24.6 Confidence intervals: Two independent means
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.
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.