One-Sample t Test

The one-sample t test is used to compare a sample mean (M) with a hypothetical population mean (μ₀) that provides some interesting standard of comparison. The null hypothesis is that the mean for the population (µ) is equal to the hypothetical population mean: μ = μ₀. The alternative hypothesis is that the mean for the population is different from the hypothetical population mean: μ _≠ μ₀. To decide between these two hypotheses, we need to find the probability of obtaining the sample mean (or one more extreme) if the null hypothesis were true. But finding this p value requires first computing a test statistic called t. (A test statistic is a statistic that is computed only to help find the p value.) The formula for t is as follows:

\[ t = \frac{M – \mu_{0}}{( \frac{SD}{ \sqrt{N}})} \]

Again, M is the sample mean and µ₀ is the hypothetical population mean of interest. SD is the sample standard deviation and N is the sample size.

The reason the t statistic (or any test statistic) is useful is that we know how it is distributed when the null hypothesis is true. As shown in Figure 13.1 “Distribution of “, this distribution is unimodal and symmetrical, and it has a mean of 0. Its precise shape depends on a statistical concept called the degrees of freedom, which for a one-sample t test is N − 1. (There are 24 degrees of freedom for the distribution shown in Figure 13.1 “Distribution of “.) The important point is that knowing this distribution makes it possible to find the p value for any t score. Consider, for example, a t score of +1.50 based on a sample of 25. The probability of a t score at least this extreme is given by the proportion of t scores in the distribution that are at least this extreme. For now, let us define extreme as being far from zero in either direction. Thus the p value is the proportion of t scores that are +1.50 or above or that are −1.50 or below—a value that turns out to be .14.

Fortunately, we do not have to deal directly with the distribution of t scores. If we were to enter our sample data and hypothetical mean of interest into one of the online statistical tools in Chapter 12 “Descriptive Statistics” or into a program like SPSS (Excel does not have a one-sample t test function), the output would include both the t score and the p value. At this point, the rest of the procedure is simple. If p is less than .05, we reject the null hypothesis and conclude that the population mean differs from the hypothetical mean of interest. If p is greater than .05, we retain the null hypothesis and conclude that there is not enough evidence to say that the population mean differs from the hypothetical mean of interest. (Again, technically, we conclude only that we do not have enough evidence to conclude that it does differ.)

If we were to compute the t score by hand, we could use a table like Table 13.2 “Table of Critical Values of “ to make the decision. This table does not provide actual p values. Instead, it provides the critical values of t for different degrees of freedom (df) when α is .05. For now, let us focus on the two-tailed critical values in the last column of the table. Each of these values should be interpreted as a pair of values: one positive and one negative. For example, the two-tailed critical values when there are 24 degrees of freedom are +2.064 and −2.064. These are represented by the red vertical lines in Figure 13.1 “Distribution of “. The idea is that any t score below the lower critical value (the left-hand red line in Figure 13.1 “Distribution of “) is in the lowest 2.5% of the distribution, while any t score above the upper critical value (the right-hand red line) is in the highest 2.5% of the distribution. This means that any t score beyond the critical value in either direction is in the most extreme 5% of t scores when the null hypothesis is true and therefore has a p value less than .05. Thus if the t score we compute is beyond the critical value in either direction, then we reject the null hypothesis. If the t score we compute is between the upper and lower critical values, then we retain the null hypothesis.

Thus far, we have considered what is called a two-tailed test, where we reject the null hypothesis if the t score for the sample is extreme in either direction. This makes sense when we believe that the sample mean might differ from the hypothetical population mean but we do not have good reason to expect the difference to go in a particular direction. But it is also possible to do a one-tailed test, where we reject the null hypothesis only if the t score for the sample is extreme in one direction that we specify before collecting the data. This makes sense when we have good reason to expect the sample mean will differ from the hypothetical population mean in a particular direction.

Here is how it works. Each one-tailed critical value in Table 13.2 “Table of Critical Values of “ can again be interpreted as a pair of values: one positive and one negative. A t score below the lower critical value is in the lowest 5% of the distribution, and a t score above the upper critical value is in the highest 5% of the distribution. For 24 degrees of freedom, these values are −1.711 and +1.711. (These are represented by the green vertical lines in Figure 13.1 “Distribution of “.) However, for a one-tailed test, we must decide before collecting data whether we expect the sample mean to be lower than the hypothetical population mean, in which case we would use only the lower critical value, or we expect the sample mean to be greater than the hypothetical population mean, in which case we would use only the upper critical value. Notice that we still reject the null hypothesis when the t score for our sample is in the most extreme 5% of the t scores we would expect if the null hypothesis were true—so α remains at .05. We have simply redefined extreme to refer only to one tail of the distribution. The advantage of the one-tailed test is that critical values are less extreme. If the sample mean differs from the hypothetical population mean in the expected direction, then we have a better chance of rejecting the null hypothesis. The disadvantage is that if the sample mean differs from the hypothetical population mean in the unexpected direction, then there is no chance at all of rejecting the null hypothesis.

The Dependent-Samples t Test

The dependent-samples t test (sometimes called the paired-samples t test) is used to compare two means for the same sample tested at two different times or under two different conditions. This makes it appropriate for pretest-posttest designs or within-subjects experiments. The null hypothesis is that the means at the two times or under the two conditions are the same in the population. The alternative hypothesis is that they are not the same. This test can also be one-tailed if the researcher has good reason to expect the difference goes in a particular direction.

