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Choose when to use a single-sample t test.
A single-sample t test is used to compare a sample mean to some specified value, like a population mean, when the population standard deviation is not known. Like all statistical tests, it has assumptions that must be met, hypotheses to be listed, and a decision rule to be set in advance of calculating the value of the test statistic.
Calculate the test statistic for a single-sample t test.
The single-sample t test calculates a t value. t is distributed very much like z—if the null hypothesis for a two-tailed test is true, the t distribution is symmetrical and centered on zero. The t value is based on the size of the difference between the sample mean and the specified value. If the difference, when standardized as a t score, is too large to be explained by sampling error, the null hypothesis is rejected.
Interpret the results of a single-sample t test.
Interpretation explains the results of a statistical test in plain language. An interpretation is subjective, but it is based on facts. Interpretation proceeds by asking questions of the data (e.g., was the null hypothesis rejected, how big is the effect) and then using the answers to address four points: (1) what was the study about, (2) what were the results, (3) what do the results mean, and (4) what should be done in future research.
Cohen’s d and r2 were introduced in this chapter and confidence intervals made a return appearance. All can be used to measure the size of the effect, the impact of the explanatory variable on the outcome variable. A confidence interval uses a sample value to estimate the range within which a population value falls; narrower confidence intervals give a more precise estimate. Cohen’s d takes the difference between two means and standardizes it. Cohen has suggested d values, for the social and behavioral sciences, representing small, medium, and large effects. r2 tells how much of the variability in the outcome variable is explained by or accounted for by, the explanatory variable.
Every year the Centers for Disease Control (CDC) conducts a survey of health practices and risk behaviors in American adults. The survey is called the BRFSS, the Behavioral Risk Factor Surveillance System. In 2013, the most recent year for which data are available, the sample consisted of almost 500,000 Americans, 18 and older, from all 50 states, Washington, D.C., Guam, and Puerto Rico.
| Hours of Sleep | Weight in Pounds | Drinks per Drinking Day | |
| Men and women | 7.05 | 176.54 | 2.21 |
| Men | 7.03 | 196.88 | 2.66 |
| Women | 7.06 | 161.79 | 1.80 |
I picked three variables in the data set—number of hours of sleep per night, weight in pounds, and number of alcoholic drinks consumed per day during the past month, on days that at least one drink was consumed—and found the mean for each for men, for women, and for both sexes combined.
Pick one of these variables and survey about 10 of your friends. Calculate the mean and the standard deviation. Then complete a single-sample t test to see whether your sample differs in its behavior from the U.S. population.