at each , and (c) increasing the of small

samples.

NXY

␳

␮

␮s

KEY TERMS

confidence interval for a single 243

estimated standard error of the

mean 236

interval estimation 243

margin of error 243

␮

␳r

crit

obt

dfs

crit

obt

one-sample t-test 234

point estimation 243

sampling distribution of 249

sampling distribution of 252

t-distribution 238

REVIEW QUESTIONS

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

1. A scientist has conducted a one-sample experiment. (a) What two parametric pro-

cedures are available to her? (b) What is the deciding factor for selecting between

them? (c) What are the other assumptions of the t-test?

2. In this chapter, you learned how to perform four different statistical procedures.

List them.

3. (a) What is the difference between and ? (b) How is their use the same?

4. (a) Why are there different values of when samples have different ?

(b) What must you compute in order to find ?

5. (a) What is the symbol for the Pearson correlation coefficient in the population?

(b) What is the symbol for the Spearman correlation coefficient in the population?

crit

Nst

crit

Application Questions 257

6. Summarize the steps involved in analyzing the results of a one-sample

experiment.

7. What is the final step when results are significant in any study?

8. Say that you have a sample mean of 44 in a study. (a) Estimate the corresponding

using point estimation. (b) What would a confidence interval for this tell you?

(d) What is the difference between reporting an estimate of using a margin of

error versus using a confidence interval?

9. (a) What is power? (b) What outcome should cause you to worry about having

sufficient power? (c) Why? (d) At what stage do you build in power?

10. (a) What are the three aspects of maximizing the power of a t-test? (b) What are

the three aspects of maximizing the power of a correlation coefficient?

APPLICATION QUESTIONS

11. We ask whether a new version of our textbook is beneficial or detrimental to students

learning statistics. On a national statistics exam, for students using other

textbooks. A sample of students using this book has the following scores:

64 69 92 77 71 99 82 74 69 88

(a) What are and for this study? (b) Compute . (c) With , what

is ? (d) What do you conclude about the use of this book? (e) Compute the

confidence interval for .

12. A researcher predicts that smoking cigarettes decreases a person’s sense of smell.

On a test of olfactory sensitivity, the for nonsmokers is 18.4. A sample of peo-

ple who smoke a pack a day produces these scores:

16 14 19 17 16 18 17 15 18 19 12 14

(a) What are and for this study? (b) Compute . (c) With ,

what is ? (d) What should the researcher conclude about this relationship?

(e) Compute the confidence interval for .

13. Foofy studies if hearing an argument in favor of an issue alters participants’

attitudes toward the issue one way or the other. She presents a 30-second

argument to 8 people. In a national survey about this issue, . She obtains

and . (a) What are and ? (b) What is ? (c) With

, what is ? (d) If appropriate, compute the confidence interval for .

(e) What conclusions should Foofy draw about the relationship?

14. In question 13, (a) What statistical principle should Foofy be concerned with?

(b) Identify three problems with her study from a statistical perspective.

15. Poindexter examined the relationship between ratio scores measuring the

quality of sneakers worn by volleyball players and their average number of

points scored per game. Studying 10 people who owned sneakers of good to

excellent quality, he computed . Without further ado, he immediately

claimed to have support for the notion that better-quality sneakers are related to

better performance on a somewhat consistent basis. He then computed and the

regression equation. Do you agree or disagree with his approach? Why?

16. Eventually, for the study in question 15, Poindexter reported that ,

. (a) What should he conclude about this relationship? (b) What otherp 7 .05

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