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CHAPTER 7

✪ The t Test for a Single Sample 223

✪ The t Test for Dependent Means 236

✪ Assumptions of the t Test for a Single Sample and the t Test for Dependent Means 247

✪ Effect Size and Power for the t Test for Dependent Means 247

✪ Controversy: Advantages and Disadvantages of Repeated-Measures Designs 250

At this point, you may think you know all about hypothesis testing. Here’s asurprise: what you know will not help you much as a researcher. Why? Theprocedures for testing hypotheses described up to this point were, of course, absolutely necessary for what you will now learn. However, these procedures in- volved comparing a group of scores to a known population. In real research practice, you often compare two or more groups of scores to each other, without any direct information about populations. For example, you may have two scores for each per- son in a group of people, such as scores on an anxiety test before and after psy- chotherapy or number of familiar versus unfamiliar words recalled in a memory experiment. Or you might have one score per person for two groups of people, such

✪ Single Sample t Tests and Dependent Means t Tests in Research Articles 252

✪ Summary 253

✪ Key Terms 254

✪ ExampleWorked-OutProblems 254

✪ Practice Problems 258

✪ Using SPSS 265

✪ Chapter Notes 268

Introduction to t Tests Single Sample and Dependent Means

Chapter Outline

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Introduction to t Tests 223

t test hypothesis-testing procedure in which the population variance is un- known; it compares t scores from a sam- ple to a comparison distribution called a t distribution.

as an experimental group and a control group in a study of the effect of sleep loss on problem solving, or comparing the self-esteem test scores of a group of 10-year-old girls to a group of 10-year-old boys.

These kinds of research situations are among the most common in psychology, where usually the only information available is from samples. Nothing is known about the populations that the samples are supposed to come from. In particular, the researcher does not know the variance of the populations involved, which is a crucial ingredient in Step ❷ of the hypothesis-testing process (determining the characteristics of the comparison distribution).

In this chapter, we first look at the solution to the problem of not knowing the population variance by focusing on a special situation: comparing the mean of a sin- gle sample to a population with a known mean but an unknown variance. Then, after describing how to handle this problem of not knowing the population variance, we go on to consider the situation in which there is no known population at all—the sit- uation in which all we have are two scores for each of a number of people.

The hypothesis-testing procedures you learn in this chapter, those in which the population variance is unknown, are examples of t tests. The t test is sometimes called “Student’s t” because its main principles were originally developed by William S. Gosset, who published his research articles anonymously using the name “Student” (see Box 7–1).

The t Test for a Single Sample Let’s begin with an example. Suppose your college newspaper reports an informal survey showing that students at your college study an average of 17 hours per week. However, you think that the students in your dormitory study much more than that. You randomly pick 16 students from your dormitory and ask them how much they study each day. (We will assume that they are all honest and accurate.) Your result is that these 16 students study an average of 21 hours per week. Should you conclude that students in your dormitory study more than the college average? Or should you conclude that your results are close enough to the college average that the small dif- ference of 4 hours might well be due to your having picked, purely by chance, 16 of the more studious residents in your dormitory?

In this example you have scores for a sample of individuals and you want to com- pare the mean of this sample to a population for which you know the mean but not the variance. Hypothesis testing in this situation is called a t test for a single sample. (It is also called a one-sample t test.) The t test for a single sample works basically the same way as the Z test you learned in Chapter 5. In the studies we considered in that chapter, you had scores for a sample of individuals (such as a group of 64 students rating the at- tractiveness of a person in a photograph after being told that the person has positive personality qualities) and you wanted to compare the mean of this sample to a popula- tion (in this case, a population of students not told about the person’s personality qual- ities). However, in the studies we considered in Chapter 5, you knew both the mean and variance of the general population to which you were going to compare your sam- ple. In the situations we are now going to consider, everything is the same, but you don’t know the population variance. This presents two important new wrinkles affect- ing the details of how you carry out two of the steps of the hypothesis-testing process.

The first important new wrinkle is in Step ❷. Because the population variance is not known, you have to estimate it. So the first new wrinkle we consider is how to estimate an unknown population variance. The other important new wrinkle affects Steps ❷ and ❸. When the population variance has to be estimated, the shape of the comparison

t test for a single sample hypothesis- testing procedure in which a sample mean is being compared to a known population mean and the population variance is unknown.

