The single-sample t test is used to compare a sample mean to a specific value like a population mean. As it deals with means, it is used when the dependent variable is measured at the interval or ratio level. It does the same thing as a single-sample z test, but it is more commonly used because it doesn’t require knowing the population standard deviation, σ. Instead, the single-sample t test uses the sample standard deviation, s. As long as there is access to the sample, it is possible to calculate s.

For an example, imagine the following scenario. Dr. Farshad, a clinical psychologist, wondered whether adults with attention deficit hyperactivity disorder (ADHD) had reflexes that differed in speed from those of the general population. She located a test for reaction time that was normed on adults in the United States. The average reaction time was 200 milliseconds (msec). This is a population mean, so the Greek letter mu is used to represent it and one would write μ = 200.

From the various ADHD treatment centers in her home state of Illinois, Dr. Farshad obtained a random sample of 141 adults who had been diagnosed with ADHD. Each adult was tested individually on the reaction-time test. The mean reaction time for the sample of 141 was 220 msec (M = 220), with a standard deviation of 27 msec (s = 27). (Note that because these are values calculated from a sample, the abbreviations for the mean and standard deviation are Roman, not Greek, letters.)

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Dr. Farshad’s question is whether adults with ADHD differ in reaction time from the general population. 220 msec is definitely different from 200 msec, so the mean reaction time of this sample of adults with ADHD is different from the population mean. However, Dr. Farshad doesn’t know if the difference is a statistically significant one! It’s possible that the sample mean is different from the population mean due to sampling error. To answer her question, she’s going to need hypothesis testing.

Let’s follow Dr. Farshad through the six steps of hypothesis testing for the single-sample t test: (1) picking a test; (2) checking the assumptions; (3) listing the hypotheses; (4) setting the decision rule; (5) calculating the test statistic; and (6) interpreting the results. Remember the mnemonic: “Tom and Harry despise crabby infants.”

Step 1 Pick a Test

The first step in hypothesis testing involves selecting the appropriate statistical test. Dr. Farshad is comparing the mean of a sample to the mean of a population, so she could use either a single-sample z test or a single-sample t test. However, she doesn’t know the population standard deviation. She’ll have to use the single-sample t test to figure out whether the mean reaction time of the sample of adults with ADHD is statistically different from the mean reaction time of the general population.

Step 2 Check the Assumptions

To determine if the sample mean is statistically different from the population mean, Dr. Farshad plans to use a single-sample t test. However, a single-sample t test can only be used if its assumptions are met. The three assumptions for the single-sample t test are the same as they were for the single-sample z test and are listed in Table 7.1.

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The first assumption involves whether the sample is a random sample from the population. This is a robust assumption. So if it is violated—and it often is—the analysis can still be completed. One just has to be careful about the population to which one generalizes the results. In this example, the sample is a random sample from the population of adults with ADHD in one state. This means that Dr. Farshad should only generalize her results to that state.

The second assumption is that the observations within the sample are independent. This assumption is not robust. If it is violated, one cannot proceed with the single-sample t test. In this example, each participant is in the sample only once and the reaction time of each case is not influenced by any other case. The cases were selected randomly and tested individually, so they are independent and the second assumption is not violated.

The third assumption is that the dependent variable, reaction time, is normally distributed in the population. This assumption is robust and the analysis can be completed if the assumption is violated as long as the sample size is large, say, 30 or more, and the deviation from normality is not too large. How does one test this assumption? One way is simply assuming that it is true—it is generally accepted that psychological characteristics (like personality) and physical characteristics (like height) are normally distributed. So, Dr. Farshad is willing to assume that reaction time is normally distributed. Another way is to make a graph of the data. In Figure 7.1, the histogram that Dr. Farshad made for the frequency distribution of the reaction-time data can be seen. Though not perfectly normal, it has a normal-ish shape. That, combined with a large sample size, leads Dr. Farshad to feel sure that this assumption has not been violated.

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The assumptions have been met, so she can proceed with the planned statistical test, the single-sample t test.

Step 3 List the Hypotheses

Before writing the hypotheses, the researcher has to know if he or she is doing a one-tailed test or a two-tailed test. Two-tailed tests, called nondirectional tests, are more commonly used and they are a bit more conservative, in the sense that they make it harder to reject the null hypothesis. So, a two-tailed test is the “default” option—if in doubt, choose to do a two-tailed test. But, here’s the difference between the two types of tests:

The hypotheses for the nondirectional (two-tailed) single-sample t test for Dr. Farshad’s study are

H₀: μ_ADHD-Adults = 200

H₁: μ_ADHD-Adults ≠ 200

The null hypothesis (H₀) says that the mean reaction time for the population of adults with ADHD is not different from some specified value. In this case, that value is the mean reaction time, 200 msec, of Americans in general. (The null hypothesis could be phrased as μ_ADHD-Adults = μ_Americans = 200.)

