A single-sample t test is a hypothesis test in which we compare a sample from which we collect data to a population for which we know the mean but not the standard deviation.

Degrees of freedom is the number of scores that are free to vary when we estimate a population parameter from a sample.

EXAMPLE 9.4

For example, the manager of a baseball team needs to assign nine players to particular spots in the batting order but only has to make eight decisions (N − 1). Why? Because only one option remains after making the first eight decisions. So before the manager makes any decisions, there are N − 1, or 9 − 1 = 8, degrees of freedom. After the second decision, there are N − 1, or 8 − 1 = 7, degrees of freedom, and so on.

As in the baseball example, there is always one score that cannot vary once all of the others have been determined. For example, if we know that the mean of four scores is 6 and we know that three of the scores are 2, 4, and 8, then the last score must be 10. So the degrees of freedom is the number of scores in the sample minus 1. Degrees of freedom is written in symbolic notation as df, which is always italicized. The formula for degrees of freedom for a single-sample t test, therefore, is:

Table : TABLE 9-1. Excerpt from the t Table When conducting hypothesis testing, we use the t table to determine critical values for a given p level, based on the degrees of freedom and whether the test is one- or two-tailed. The full table is in Appendix B.

MASTERING THE CONCEPT

9.3: As sample size increases, the t distributions more and more closely approximate the z distribution. You can think of the z statistic as a single-blade Swiss Army knife and the t statistic as a multiblade Swiss Army knife that includes the single blade that is the z statistic.

EXAMPLE 9.5

The study: A researcher knows the mean number of calories a rat will consume in half an hour if unlimited food is available. He wonders whether a new food will lead rats to consume a different number of calories—either more or fewer. He studies 38 rats and uses a p level of 0.05.

The cutoff(s): This is a two-tailed test because the research hypothesis allows for change in either direction. There are 38 rats, so the degrees of freedom is:

We want to look in the t table under two-tailed tests, in the column for 0.05 and in the row for a df of 37; however, there is no df of 37. In this case, we err on the side of being more conservative and choose the more extreme (i.e., larger) of the two possible critical t values, which is always the smaller df. Here, we look next to 35, where we see a value of 2.030. Because this is a two-tailed test, we will have critical values of −2.030 and 2.030.

EXAMPLE 9.6

Chapter 4 presented data that included the mean number of sessions attended by clients at a university counseling center. We noted that one study reported a mean of 4.6 sessions (Hatchett, 2003). Let’s imagine that the counseling center hoped to increase participation rates by having students sign a contract to attend at least 10 sessions. Five students sign the contract and attend 6, 6, 12, 7, and 8 sessions, respectively. The researchers are interested only in their university, so treat the mean of 4.6 sessions as a population mean.

STEP 1: Identify the populations, distribution, and assumptions.

Population 1: All clients at this counseling center who sign a contract to attend at least 10 sessions. Population 2: All clients at this counseling center who do not sign a contract to attend at least 10 sessions.

The comparison distribution will be a distribution of means. The hypothesis test will be a single-sample t test because we have only one sample and we know the population mean but not the population standard deviation.

This study meets one of the three assumptions and may meet the other two: (1) The dependent variable is scale. (2) We do not know whether the data were randomly selected, however, so we must be cautious with respect to generalizing to other clients at this university who might sign the contract. (3) We do not know whether the population is normally distributed, and there are not at least 30 participants. However, the data from the sample do not suggest a skewed distribution.

STEP 2: State the null and research hypotheses.

Null hypothesis: Clients at this university who sign a contract to attend at least 10 sessions attend the same number of sessions, on average, as clients who do not sign such a contract—H₀: μ₁ = μ₂.

Research hypothesis: Clients at this university who sign a contract to attend at least 10 sessions attend a different number of sessions, on average, than do clients who do not sign such a contract—H₁: μ₁ ≠ μ₂.

STEP 3: Determine the characteristics of the comparison distribution.

Calculations:

The numerator of the standard deviation formula is the sum of squares:

STEP 4: Determine the critical values, or cutoffs.

