The anthropologist has the following hypothesis she would like to test; namely, that there is no difference among the two tribes in terms of heights.

The observed mean difference is Y= 59.4 - 61.3 = -1.9 inches. The standard error of Y is given by the following formula [(1.8^2)/25 + (2.4^2)/27]^(1/2). So, the two sample z-statistic is z=( observed difference - expected difference ) / ( SE for difference ) = -1.9/.5856=-3.2

In other words the difference between the heights of tribe 1 and tribe 2 is about 3.2 SE's below the value expected under the null hypothesis. Hence, we reject the null hypothesis and accept the alternative hypothesis that the difference is real.

Remarks:

  1. The test assumes simple random samples, which is NOT the case here.
  2. The sample is large enough so due to the Central Limit Theorem the probability histogram for each sample average and consequently of their difference follows the normal curve.

Although, the problem assumes simple random samples that is not the case in this problem. The anthropologist got samples of convenience with selection bias being present. So, the means of both samples are biased and the likely amount by which the biased sample estimates differ from the truth in the two populations are given by the formula [bias^2 + standard error^2]^(1/2).

The standard error of both means underestimate the correct answer. The correct answer is hard to assess in the absence of any information regarding the bias component.

However, since both samples were gathered the same way the biases built in each of them cancel out to a great extent and so the inference made on their mean difference is probably highly accurate, while each individual mean is not.