The alternative hypothesis (H₁) says that the mean reaction time for the population of adults with ADHD is something other than the specified value of 200 msec. It doesn’t say whether it is faster or slower, just that the mean reaction time is different. This means that the observed sample mean should be different enough from 200 msec that sampling error does not explain the difference. (The alternative hypothesis could be phrased as μ_ADHD-Adults ≠ μ_Americans.)

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If Dr. Farshad had believed that adults with ADHD had, for example, slower reaction times than the general population, then she would have planned a one-tailed test. In such a situation, the hypotheses would have been

H₀: μ_ADHD-Adults ≤ 200

H₁: μ_ADHD-Adults > 200

Never forget that the alternative hypothesis expresses what the researcher believes to be true and that the researcher wants to be forced to reject the null hypothesis, so he or she has to accept the alternative hypothesis. In this one-tailed test example, the researcher’s belief is that the reaction time is longer (i.e., slower) for people with ADHD. Once the alternative hypothesis is formed, the null hypothesis is written to make both hypotheses all-inclusive and mutually exclusive. Here, the null hypothesis would be that people with ADHD have reaction times the same as or faster than those of the general population.

Step 4 Set the Decision Rule

Setting the decision rule involves determining when to reject the null hypothesis and when to fail to reject the null hypothesis. To set the decision rule, find the critical value of the test statistic, the boundary between the rare zone and the common zone of the sampling distribution. For the single-sample t test, this will be a critical value of t.

What is t? t is the statistic that is calculated in the next step of the six-step hypothesis test procedure. The statistic t is a lot like the statistic z in the single-sample z test:

The point that separates “differs enough from zero” from “doesn’t differ enough from zero” is called the critical value of t, abbreviated t_cv. To find t_cv, three pieces of information are needed:

The first question, whether a one-tailed or a two-tailed test is being done, was already answered when writing the hypotheses. Dr. Farshad is examining whether adults with ADHD have faster or slower reaction times than the general population. The reaction-time study calls for a two-tailed test because the hypotheses didn’t specify a direction for the difference.

The second question in determining t_cv involves how willing one is to make a Type I error. A Type I error occurs when the researcher concludes, mistakenly, that the null hypothesis should be rejected. A common convention in statistics is to choose to have no more than a 5% chance of making a Type I error. As alpha (α) is the probability of a Type I error, statisticians phrase this as “setting alpha at .05” or as “α = .05.” Dr. Farshad has chosen to follow convention and set alpha at .05.

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Overall, she wants her chance of making a Type I error to be no more than 5%. She has chosen to do a two-tailed test, so she will need to split that 5% in two and put 2.5% in each tail of the sampling distribution. This means Dr. Farshad will have two critical values of t, one positive and one negative. (If the test were one-tailed, all 5% of the rare zone would fall on one side and there would be only one critical value of t.)

The third question for determining t_cv, how large the sample size is, matters because the shape of the t distribution changes as the sample size changes. The tail of a t distribution is larger when the sample size is smaller. As a result, the critical value of t is farther away from zero when the sample size is small.

This is difficult to visualize without a concrete example, so look at Figure 7.2 in which two sampling distributions of t are superimposed—one for N = 6 (the dotted line) and one for N = 60 (the solid line). Which line is on top—the dotted line or the solid line—depends on whether one is looking at the center of the distribution or the tails.

Look at the tails. There, the dotted line (the one for the t distribution when N = 6) is above the solid line for the t distribution when N = 60. What does it mean for one line to be on top of the other? The y-axis measures frequency, so when one line is above another, this means it has a higher frequency at that point.

Focus on the positive side, the right-hand side, of the distribution. (Because the t distribution is symmetric, both sides are the same.) Note that around t = 1.5, the two lines cross. From that point on, the frequencies at each t value are higher for the N = 6 distribution than for the N = 60 distribution. Which distribution has more cases with t values above 1.50, N = 6 or N = 60? The answer is that the N = 6 distribution has a higher percentage of cases in the tail.

The implication of this is important: the total frequency of scores in the tail is higher for the distribution with the smaller sample size (N = 6). To cut off the extreme 2.5% of the scores for each distribution, the cut-off point will fall farther away from zero for the N = 6 distribution than for the N = 60 distribution. In this example, the cut-off points (the critical values of t) are 2.57 for N = 6 and 2.00 for N = 60. The critical value falls closer to zero when the sample size is larger, and it falls farther away from zero when the sample size is smaller.

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Sample size affects the critical value of t, and the critical value determines the rare and common zones in the sampling distribution. (Remember, when a result falls in the rare zone, the null hypothesis is rejected; when a result falls in the common zone, the null hypothesis is not rejected.) So, sample size affects the ability to reject the null hypothesis:

The goal of research almost always is to reject the null hypothesis, so having a larger sample size is an advantage.

