## 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

approximate95% 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.

**Example 24.1 (Gray whales) **A study of gray whales (*Eschrichtius robustus*)
measured (among other things) the length of whales at birth (Agbayani et al. 2020).
The data are shown below.

Sex | Mean (in m) | Standard deviation (in m) | Sample size |
---|---|---|---|

Female | 4.66 | 0.379 | 26 |

Male | 4.60 | 0.305 | 30 |

How much longer are female gray whales than males, on average?

Let’s define the *difference* as the mean length of female gray whales *minus* the mean length of male gray whales.
Then we wish to estimate the difference \(\mu_F - \mu_M\),
where \(F\) and \(M\) represent female and male gray whales respectively.
In this situation, this is the *parameter* of interest.
The best estimate of this difference is \(\bar{x}_F - \bar{x}_M = 4.66 - 4.60 = 0.06\).

We know that this value is likely to vary from sample to sample, and hence it has a standard error.

We cannot easily determine the standard error of this difference from the above information
(though it is possible), so we must be *given* this information:
\(\text{s.e.}(\bar{x}_F - \bar{x}_M) = 0.0929\).

Then the approximate 95% CI is from

\[ 0.06 - (2 \times 0.0929) = -0.125747 \] to \[ 0.06 + (2 \times 0.0929) = 0.245747, \] so the CI is from \(-0.12\) m to \(0.25\) m.

Notice that one of these limits is a *negative* value.
This does not mean a *negative* length for a whale; that would be silly.
Remember that this CI is for the *difference* between the mean lengths,
and a *negative* length just says the mean length for males is greater than the mean length for females.

So we could say:

The population mean difference between the length of female and male gray whales at birth has a 95% chance of being between \(0.12\) m longer for male whales to \(0.25\) m longer for female whales.

### References

*Eschrichtius robustus*). Journal of Mammalogy. 2020;101(3):742–54.