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
${\overline{x}}_P-{\overline{x}}_C$ 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:

$$\text{statistic}\pm (2\times \text{s.e.}(\text{statistic}\left)\right).$$ When the statistic is ${\overline{x}}_P-{\overline{x}}_C$, the approximate 95% CI is

$$({\overline{x}}_P-{\overline{x}}_C)\pm (2\times \text{s.e.}({\overline{x}}_P-{\overline{x}}_C\left)\right).$$

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

$$\begin{array}{c}51.59375\pm (2\times 19.61213),\end{array}$$ 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.