For a two-tailed test with a p level of 0.05 and df of 4, the critical values are −2.776 and 2.776 (as seen in the curve in Figure 9-2).

STEP 5: Calculate the test statistic.

STEP 6: Make a decision.

Reject the null hypothesis. It appears that counseling center clients who sign a contract to attend at least 10 sessions do attend more sessions, on average, than do clients who do not sign such a contract (Figure 9-3).

After completing the hypothesis test, we want to present the primary statistical information in a report. There is a standard American Psychological Association (APA) format for the presentation of statistics across the behavioral sciences so that the results are easily understood by the reader.

1. Write the symbol for the test statistic (e.g., t).
2. Write the degrees of freedom, in parentheses.
3. Write an equal sign and then the value of the test statistic, typically to two decimal places.
4. Write a comma and then indicate the p value by writing “p = ” and then the actual value. (Unless we use software to conduct the hypothesis test, we will not know the actual p value associated with the test statistic. In this case, we simply state whether the p value is beyond the critical value by saying p < 0.05 when we reject the null hypothesis or p > 0.05 when we fail to reject the null hypothesis.)

In the counseling center example, the statistics would read:

The statistics typically follow a statement about the finding; for example, “It appears that counseling center clients who sign a contract to attend at least 10 sessions do attend more sessions, on average, than do clients who do not sign such a contract, t(4) = 2.87, p < 0.05.” The report would also include the sample mean and standard deviation (not standard error) to two decimal points. Here, the descriptive statistics would read: (M = 7.80, SD = 2.49). By convention, we use SD instead of s to symbolize the standard deviation.

EXAMPLE 9.7

We can calculate a confidence interval with the single-sample t test data. The population mean was 4.6. We used the sample to estimate the population standard deviation to be 2.490 and the population standard error to be 1.114. The five students in the sample attended a mean of 7.8 sessions.

When we conducted hypothesis testing, we centered the curve around the mean according to the null hypothesis—the population mean of 4.6. Now we can use the same information to calculate the 95% confidence interval around the sample mean of 7.8.

STEP 1: Draw a picture of a t distribution that includes the confidence interval.

We draw a normal curve (Figure 9-4) that has the sample mean, 7.8, at its center (instead of the population mean, 4.6).

STEP 2: Indicate the bounds of the confidence interval on the drawing.

We draw a vertical line from the mean to the top of the curve. For a 95% confidence interval, we also draw two much smaller vertical lines that indicate the middle 95% of the t distribution (2.5% in each tail, for a total of 5%). We then write the appropriate percentages under the segments of the curve.

STEP 3: Look up the t statistics that fall at each line marking the middle 95%.

For a two-tailed test with a p level of 0.05 and a df of 4, the critical values are −2.776 and 2.776. We can now add these t statistics to the curve, as seen in Figure 9-5.

STEP 4: Convert the t statistics back into raw means.

As we did with the z test, we can use formulas for this conversion, but first we identify the appropriate mean and standard deviation. There are two important points to remember. First, we center the interval around the sample mean, so we use the sample mean of 7.8 in the calculations. Second, because we have a sample mean (rather than an individual score), we use a distribution of means. So we use the standard error of 1.114 as the measure of spread.

Using this mean and standard error, we calculate the raw mean at each end of the confidence interval, and add them to the curve, as in Figure 9-6. The formulas are exactly the same as for the z test except that z is replaced by t, and σ_M is replaced by s_M.

The 95% confidence interval, reported in brackets as is typical, is [4.71, 10.89].

STEP 5: Verify that the confidence interval makes sense.

The sample mean should fall exactly in the middle of the two ends of the interval.

We have a match. The confidence interval ranges from 3.09 below the sample mean to 3.09 above the sample mean. If we were to sample five students from the same population over and over, the 95% confidence interval would include the population mean 95% of the time. Note that the population mean, 4.6, does not fall within this interval. This means it is not plausible that this sample of students who signed contracts came from the population according to the null hypothesis—students seeking treatment at the counseling center who did not sign a contract. We conclude that the sample comes from a different population, one in which students attend more mean sessions than does the general population. As with the z test, the conclusions from the hypothesis test and the confidence interval are the same, but the confidence interval gives us more information—an interval estimate, not just a point estimate.