To find the critical value of t for a given sample size, use Appendix Table 3, a table of critical values of t. A portion of Appendix Table 3 is shown in Table 7.2.

There are several things to note in Table 7.2. First, note that there are different rows for the critical values. The rows represent different critical values of t based on sample size. The heading for the rows is “df,” which stands for degrees of freedom. Degrees of freedom represent the number of values in a sample that are free to vary. For example, if the mean of three cases is 10, then the values for only two cases are free to vary. If one case has a score of 9 and another case has a score of 11, then the third case has to have a score of 10. In that example, there are three values, and 2 degrees of freedom—once two values are known, the third is determined.

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Degrees of freedom and sample size are yoked together. As the sample size becomes larger, the degrees of freedom increase. And, other things being equal, larger sample sizes are better because there is a greater likelihood that the sample represents the population. In fact, when the sample size is infinitely large, the t distribution is the same as the z distribution because the whole population is being sampled.

For a single-sample t test, the degrees of freedom are calculated as the sample size minus 1. This is shown in Equation 7.1.

For the reaction-time study, there are 141 participants in the sample, so degrees of freedom are calculated like this:

df = 141 – 1

= 140

Look at Table 7.2 again. The second thing to note in the table of critical values of t is that there are four columns of critical values. Which column to use depends on two factors: (1) if a one-tailed test or two-tailed test is being done, and (2) where alpha, the willingness to make a Type I error, is set.

The critical values of t in the column for a two-tailed test with a 5% chance of Type I error (i.e., α = .05) have been bolded to make them easier to find because they are the most commonly used.

With the reaction-time study as a two-tailed test, Dr. Farshad would reject the null hypothesis if adults with ADHD had slower reaction times than the general population or if adults with ADHD had faster reaction times than the general population. That is what a two-tailed test means.

Under these conditions (df = 140, α = .05, two-tailed), Dr. Farshad finds that the critical value of t is ±1.977 (i.e., –1.977 and 1.977). These values are marked in Figure 7.3, along with the rare and common zones.

The critical value of t would be different if this were a one-tailed test. If Dr. Farshad had reason, in advance of collecting data, to believe that adults with ADHD had slower reaction times, then she could do a one-tailed test. In such a case, she would only reject the null hypothesis if adults with ADHD had slower reaction times than the general population. For df = 140 and with α = .05, the one-tailed critical value of t would be 1.656. This critical value of 1.656 is closer to zero, the midpoint of the sampling distribution, than was the critical value (1.977) for the two-tailed test. This means the rare zone is more easily reached and makes it easier to reject the null hypothesis for the one-tailed test.

Figure 7.4 shows the larger rare zone on the right for the one-tailed test. The rare zone for the two-tailed test is marked with /// and the rare zone for the one-tailed test is marked with \\\. Though the total percentage of the curve that is the rare zone is the same for the two tests, notice how more area of the rare zone is marked off on one side by the one-tailed test (\\\) than by the two-tailed test (///). As a result, it is easier to reject the null hypothesis with a one-tailed test than a two-tailed test, as long as the difference is in the expected direction. This is an advantage for one-tailed tests. The advantage disappears if there is a difference, but it is not in the hypothesized direction.

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Dr. Farshad was conducting an exploratory study to see if adults with ADHD differed in reaction time—in either direction—from the general population. So, she’s doing a two-tailed test. Here’s the general version of the decision rule for a two-tailed test:

If t ≤ –t_cv or if t ≥ t_cv, reject H₀.

If –t_cv < t < t_cv, fail to reject H₀.

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When the first statement is true, the observed value of t falls in the rare zone, and the null hypothesis is rejected. When the second statement is true, t falls in the common zone, and the researcher will fail to reject the null hypothesis.

For the reaction-time study, the specific decision rule is:

Step 5 Calculate the Test Statistic

It is time to calculate the t value that compares the sample mean for reaction time (220 msec) to the specified value, the population mean for a reaction time of 200 msec. The formula is shown in Equation 7.2.

The numerator in the single-sample t test formula subtracts the population mean (μ) from the sample mean (M). The difference is then divided by the estimated standard error of the mean (s_M). Before using Equation 7.2, one needs to know the estimated standard error of the mean. This is calculated using Equation 5.2:

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Once the estimated standard error of the mean has been calculated, all the values needed to complete Equation 7.2 are available: M = 220, μ = 200, and s_M = 2.27. Here are the calculations to find the t value:

Step 5 is done and Dr. Farshad knows the t value: t = 8.81. The sixth and final step of hypothesis testing is interpretation. We’ll turn to that after a little more practice with the first five steps of hypothesis testing in Worked Example 7.1.

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