# 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.