ECON 413
Fall 2004
Comparing Two Means
An important type of statistical inference concerns the comparison of two population
means. For example, we may want to find out if boys are better than girls at mathematics.
Here, the question we are interested in is not whether all boys are better than girls, but
rather whether boys are better than girls on average. Assuming that proclivity toward
mathematics can be tested using standardized tests, one may compare the average test
scores of a sample of boys and a sample of girls to learn about the difference between the
population mean scores of boys and girls.
Let µ1 and µ2 denote the mean of a variable
x
for the populations 1 and 2, respectively,
and let σ1 and σ2 denote the respective standard deviations. Suppose we take independent
random samples of size n1 and n2 from the two populations and calculate the two sample
averages 1
x
and 2
x
. Assuming that both populations are normally distributed, the
difference between the sample averages 12
()
x
x
−
will be normally distributed with mean
12
()
µ
µ
− and standard deviation
12
22
12
()
12
xx nn
σσ
σ
−
⎛⎞
=+
⎜⎟
⎝⎠
This means that the standardized difference
12 12
22
12
12
()(xx
z
nn
)
µ
µ
σσ
−−−
=⎛⎞
+
⎜⎟
⎝⎠
will be N(0,1).This is the two-sample z statistic.
We can test hypotheses about the differences of the means by using this statistic. We can
also construct confidence intervals in the usual way. For example, a level C confidence
interval for 12
()
µ
µ
− can be constructed as
22
12
12
12
()*xx z nn
σσ
⎛⎞
−± +
⎜⎟
⎝⎠
However, in practice, we usually don’t know the values of the population standard
deviations and we need to estimate them using the sample standard deviations. To
construct confidence intervals and to test hypotheses in this situation, we can use the two-
sample t statistic:
