The Normal Distribution and Single-Sample Significance Tests

Abstract

In this chapter, we focus on characteristics of the normal distribution and single-sample significance tests that are used for variables measured at the ratio and interval levels. Specifically, this chapter reviews percentages under the normal curve, application of the 68-95-99.7 rule, and how to conduct a significance test in R for the following: (1) comparing a sample mean to a known population (single-sample z-test for means), (2) comparing a sample mean to an unknown population (single-sample t-test), and (3) comparing a sample proportion to a population proportion (single-sample z-test for proportions). In doing so, the chapter walks through criminal justice-related examples, lays out the null and alternative hypotheses for presented examples, and shows the user how to make a determination about the null hypothesis for the aforementioned tests from R output. Additionally, you will learn how to write your own functions in R.

Key Terms

68-95-99.7 rule

Empirical rule that states that 68% of the cases in a normal distribution should fall within 1 standard deviation of the mean (so within a z-score of −1 and +1); 95% of the cases in the distribution should fall within 2 standard deviations of the mean (so within a z-score of −2 and +2); and 99.7% of the cases in the distribution should fall within 3 standard deviations of the mean (so within a z-score of −3 and +3). In the real world, you will likely not find a distribution where this rule is exact.

Confidence interval

An interval of values around a statistic (usually a point estimate). If we were to draw repeated samples and calculate a 95% confidence interval for each, then in only 5 in 100 of these samples would the interval fail to include the true population parameter. In the case of a 99% confidence interval, only 1 in 100 samples would fail to include the true population parameter.

Normal distribution

A bell-shaped frequency distribution, symmetrical in form. Its mean, mode, and median are always the same. The percentage of cases between the mean and points at a measured distance from the mean is fixed.

Single-sample t -test

A test of statistical significance that is used to examine whether a sample is drawn from a specific population with a known or hypothesized mean. In a t-test, the standard deviation of the population to which the sample is being compared is unknown.

Single-sample z -test

A test of statistical significance that is used to examine whether a sample is drawn from a specific population with a known or hypothesized mean. In a z-test, the standard deviation of the population to which the sample is being compared is either known or—as in the case of a proportion—is defined by the null hypothesis.

Standard deviation unit

A unit of measurement used to describe the deviation of a specific score or value from the mean in a z distribution.

z -score

Score that represents an observation in standard deviation units from the mean.