Statistical Tests for Experiments with Single Samples

Sometimes, the research question is as simple as investigating whether the value of a certain property of a group differs from a hypothetical value (specified by the null hypothesis). Depending on what information is available, one can choose between the following types of tests:

In this section, we are going to review these tests by working through a few toy problems.


The z-test is often used to compare a set of experimental measurements to a given constant as specified by the null hypothesis. For example, suppose that we know the mean and standard deviation of neuronal density of the frontal cortex, our research interest is to investigate whether Einstein's neuronal density is significantly different from the average -- the test statistic under the null hypothesis. The z-test is appropriate in such contexts.

Note that conducting a z-test requires that information other than the variable of interest is available. For instance, the standard deviation of the neuronal density in the target population (not the sample!) must be known -- or at least can be estimated with reasonably high accuracy (i.e. if the sample size of your experiment is on the order of thousands). If, however, the sample size of your experiment is small and no information regarding the standard deviation of the population is available, the single-sample t-test is recommended.

Example Problem

The first step is always to identify the null hypothesis. In this case, the null hypothesis can be:

In mathematical terms, the null hypothesis is that mu = mu0, where mu stands for the population mean height of males who take the growth drug during their teen years, and mu0 the population mean height of American males.

We will assume that the zobt alpha level is set to 0.05 and the test is two-tailed.

Solve the Problem by Hand

To determine whether or not we should reject the null hypothesis, we need to calculate the z statistic zobt and then determine its significance. The formula for calculating zobt is:

z_{obt} = \frac{\overline{X}_{obt} - \mu_{0}}{\sigma / \sqrt{N}}

where X-barobt is the sample mean, mu0 the population mean of American males, $\sigma$ the population standard deviation, and N the number of observations (sample size). Plugging in the numbers, it is easy to get:

z_{obt} = \frac{71 - 70}{9 / \sqrt{16}} = 0.444

Finally, we need to determine whether the effect is significant based on the calculated zobt. One can look up in the z table (usually the first Appendix table in a stats textbook) and find out the critical value for alpha = 0.05 and a two-tailed test is 1.96. The critical z score is clearly larger than the calculated zobt; therefore, we retain the null hypothesis.

Conclusion: from the current experiment, there is not enough evidence to show that the growth drug is effective. (Be careful not to conclude that the null hypothesis is TRUE. Failure to reject to the null hypothesis does not mean the null hypothesis is correct.)

Single-Sample t-test

The single-sample t-test is more often used since we, as experimenters, rarely know the true standard deviation of the target population. The only available information has to come from the sample data collected in the experiment. For example, Smokin' Joe wants to know whether marijuana can increase appetite and conducted an experiment:

Example Problem

In other words, Smokin' Joe wants to test whether the difference in calories eaten by AIDS patients (i.e. the property of a group) is different from 0 (i.e. a hypothetical value). If it is significantly different from 0, we can reject the null hypothesis which states that marijuana has no effect on appetite. Formally, the null hypothesis is:

Note here the only information available is the difference scores of the 10 subjects. We do not know the true standard deviation of the population of difference scores. Therefore, a z-test is not applicable in this case.

Approximate Population SD with Sample SD

One possible solution to the problem, which allows us to use the single-sample t-test, is to approximate the population standard deviation $\sigma$ with sample standard deviation s. Recall that the equation to calculate the z statistic zobt is:

z_{obt} = \frac{\overline{X}_{obt} - \mu_0}{\sigma / \sqrt{N}}

In the single-sample t-test, we simply replace $\sigma$ with the standard deviation of the sample s to get tobt:

t_{obt} = \frac{\overline{X}_{obt} - \mu_0}{s / \sqrt{N}}

Once we have calculated tobt, we need to determine the significance of the statistic. If you are working with a stats program, the significance p value, including the tobt, is calculated automatically for you. If you are working with pen and paper, you will need to look up in the t distribution table (usually in the appendix of any statistics textbook) for the corresponding tcrit value. In looking for the correct critical value to compare with, we also need to determine the degrees of freedom (df) of the current test. For any single-sample t-test, the df is N-1, where N is the sample size. It is then easy to locate the critical value in the table by choosing the right test (one-tailed or two-tailed?), alpha level, and the df.

Solve the Problem by Hand

Now we proceed to solve the example problem. Let's do a two-tailed test with alpha value set to 0.05. To calculate the tobt, we calculate the following items first:

Note: always carry at least three decimal places if working by hand. Rounding excessively in intermediate steps can lead to very very very inaccurate results.

Now we plug these numbers into the above equation:

t_{obt} = \frac{\overline{X}_{obt} - \mu_0}{s / \sqrt{N}} = \frac{91.5-0}{117.449 / 3.162} = 2.463

Looking up in the critical values table for the t distribution, we find that tcrit = 2.262 given that df is 9 and alpha is 0.05 for a two-tailed test. Since our t statistic is greater than the critical value, we reject the null hypothesis. We conclude that Smokin' Joe's experimental results support the hypothesis that taking the THC pill increases appetite.

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StatsWiki: OneSampleOneVariable (last edited 2012-01-23 18:14:22 by cpe-69-207-83-233)

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