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5.2 - Writing Hypotheses

The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)).

The statement that there is not a difference in the population(s), denoted as \(H_0\) The statement that there is some difference in the population(s), denoted as \(H_a\) or \(H_1\)

When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

  1. At this point we can write hypotheses for a single mean (\(\mu\)), paired means(\(\mu_d\)), a single proportion (\(p\)), the difference between two independent means (\(\mu_1-\mu_2\)), the difference between two proportions (\(p_1-p_2\)), a simple linear regression slope (\(\beta\)), and a correlation (\(\rho\)).
  2. The research question will give us the information necessary to determine if the test is two-tailed (e.g., "different from," "not equal to"), right-tailed (e.g., "greater than," "more than"), or left-tailed (e.g., "less than," "fewer than").
  3. The research question will also give us the hypothesized parameter value. This is the number that goes in the hypothesis statements (i.e., \(\mu_0\) and \(p_0\)). For the difference between two groups, regression, and correlation, this value is typically 0.

Hypotheses are always written in terms of population parameters (e.g., \(p\) and \(\mu\)). The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).

One Group Mean
Research Question Is the population mean different from \( \mu_ \)? Is the population mean greater than \(\mu_\)? Is the population mean less than \(\mu_\)?
Null Hypothesis, \(H_\) \(\mu=\mu_ \) \(\mu=\mu_ \) \(\mu=\mu_ \)
Alternative Hypothesis, \(H_\) \(\mu\neq \mu_ \) \(\mu> \mu_ \) \(\mu <\mu_\)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional