Chapter 4 One-Sample

Use one-sample tests to either describe a single variable’s frequency or central tendency, or to compare the frequency or central tendency to a hypothesized distribution or value.

If the data generating process produces continuous outcomes (interval or ratio), and the outcomes are symmetrically distributed, the sample mean, \(\bar{x}\), is a random variable centered at the population mean, \(\mu\). You can then use a theoretical distribution (normal or student t) to estimate a 95% confidence interval (CI) around \(\mu\), or compare \(\bar{x}\) to an hypothesized population mean, \(\mu_0\). If you (somehow) know the population variance, or the Central Limit Theorem (CLT) conditions hold, you can assume the random variable is normally distributed and use the z-test, otherwise assume the random variable has student t distribution and use the t-test.4 If the data generating process produces continuous outcomes that are not symmetrically distributed, use a non-parametric test like the Wilcoxon median test.

If the data generating process produces discrete outcomes (counts), the sample count, \(x\), is a random variable from a Poisson, binomial, normal, or multinomial distribution, or a random variable from a theoretical outcome.

  • For counts over a fixed time or space, treat the count as a random variable from a Poisson distribution with expected value \(\lambda\) and variance \(\lambda\).

  • For counts within a fixed total that are then classified into two levels (usually yes/no), then treat the count as a random variable from a binomial distribution with expected value \(n\pi\) and variance \(n\pi(1-\pi)\).

  • For binomial distributions where \(n\ge30\) and the frequency counts of both levels is \(\ge\) 5, treat the proportion as a random variable from the normal distribution with expected valued \(\pi\) and variance \(\frac{\pi(1-\pi)}{n}\).

  • For counts within a fixed total that are then classified into three or more levels, treat the count as a random variable from the multinomial distribution with expected value \(n\pi_j\) and variance \(n\pi_j(1-\pi_j)\).

Whatever the source of the expected values, you use either the chi-squared goodness-of-fit test or G test to test whether the observed values fit the expected values from the distribution. In the special case of binary outcomes with small (n < 1,000), you can use Fisher’s exact test instead. The discrete variable tests are discussed in PSU STATS 504.

  1. The t-test returns nearly the same result as the z-test when the CLT holds, so in practice no one bothers with the z-test except as an aid to teach the t-test.↩︎