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.