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distribution is not quite a normal curve; so the second new wrinkle we consider is the shape of the comparison distribution (for Step ❷) and how to use a special table to find the cutoff (Step ❸) on what is a slightly differently shaped distribution.

Let’s return to the amount of studying example. Step ❶ of the hypothesis-testing procedure is to restate the problem as hypotheses about populations. There are two populations:

Population 1: The kind of students who live in your dormitory. Population 2: The kind of students in general at your college.

The research hypothesis is that Population 1 students study more than Population 2 students; the null hypothesis is that Population 1 students do not study more than Population 2 students. So far, the problem is no different from those in Chapter 5.

Step ❷ is to determine the characteristics of the comparison distribution. In this example, its mean will be 17, what the survey found for students at your college generally (Population 2).

224 Chapter 7

professor of mathematics and not a proper brewer at all. To his statistical colleagues, mainly at the Biometric Lab- oratory at University College in London, he was a mere brewer and not a proper mathematician.

So Gosset discovered the t distribution and invented the t test—simplicity itself (compared to most of statistics)—for situations when samples are small and the variability of the larger population is unknown. How- ever, the Guinness brewery did not allow its scientists to publish papers, because one Guinness scientist had re- vealed brewery secrets. To this day, most statisticians call the t distribution “Student’s t” because Gosset wrote under the anonymous name “Student.” A few of his fel- low statisticians knew who “Student” was, but apparently meetings with others involved the secrecy worthy of a spy novel. The brewery learned of his scientific fame only at his death, when colleagues wanted to honor him.

In spite of his great achievements, Gosset often wrote in letters that his own work provided “only a rough idea of the thing” or so-and-so “really worked out the com- plete mathematics.” He was remembered as a thoughtful, kind, humble man, sensitive to others’ feelings. Gosset’s friendliness and generosity with his time and ideas also resulted in many students and younger colleagues mak- ing major breakthroughs based on his help.

To learn more about William Gosset, go to http:// www-history.mcs.st-andrews.ac.uk/Biographies/Gosset. html.

Sources: Peters (1987); Salsburg (2001); Stigler (1986); Tankard (1984).

BOX 7–1 William S. Gosset, Alias “Student”: Not a Mathematician, But a Practical Man

William S. Gosset graduated from Oxford University in 1899 with degrees in mathe- matics and chemistry. It hap- pened that in the same year the Guinness brewers in Dublin, Ireland, were seeking a few young scientists to take a first-ever scientific look at beer making. Gosset took one of these jobs and soon had

immersed himself in barley, hops, and vats of brew. The problem was how to make beer of a consistently

high quality. Scientists such as Gosset wanted to make the quality of beer less variable, and they were especially in- terested in finding the cause of bad batches. A proper sci- entist would say, “Conduct experiments!” But a business such as a brewery could not afford to waste money on ex- periments involving large numbers of vats, some of which any brewer worth his hops knew would fail. So Gosset was forced to contemplate the probability of, say, a certain strain of barley producing terrible beer when the experi- ment could consist of only a few batches of each strain. Adding to the problem was that he had no idea of the vari- ability of a given strain of barley—perhaps some fields planted with the same strain grew better barley. (Does this sound familiar? Poor Gosset, like today’s psychologists, had no idea of his population’s variance.)

Gosset was up to the task, although at the time only he knew that. To his colleagues at the brewery, he was a

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The next part of Step ❷ is finding the variance of the distribution of means. Now you face a problem. Up to now in this book, you have always known the variance of the population of individuals. Using that variance, you then figured the variance of the distribution of means. However, in the present example, the variance of the number of hours studied for students at your college (the Population 2 students) was not reported in the newspaper article. So you email the paper. Unfortunately, the reporter did not figure the variance, and the original survey results are no longer available. What to do?

Basic Principle of the t Test: Estimating the Population Variance from the Sample Scores If you do not know the variance of the population of individuals, you can estimate it from what you do know—the scores of the people in your sample.

In the logic of hypothesis testing, the group of people you study is considered to be a random sample from a particular population. The variance of this sample ought to reflect the variance of that population. If the scores in the population have a lot of variation, then the scores in a sample randomly selected from that population should also have a lot of variation. If the population has very little variation, the scores in a sample from that population should also have very little variation. Thus, it should be possible to use the variation among the scores in the sample to make an informed guess about the spread of the scores in the population. That is, you could figure the variance of the sample’s scores, and that should be similar to the variance of the scores in the population. (See Figure 7–1.)

