University of York
Department of Health Sciences
Applied Biostatistics
Suggested answers to exercise: Comparing the means of small
samples
Question 1
a) What method could be used to calculate P in this study? In this study we have means
from two independent samples. Hence we can use an unpaired or two sample t test to test
the null hypothesis that the means are the same in the populations from which samples
were drawn.
b) What conditions, if any, do the data have to fulfil for the method to be valid? This
method would assume that the data are from Normal distributions, with the same
variance.
c) Are they likely to be fulfilled here? These variables are measurements of skeletal size
which often follow a Normal distribution. The standard deviations are similar. They are
also small in comparison with the means, providing no evidence of skewness.
d) From these P values, can we conclude that the length of femoral neck in elderly women
has increased over time? There is good evidence that the length of the femoral neck is
different in the populations which these samples represent because the difference was
statistically significant. To conclude that there is a change over time we must assume
that the samples are truly representative of elderly white women in these two decades.
e) Can we conclude that the width of femoral neck in elderly women has not increased over
time? There is no evidence that the width of the femoral neck is different in the
populations which these samples represent because the difference was not significant.
However, this does not mean that it has not changed. The sample may be too small to
detect a change which has occurred. A confidence interval would be more informative.
The difference is 0.5 with 95% confidence -1.0 to 2.0. The clinical implication is that
elderly women in the 1990s had longer, but not thicker, femoral necks than similar
women in the 1950s. Longer bones may be more likely to break and so this may explain
any increase in fractured neck of femur.
Question 2
a) What is meant by no significant difference? No significant difference is the result of a
significance test comparing PEFR in children at two points in time after exposure. The
null hypothesis is that there is no change in PEFR in the population from which the
sample of children was drawn. The data was used to give a test statistic which has a
known distribution if the null hypothesis is true. We then find the probability of data as
or more extreme as that observed if the null hypothesis were true. If the probability is
small then we have good evidence against the null hypothesis. Conventionally, we use
P=0.05 as the cut-off. Here the probability must be >0.05, therefore we conclude that the
study failed to detect a difference. This does not necessarily mean that no difference
exists since we cannot prove that the null hypothesis is true on the basis of a significance
test.
