## 30.7 Statistical validity conditions: Two independent means

As usual,
these results apply
under certain conditions,
which are the same as those for forming a
CI for the *difference* between two means.

The test above is statistically valid
if *one* of these conditions is true:

*Both*sample sizes are at least 25;*or*- Either sample size is smaller than 25,
**and**both*populations*have an approximate normal distribution.

The sample size of 25 is a rough figure here, and some books give other values (such as 30).
We can explore the histograms of the *samples*
to determine if normality of the *populations* seems reasonable.

In addition to the statistical validity condition, the test will be

**internally valid**if the study was well designed; and**externally valid**if the sample is a simple random sample and is internally valid.

**Example 30.1 (Statistical validity) **For the reaction-time data,
both samples sizes are \(n=32\).
This means that the results will be statistically valid.

**Example 30.2 (Gray whales) **A study of gray whales (*Eschrichtius robustus*)
measured (among other things) the length of adult whales (Agbayani et al. 2020).
The data are shown below.

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

Female | 12.70 | 0.611 | 260 |

Male | 12.07 | 0.705 | 139 |

Are adult female gray whales longer 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;
this is the *parameter* of interest.
The best estimate of this difference is \(\bar{x}_F - \bar{x}_M = 12.70 - 12.07 = 0.63\) m.

The hypotheses are:

- \(H_0\): \(\mu_F - \mu_M = 0\)
- \(H_1\): \(\mu_F - \mu_M \ne 0\)

We know that the difference between the sample means 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:
\(\text{s.e.}(\bar{x}_F - \bar{x}_M) = 0.07079\).

The test statistic is

\[
t = \frac{(\bar{x}_F - \bar{x}_M) - (\mu_F - \mu_M)}{\text{s.e.}(\bar{x}_F - \bar{x}_M)}
= \frac{0.63 - 0}{0.07079} = 8.90,
\]
which is *very* large.
This means that the \(P\)-value will be very small (using the 68-95–99.7 rule).

We write:

There is very strong evidence (\(t = 8.90\); two-tailed \(P < 0.001\)) that the mean length of adult gray whales is different for females (mean: 12.70 m; standard deviation: 0.611 m) and males (mean: 12.07 m; standard deviation: 0.705 m; 95% CI for the difference: 0.48 m to 0.77 m).

Since both sample sizes are large, the test is statistically valid.

(Check that you can compute the correct CI!)### References

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