A Level Maths Notes: S4 - Hypothesis Testing for the Equality of the Means of Two Populations if the Common Variance is Not known
When independent samples of size
and
are
taken from two normally distributed populations with means
and
and
known population standard deviations
and
the
random variable
is
normally distributed with mean
and
variance
is
normally distributed. We can test for the equality ofandby
doing a hypothesis test using
If
then
(1)
If
is
not known then we cannot use the last expression above. We can
however work out an estimate for the varianceusing
the sample variances
and
We
can use these as two estimates of
and
pool them by weighting them according to their sample size or degrees
of freedom:
so
Replacing
in
(1) with the pooled standard deviation
gives
the random variable
which
has a
distribution
with
degrees
of freedom.
The assumption
must
be made but in practice we can use this test if
and
are
not dissimilar – they do not differ by a factor of more than about
2.
Example: Test for the equality of the two means of the two sets of data at the 10% level:
A: 51.4, 76.7, 73.7, 66.2, 65.5, 49.7, 65.8, 62.1, 75.8, 62.0, 72.0, 55.0, 79.7, 65.4, 73.3
B: 86.0, 59.7, 68.6, 98.6, 87.7, 69.0, 80.2, 78.1, 69.8, 77.2
No assumption is made as two which is greater so we carry out a two tailed test:
and
Reject the null hypothesis.