8.2.3.1 - One Sample Mean t Test, Formulas

Five Step Hypothesis Testing Procedure

1. Check assumptions and write hypotheses

Data must be quantitative. In order to use the t distribution to approximate the sampling distribution either the sample size must be large (\(\ge\ 30\)) or the population must be known to be normally distributed. The possible combinations of null and alternative hypotheses are:

Research Question Is the mean different from \( \mu_{0} \)? Is the mean greater than \(\mu_{0}\)? Is the mean less than \(\mu_{0}\)?
Null Hypothesis, \(H_{0}\) \(\mu=\mu_{0} \) \(\mu=\mu_{0} \) \(\mu=\mu_{0} \)
Alternative Hypothesis, \(H_{a}\) \(\mu\neq \mu_{0} \) \(\mu> \mu_{0} \) \(\mu<\mu_{0} \)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional

where \( \mu_{0} \) is the hypothesized population mean.

2. Calculate the test statistic

For the test of one group mean we will be using a \(t\) test statistic:

Test Statistic: One Group Mean

\(t=\frac{\overline{x}-\mu_0}{\frac{s}{\sqrt{n}}}\)

\(\overline{x}\) = sample mean
\(\mu_{0}\) = hypothesized population mean
\(s\) = sample standard deviation
\(n\) = sample size

Note that structure of this formula is similar to the general formula for a test statistic: \(\frac{sample\;statistic-null\;value}{standard\;error}\)

3. Determine the p-value

When testing hypotheses about a mean or mean difference, a \(t\) distribution is used to find the \(p\)-value. These \(t\) distributions are indexed by a quantity called degrees of freedom, calculated as \(df = n – 1\) for the situation involving a test of one mean or test of mean difference. The \(p\)-value can be found using Minitab Express.

4. Make a decision

If \(p \leq \alpha\) reject the null hypothesis.

If \(p>\alpha\) fail to reject the null hypothesis.

5. State a "real world" conclusion

Based on your decision in Step 4, write a conclusion in terms of the original research question.

The new few pages will walk you through examples before giving you the opportunity to do two on your own.