There is, however, one small hitch. The variance of a sample will generally be slightly smaller than the variance of the population from which it is taken. For this reason, the variance of the sample is a biased estimate of the population variance.1

It is a biased estimate because it consistently underestimates the actual variance of the population. (For example, if a population has a variance of 180, a typical sample

Introduction to t Tests 225

(c)(b) (a)

Figure 7–1 The variation in samples (as in each of the lower distributions) is similar to the variations in the populations they are taken from (each of the upper distributions).

biased estimate estimate of a popula- tion parameter that is likely systemati- cally to overestimate or underestimate the true value of the population parame- ter. For example, would be a biased estimate of the population variance (it would systematically underestimate it).

SD2

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unbiased estimate of the population variance ( ) estimate of the popula- tion variance, based on sample scores, which has been corrected so that it is equally likely to overestimate or under- estimate the true population variance; the correction used is dividing the sum of squared deviations by the sample size minus 1, instead of the usual procedure of dividing by the sample size directly.

S2 of 20 scores might have a variance of only 171.) If we used a biased estimate of the population variance in our research studies, our results would not be accurate. There- fore, we need to identify an unbiased estimate of the population variance.

Fortunately, you can figure an unbiased estimate of the population variance by slightly changing the ordinary variance formula. The ordinary variance formula is the sum of the squared deviation scores divided by the number of scores. The changed for- mula still starts with the sum of the squared deviation scores, but divides this by the number of scores minus 1. Dividing by a slightly smaller number makes the result slightly larger. Dividing by the number of scores minus 1 makes the variance you get just enough larger to make it an unbiased estimate of the population variance. (This unbiased estimate is our best estimate of the population variance. However, it is still an estimate, so it is unlikely to be exactly the same as the true population variance. But we can be certain that our unbiased estimate of the population variance is equally likely to be too high as it is to be too low. This is what makes the estimate unbiased.)

The symbol we will use for the unbiased estimate of the population variance is . The formula is the usual variance formula, but now dividing by :

(7–1)

(7–2)

Let’s return again to the example of hours spent studying and figure the estimated population variance from the sample’s 16 scores. First, you figure the sum of squared deviation scores. (Subtract the mean from each of the scores, square those deviation scores, and add them.) Presume in our example that this comes out to To get the estimated population variance, you divide this sum of squared deviation scores by the number of scores minus 1; that is, in this example, you divide 694 by

; 694 divided by 15 comes out to 46.27. In terms of the formula,

At this point, you have now seen several different types of standard deviation and variance (that is, for a sample, for a population, and unbiased estimates); and each of these types has used a different symbol. To help you keep them straight, a summary of the types of standard deviation and variance is shown in Table 7–1.

Degrees of Freedom The number you divide by (the number of scores minus 1) to get the estimated pop- ulation variance has a special name. It is called the degrees of freedom. It has this name because it is the number of scores in a sample that are “free to vary.” The idea is that, when figuring the variance, you first have to know the mean. If you know the mean and all but one of the scores in the sample, you can figure out the one you don’t know with a little arithmetic. Thus, once you know the mean, one of the scores in the sample is not free to have any possible value. So in this kind of situation the degrees of freedom are the number of scores minus 1. In terms of a formula,

(7–3)

df is the degrees of freedom.

df = N – 1

S2 = a (X – M)2 N – 1 =

SS

N – 1 = 694

16 – 1 =

694

15 = 46.27

16 – 1

694 (SS = 694).

S = 2S2

S2 = a (X – M)2 N – 1 =

SS

N – 1

N – 1S2 The estimated population variance is the sum of the squared deviation scores di- vided by the number of scores minus 1.

The estimated population standard deviation is the square root of the estimated population variance.

degrees of freedom (df ) number of scores free to vary when estimating a population parameter; usually part of a formula for making that estimate—for example, in the formula for estimating the population variance from a single sample, the degrees of freedom is the number of scores minus 1.

Table 7–1 Summary of Different Types of Standard Deviation and Variance

Statistical Term Symbol

Sample standard deviation SD

Population standard deviation

Estimated population S standard deviation

Sample variance SD2

Population variance

Estimated population variance S 2 �2

�

The degrees of freedom are the number of scores in the sample minus 1.

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In our example, . (In some situations you learn about in later chapters, the degrees of freedom are figured a bit differently. This is because in those situations, the number of scores free to vary is different. For all the situations you learn about in this chapter, .)

