24.6 Two independent means: Confidence intervals

Being able to describe the sampling distribution implies that we have some idea of how the values of
\(\bar{x}_P - \bar{x}_C\) are likely to vary from sample to sample. Then, finding an approximate 95% CI for the difference between the mean reaction times is similar to the process used in Chap. 22. Approximate 95% CIs all have the same form:

\[ \text{statistic} \pm (2\times\text{s.e.}(\text{statistic})). \] When the statistic is \(\bar{x}_P - \bar{x}_C\), the approximate 95% CI is

\[ (\bar{x}_P - \bar{x}_C) \pm (2 \times \text{s.e.}(\bar{x}_P - \bar{x}_C)). \]

In this case (using more decimal places than in the summary table in Table 24.2), the CI is

\[\begin{eqnarray*} 51.59375 \pm (2 \times 19.61213), \end{eqnarray*}\] or \(51.59375\pm 19.61213\). After rounding appropriately, an approximate 95% CI for the difference is from \(12.37\) to \(90.82\) milliseconds. We write:

Based on the sample, an approximate 95% CI for the difference in reaction time while driving, for those using a phone and those not using a phone, is from \(12.37\) to \(90.82\) milliseconds (higher for those using a phone).

The plausible values for the difference between the two population means are between \(12.37\) to \(90.82\) milliseconds.

Stating the CI is insufficient; you must also state the direction in which the differences were calculated, so readers know which group had the higher mean.