It helps to think of the dependent-samples t test as a special case of the one-sample t test. However, the first step in the dependent-samples t test is to reduce the two scores for each participant to a single difference score by taking the difference between them. At this point, the dependent-samples t test becomes a one-sample t test on the difference scores. The hypothetical population mean (µ₀) of interest is 0 because this is what the mean difference score would be if there were no difference on average between the two times or two conditions. We can now think of the null hypothesis as being that the mean difference score in the population is 0 (µ₀ = 0) and the alternative hypothesis as being that the mean difference score in the population is not 0 (µ₀ ≠ 0).

The Independent-Samples t Test

The independent-samples t test is used to compare the means of two separate samples (M₁ and M₂). The two samples might have been tested under different conditions in a between-subjects experiment, or they could be preexisting groups in a correlational design (e.g., women and men, extroverts and introverts). The null hypothesis is that the means of the two populations are the same: µ₁ = µ₂. The alternative hypothesis is that they are not the same: µ₁ ≠ µ₂. Again, the test can be one-tailed if the researcher has good reason to expect the difference goes in a particular direction.

The t statistic here is a bit more complicated because it must take into account two sample means, two standard deviations, and two sample sizes. The formula is as follows:

\[ t = \frac{ M_{1} – M_{2} }{ \sqrt{ \frac{ SD_{1}^{2}}{n_{1}} + \frac{ SD_{2}^{2}}{n_{2}}}} \]

Notice that this formula includes squared standard deviations (the variances) that appear inside the square root symbol. Also, lowercase n₁ and n₂ refer to the sample sizes in the two groups or condition (as opposed to capital N, which generally refers to the total sample size). The only additional thing to know here is that there are N − 2 degrees of freedom for the independent-samples t test.

One-Way ANOVA

The one-way ANOVA is used to compare the means of more than two samples (M1, M₂…M_G) in a between-subjects design. The null hypothesis is that all the means are equal in the population: µ₁= µ₂ =…= µ_G. The alternative hypothesis is that not all the means in the population are equal.

The test statistic for the ANOVA is called F. It is a ratio of two estimates of the population variance based on the sample data. One estimate of the population variance is called the mean squares between groups (MS_B) and is based on the differences among the sample means. The other is called the mean squares within groups (MS_W) and is based on the differences among the scores within each group. The F statistic is the ratio of the MS_B to the MS_W and can therefore be expressed as follows:

F=MSBMSW.

Again, the reason that F is useful is that we know how it is distributed when the null hypothesis is true. As shown in Figure 13.2 “Distribution of the “, this distribution is unimodal and positively skewed with values that cluster around 1. The precise shape of the distribution depends on both the number of groups and the sample size, and there is a degrees of freedom value associated with each of these. The between-groups degrees of freedom is the number of groups minus one: df_B = (G − 1). The within-groups degrees of freedom is the total sample size minus the number of groups: df_W = N − G. Again, knowing the distribution of F when the null hypothesis is true allows us to find the p value.

The online tools in Chapter 12 “Descriptive Statistics” and statistical software such as Excel and SPSS will compute F and find the p value. If p is less than .05, then we reject the null hypothesis and conclude that there are differences among the group means in the population. If p is greater than .05, then we retain the null hypothesis and conclude that there is not enough evidence to say that there are differences. In the unlikely event that we would compute F by hand, we can use a table of critical values like Table 13.3 “Table of Critical Values of “ to make the decision. The idea is that any F ratio greater than the critical value has a p value of less than .05. Thus if the F ratio we compute is beyond the critical value, then we reject the null hypothesis. If the F ratio we compute is less than the critical value, then we retain the null hypothesis.

ANOVA Elaborations

Factorial ANOVA

When more than one independent variable is included in a factorial design, the appropriate approach is the factorial ANOVA. Again, the basics of the factorial ANOVA are the same as for the one-way and repeated-measures ANOVAs. The main difference is that it produces an F ratio and p value for each main effect and for each interaction. Returning to our calorie estimation example, imagine that the health psychologist tests the effect of participant major (psychology vs. nutrition) and food type (cookie vs. hamburger) in a factorial design. A factorial ANOVA would produce separate F ratios and p values for the main effect of major, the main effect of food type, and the interaction between major and food. Appropriate modifications must be made depending on whether the design is between subjects, within subjects, or mixed.

Example Test of Pearson’s r

Imagine that the health psychologist is interested in the correlation between people’s calorie estimates and their weight. He has no expectation about the direction of the relationship, so he decides to conduct a two-tailed test. He computes the correlation for a sample of 22 college students and finds that Pearson’s r is −.21. The statistical software he uses tells him that the p value is .348. It is greater than .05, so he retains the null hypothesis and concludes that there is no relationship between people’s calorie estimates and their weight. If he were to compute Pearson’s r by hand, he could look at Table 13.5 “Table of Critical Values of Pearson’s “ and see that the critical value for 22 − 2 = 20 degrees of freedom is .444. The fact that Pearson’s r for the sample is less extreme than this critical value tells him that the p value is greater than .05 and that he should retain the null hypothesis.