The formula for the estimated population variance is often written using df in- stead of :

(7–4)

The Standard Deviation of the Distribution of Means Once you have figured the estimated population variance, you can figure the stan- dard deviation of the comparison distribution using the same procedures you learned in Chapter 5. Just as before, when you have a sample of more than one, the compar- ison distribution is a distribution of means, and the variance of a distribution of means is the variance of the population of individuals divided by the sample size. You have just estimated the variance of the population. Thus, you can estimate the variance of the distribution of means by dividing the estimated population variance by the sample size. The standard deviation of the distribution of means is the square root of its variance. Stated as formulas,

(7–5)

(7–6)

Note that, with an estimated population variance, the symbols for the variance and standard deviation of the distribution of means use S instead of .

In our example, the sample size was 16 and we worked out the estimated popu- lation variance to be 46.27. The variance of the distribution of means, based on that estimate, will be 2.89. That is, 46.27 divided by 16 equals 2.89. The standard devia- tion is 1.70, the square root of 2.89. In terms of the formulas,

The Shape of the Comparison Distribution When Using an Estimated Population Variance: The t Distribution In Chapter 5 you learned that when the population distribution follows a normal curve, the shape of the distribution of means will also be a normal curve. However, this changes when you do hypothesis testing with an estimated population variance. When you are using an estimated population variance, you have less true informa- tion and more room for error. The mathematical effect is that there are likely to be slightly more extreme means than in an exact normal curve. Further, the smaller your

SM = 2S2M = 22.89 = 1.70 S2M =

S2

N =

46.27

16 = 2.89

�

SM = 2S2M

S2M = S2

N

S2 = a (X – M)2

df =

SS

df

N – 1

df = N – 1

df = 16 – 1 = 15

T I P F O R S U C C E S S Be sure that you fully understand the difference between and . These terms look quite similar, but they are quite different. is the estimated variance of the popula- tion of individuals. is the esti- mated variance of the distribution of means (based on the estimated variance of the population of indi- viduals, ).S2

S2M

S2

SM 2S2

The estimated population variance is the sum of squared deviations divided by the de- grees of freedom.

The variance of the distribu- tion of means based on an es- timated population variance is the estimated population variance divided by the num- ber of scores in the sample.

The standard deviation of the distribution of means based on an estimated population vari- ance is the square root of the variance of the distribution of means based on an estimated population variance.

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sample size, the bigger this tendency. This is because, with a smaller sample size, your estimate of the population variance is based on less information.

The result of all this is that, when doing hypothesis testing using an estimated variance, your comparison distribution will not be a normal curve. Instead, the com- parison distribution will be a slightly different curve called a t distribution.

Actually, there is a whole family of t distributions. They vary in shape according to the degrees of freedom you used to estimate the population variance. However, for any particular degrees of freedom, there is only one t distribution.

Generally, t distributions look to the eye like a normal curve—bell-shaped, sym- metrical, and unimodal. A t distribution differs subtly in having heavier tails (that is, slightly more scores at the extremes). Figure 7–2 shows the shape of a t distribution compared to a normal curve.

This slight difference in shape affects how extreme a score you need to reject the null hypothesis. As always, to reject the null hypothesis, your sample mean has to be in an extreme section of the comparison distribution of means, such as the top 5%. However, if the comparison distribution has more of its means in the tails than a normal curve would have, then the point where the top 5% begins has to be farther out on this comparison distribution. The result is that it takes a slightly more extreme sample mean to get a significant result when using a t distribution than when using a normal curve.

Just how much the t distribution differs from the normal curve depends on the de- grees of freedom, the amount of information used in estimating the population vari- ance. The t distribution differs most from the normal curve when the degrees of freedom are low (because your estimate of the population variance is based on a very small sample). For example, using the normal curve, you may recall that 1.64 is the cutoff for a one-tailed test at the .05 level. On a t distribution with 7 degrees of free- dom (that is, with a sample size of 8), the cutoff is 1.895 for a one-tailed test at the .05 level. If your estimate is based on a larger sample, say a sample of 25 (so that ), the cutoff is 1.711, a cutoff much closer to that for the normal curve. If your sample size is infinite, the t distribution is the same as the normal curve. (Of course, if your sample size were infinite, it would include the entire population!) But even with sam- ple sizes of 30 or more, the t distribution is nearly identical to the normal curve.

