Testing for similarity in two populations

Question

The table shows the half-lives (in days) of methylmercury (CH203 3 ) the systems of six women and nine men.

Females: 56 , 45 , 76 , 97 , 43

Males: 85 , 56, 35 , 56 , 34 , 65 , 34 , 67 , 94

Test at the alpha = 0:05 if men and women metabolize CH203 3 at the same rate using two different suitable tests.

The two tests I have been thinking are testing the variance ratio using F distribution and two sample T test. For the two sample t test I am not sure if I should use pooled or non pooled data.

Also are there any other methods ?

Answer 1

Here is a strip chart made in R, showing your data for women (bottom) and men.

f = c(56, 45, 76, 97, 43)
m = c(85, 56, 35, 56, 34, 65, 34, 67, 94)
stripchart(list(f,m), ylim=c(.5,2.5), meth="stack")

The two samples do not seem to differ in location or spread. It seems unlikely that any test is going to show a significant difference for these data. Below we show results for a Welch 2-sample t test and a nonparametric two-sample Wilcoxon test.

Welch t. A Welch two-sample t test shows a large P-value confirming that there is no significant difference between sample means 63.4 for women and 58.4 for men.

t.test(f,m)

        Welch Two Sample t-test

data:  f and m
t = 0.39396, df = 8.0631, p-value = 0.7038
alternative hypothesis: 
  true difference in means is not equal to 0
95 percent confidence interval:
 -24.01156  33.92267
sample estimates:
mean of x mean of y 
 63.40000  58.44444 

More generally, if there is any doubt that the men and women have the same population variances for such tests, you should use the Welch two-sample test, which does not assume equal variances.

It is not good practice to use an F-test to see whether variances differ significantly, and then to use that result to 'branch' to either the Welch t test (if different) and the pooled t test (if not).

Wilcoxon rank sum test. From such small samples it is difficult to say whether populations are normal. The plots show no marked skewness or far outliers, which might indicate non-normal populations. If prior experience with such assays have shown non-normal data, you might want to do a two-sample Wilcoxon (rank sum) test. This test does not assume that data are normal but does assume that the two population distributions are of the same shape (including roughly equal variability) and are continuous (no ties).

For your data, a two-sample Wilcoxon test finds no significant difference in location, but there is a warning message that the P-value may not be exactly correct, on account of ties (obvious in the plots).

wilcox.test(f,m)

        Wilcoxon rank sum test with continuity correction

data:  f and m
W = 26, p-value = 0.6875
alternative hypothesis: true location shift is not equal to 0

Warning message:
In wilcox.test.default(f, m) : 
  cannot compute exact p-value with ties

Note: In order to make sure you get practice with various kinds of two-sample tests, it may be OK for a textbook exercise to ask you to do two tests. However, in statistical practice, it is not OK to do multiple tests and then to report the one that gives the smallest P-value. Good practice is to choose the one test that best fits the situation and to report its result.