30.2 Hypotheses and notation: Two independent means
Since two groups are being compared, distinguishing between the statistics for the two groups (say, Group A and Group B) is important (recapping Sect. 24.3).
One way is to use subscripts (see Table 30.1). Using this notation, the parameter in the RQ is the difference between population means: \(\mu_A-\mu_B\).
As usual, the population values are unknown, so this is estimated using the statistic \(\bar{x}_A-\bar{x}_B\).
| Group A | Group B | |
|---|---|---|
| Population means: | \(\mu_A\) | \(\mu_B\) |
| Sample means: | \(\bar{x}_A\) | \(\bar{x}_B\) |
| Standard deviations: | \(s_A\) | \(s_B\) |
| Standard errors: | \(\displaystyle\text{s.e.}(\bar{x}_A) = \frac{s_A}{\sqrt{n_A}}\) | \(\displaystyle\text{s.e.}(\bar{x}_B) = \frac{s_B}{\sqrt{n_B}}\) |
| Sample sizes: | \(n_A\) | \(n_B\) |
For the reaction-time data, the differences are computed as the mean reaction time for phone users, minus the mean reaction time for non-phone users: \(\mu_P - \mu_C\). By this definition, the differences refer to how much greater (on average) the reaction times are when students are using phones.
The parameter is \(\mu_P - \mu_C\), the difference between the population mean reaction times (using phone, minus not using a phone).
As always (Sect. 28.2), the null hypothesis is the default ‘no difference, no change, no relationship’ position; hence the null hypothesis is that there is ‘no difference’ between the population means of the two groups:
- \(H_0\): \(\mu_P - \mu_C=0\) (or \(\mu_P = \mu_C\)).
This hypothesis proposes that any difference between the sample means is due to sampling variation. This becomes the initial assumption.
From the RQ, the alternative hypothesis will be two-tailed:
- \(H_1\): \(\mu_P - \mu_C\ne 0\) (or \(\mu_P \ne \mu_C\)).