Shortly, you will learn how to find the cutoff using a t distribution, but let’s first return briefly to the example of how much students in your dorm study each week. You finally have everything you need for Step ❷ about the characteristics of the comparison distribution. We have already seen that the distribution of means in this example has a mean of 17 hours and a standard deviation of 1.70. You can now add that the shape of the comparison distribution will be a t distribution with 15 degrees of freedom.2

df = 24

Normal distribution

t distribution

Figure 7–2 A t distribution (dashed blue line) compared to the normal curve (solid black line).

t distribution mathematically defined curve that is the comparison distribution used in a t test.

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Introduction to t Tests 229

The Cutoff Sample Score for Rejecting the Null Hypothesis: Using the t Table Step ❸ of hypothesis testing is determining the cutoff for rejecting the null hypothesis. There is a different t distribution for any particular degrees of freedom. However, to avoid taking up pages and pages with tables for each possible t distribution, you use a simplified table that gives only the crucial cutoff points. We have included such a t table in the Appendix (Table A–2). Just as with the normal curve table, the t table shows only positive t scores. If you have a one-tailed test, you need to decide whether your cutoff score is a positive t score or a negative t score. If your one-tailed test is test- ing whether the mean of Population 1 is greater than the mean of Population 2, the cut- off t score is positive. However, if your one-tailed test is testing whether the mean of Population 1 is less than the mean of Population 2, the cutoff t score is negative.

In the hours-studied example, you have a one-tailed test. (You want to know whether students in your dorm study more than students in general at your college study.) You will probably want to use the 5% significance level, because the cost of a Type I error (mistakenly rejecting the null hypothesis) is not great. You have 16 partic- ipants, making 15 degrees of freedom for your estimate of the population variance.

Table 7–2 shows a portion of the t table from Table A–2 in the Appendix. Find the column for the .05 significance level for one-tailed tests and move down to the row for 15 degrees of freedom. The crucial cutoff is 1.753. In this example, you are testing whether students in your dormitory (Population 1) study more than students in general at your college (Population 2). In other words, you are testing whether

Table 7–2 Cutoff Scores for t Distributions with 1 Through 17 Degrees of Freedom (Highlighting Cutoff for Hours-Studied Example)

One-Tailed Tests Two-Tailed Tests

df .10 .05 .01 .10 .05 .01

1 3.078 6.314 31.821 6.314 12.706 63.657

2 1.886 2.920 6.965 2.920 4.303 9.925

3 1.638 2.353 4.541 2.353 3.182 5.841

4 1.533 2.132 3.747 2.132 2.776 4.604

5 1.476 2.015 3.365 2.015 2.571 4.032

6 1.440 1.943 3.143 1.943 2.447 3.708

7 1.415 1.895 2.998 1.895 2.365 3.500

8 1.397 1.860 2.897 1.860 2.306 3.356

9 1.383 1.833 2.822 1.833 2.262 3.250

10 1.372 1.813 2.764 1.813 2.228 3.170

11 1.364 1.796 2.718 1.796 2.201 3.106

12 1.356 1.783 2.681 1.783 2.179 3.055

13 1.350 1.771 2.651 1.771 2.161 3.013

14 1.345 1.762 2.625 1.762 2.145 2.977

15 1.341 1.753 2.603 1.753 2.132 2.947

16 1.337 1.746 2.584 1.746 2.120 2.921

17 1.334 1.740 2.567 1.740 2.110 2.898

t table table of cutoff scores on the t distribution for various degrees of freedom, significance levels, and one- and two-tailed tests.

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students in your dormitory have a higher t score than students in general. This means that the cutoff t score is positive. Thus, you will reject the null hypothesis if your sample’s mean is 1.753 or more standard deviations above the mean on the compar- ison distribution. (If you were using a known variance, you would have found your cutoff from a normal curve table. The Z score to reject the null hypothesis based on the normal curve would have been 1.645.)

One other point about using the t table: In the full t table in the Appendix, there are rows for each degree of freedom from 1 through 30, then for 35, 40, 45, and so on up to 100. Suppose your study has degrees of freedom between two of these higher values. To be safe, you should use the nearest degrees of freedom to yours given on the table that is less than yours. For example, in a study with 43 degrees of freedom, you would use the cutoff for .