EXAMPLE 9.8

Let’s calculate the effect size for the counseling center study. Similar to what we did with the z test, we simply use the formula for the t statistic, substituting s for s_M (and μ for μ_M, even though these means are always the same). This means we use 2.490 instead of 1.114 in the denominator. Cohen’s d is based on the spread of the distribution of individual scores, rather than the distribution of means.

The effect size, d = 1.29, tells us that the sample mean and the population mean are 1.29 standard deviations apart. According to the conventions we learned in Chapter 8 (that 0.2 is a small effect, 0.5 is a medium effect, and 0.8 is a large effect), this is a large effect. We can add the effect size when we report the statistics as follows: t(4) = 2.87, p < 0.05, d = 1.29.

Next Steps

Dot Plots

When we conduct hypothesis tests such as the single-sample t test, we must be concerned with the shape of the distribution of the underlying populations. When we have a sample size greater than 30, our comparison distribution can be assumed to be normal and we can proceed with hypothesis testings. With smaller samples, however, we often use the shapes of our samples to assess the shapes of the populations from which they are drawn. In earlier chapters, we learned to construct frequency histograms and frequency polygons to examine the shape of the data in a sample, but these graphs do not allow us to see every single data point. In Next Steps in Chapter 2, we explored stem-and-leaf plots. Here, we’ll introduce an alternative, the dot plot.

The dot plot is a graph that displays all the data points in a sample, with the range of scores along the x-axis and a dot for each data point above the appropriate value. Dot plots serve a function similar to that of stem-and-leaf plots. They both allow us to view the overall shape of a sample, and they both retain all of the individual data points. Moreover, a dot plot is easy on the eyes because the dots for one group are placed directly above those for the other group, allowing us to view two groups simultaneously, a useful feature for when we compare two groups, as we’ll do in chapters 10 and 11.

To demonstrate a dot plot, we’ll use the same data we used in Next Steps in Chapter 2—the numbers of minutes students typically spend in the shower. Here are the data for 30 women in our statistics classes, already arranged in order from lowest to highest:

Here are the scores for 30 men in our statistics classes who also reported how many minutes they typically spend in the shower:

To create a dot plot, there are three basic steps.

STEP 1: Determine the lowest score and highest score of the sample.

These scores are 5 and 60 for women, and 5 and 25 for men.

STEP 2: Draw an x-axis and label it, including the values from the lowest through highest scores.

See Figure 9.7.

STEP 3: Place a dot above the appropriate value for every score.

See Figure 9.7.

Figure 9-7 displays a stacked dot plot for the samples of male and female students’ minutes spent in the shower. With the dot plot, we can easily observe the slightly higher central tendency and larger spread for the women than the men, as well as the potential outlier in the female sample.

CHECK YOUR LEARNING

Clarifying the Concepts

• 9-5 Explain the term degrees of freedom.
• 9-6 Why is a single-sample t test more useful than a z test?

Calculating the Statistics

• 9-7 Compute degrees of freedom for each of the following:
  1. An experimenter times how long it takes 35 rats to run through a maze with eight pathways.
  2. Test scores for 14 students are collected and averaged over four semesters.
• 9-8 Identify the critical t value for each of the following tests:
  1. A two-tailed test with alpha of 0.05 and 11 degrees of freedom
  2. A one-tailed test with alpha of 0.01 and N of 17

Applying the Concepts

• 9-9 Let’s assume that according to university summary statistics, the average student misses 3.7 classes during a semester. Imagine the data you have been working with (6, 3, 7, 6, 4, 5) for the number of classes missed by a group of students. Conduct all six steps of hypothesis testing, using a two-tailed test with a p level of 0.05. (Note: You completed the work for step 3 in Check Your Learning 9-2 and 9-3.)