The Sample Mean’s Score on the Comparison Distribution: The t Score Step ❹ of hypothesis testing is figuring your sample mean’s score on the comparison distribution. In Chapter 5, this meant finding the Z score on the comparison distribution—the number of standard deviations your sample’s mean is from the mean on the distribution. You do exactly the same thing when your comparison distri- bution is a t distribution. The only difference is that, instead of calling this a Z score, because it is from a t distribution, you call it a t score. In terms of a formula,

(7–7)

In the example, your sample’s mean of 21 is 4 hours from the mean of the distri- bution of means, which amounts to 2.35 standard deviations from the mean (4 hours divided by the standard deviation of 1.70 hours).3 That is, the t score in the example is 2.35. In terms of the formula,

Deciding Whether to Reject the Null Hypothesis Step ➎ of hypothesis testing is deciding whether to reject the null hypothesis. This step is exactly the same with a t test, as it was in the hypothesis-testing situations dis- cussed in previous chapters. In the example, the cutoff t score was 1.753 and the actual t score for your sample was 2.35. Conclusion: reject the null hypothesis. The research hypothesis is supported that students in your dorm study more than students in the college overall.

Figure 7–3 shows the various distributions for this example.

Summary of Hypothesis Testing When the Population Variance Is Not Known Table 7–3 compares the hypothesis-testing procedure we just considered (for a t test for a single sample) with the hypothesis-testing procedure for a Z test from Chapter 5. That is, we are comparing the current situation in which you know the population’s mean but not its variance to the Chapter 5 situation, where you knew the population’s mean and variance.

t = M – �

SM =

21 – 17 1.70

= 4

1.70 = 2.35

t = M – �

SM

df = 40

The t score is your sample’s mean minus the population mean, divided by the standard deviation of the distribution of means.

t score on a t distribution, number of standard deviations from the mean (like a Z score, but on a t distribution).

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Introduction to t Tests 231

Comparison distribution

(t)

Population (normal)

17

15.3013.60 17.00 18.70 20.40 –1–2 0 1 2

21

Sample

Raw Scores: t Scores:

Figure 7–3 Distribution for the hours-studied example.

Table 7–3 Hypothesis Testing with a Single Sample Mean When Population Variance Is Unknown (t Test for a Single Sample) Compared to When Population Variance Is Known (Z Test)

Another Example of a t Test for a Single Sample Consider another fictional example. Suppose a researcher was studying the psychologi- cal effects of a devastating flood in a small rural community. Specifically, the researcher was interested in how hopeful (versus unhopeful) people felt after the flood. The

Steps in Hypothesis Testing Difference From When Population Variance Is Known

❶ Restate the question as a research hypothesis and a null hypothesis about the populations.

No difference in method.

❷ Determine the characteristics of the comparison distribution:

Population mean No difference in method.

Standard deviation of the distribution of sample means

No difference in method (but based on estimated population variance).

Population variance Estimate from the sample.

Shape of the comparison distribution Use the t distribution with .df = N – 1 ❸ Determine the significance cutoff. Use the t table.

❹ Determine your sample’s score on the comparison distribution.

No difference in method (but called a t score).

❺ Decide whether to reject the null hypothesis. No difference in method.

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researcher randomly selected 10 people from this community to complete a short ques- tionnaire. The key item on the questionnaire asked how hopeful they felt, using a 7-point scale from extremely unhopeful (1) to neutral (4) to extremely hopeful (7). The re- searcher wanted to know whether the ratings of hopefulness for people who had been through the flood would be consistently above or below the neutral point on the scale (4).

Table 7–4 shows the results and figuring for the t test for a single sample; Figure 7–4 shows the distributions involved. Here are the steps of hypothesis testing.

❶ Restate the question as a research hypothesis and a null hypothesis about the populations. There are two populations:

Population 1: People who experienced the flood. Population 2: People who are neither hopeful nor unhopeful.

The research hypothesis is that the two populations will score differently. The null hypothesis is that they will score the same.

❷ Determine the characteristics of the comparison distribution. If the null hy- pothesis is true, the mean of both populations is 4. The variance of these popu- lations is not known, so you have to estimate it from the sample. As shown in Table 7–4, the sum of the squared deviations of the sample’s scores from the sample’s mean is 32.10. Thus, the estimated population variance is 32.10 divided by 9 degrees of freedom (10 – 1), which comes out to 3.57.

The distribution of means has a mean of 4 (the same as the population mean). Its variance is the estimated population variance divided by the sample size (3.57

Table 7–4 Results and Figuring for a Single-Sample t Test for a Study of 10 People’s Ratings of Hopefulness Following a Deva