6.7 Relationship Between Confi dence Intervals and Hypothesis Tests 79quantity for construction confi dence interval and testing hypotheses
about the unknown parameter p.
For a confi dence interval, the central limit theorem can be applied
for large n. So let Z=−−()X np 12 //⎡⎣ npˆˆ() 1 −p⎦⎤, where p^
is the
sample estimate of p , and X is the number of successes. The estimate
we use is p^
= X/n. Z has an approximate normal distribution with mean
0 and variance 1. This is the continuity - corrected version. Removing
the term − 1/2 from the numerator gives an approximation without the
continuity correction. Here Z can be used to invert to make a confi dence
interval statement about p using the standard normal distribution.
However, for hypothesis testing, we can take advantage of the fact that
p = p 0 under the null hypothesis to construct a more powerful test.
p 0 is used in place of p^
and p in the defi nition of Z. So we have
Z=− −()X np 00012 //⎡⎣ np() 1 −p ⎦⎤. Under the null hypothesis, this
continuity - corrected version has an approximate standard normal
distribution.
6.7 RELATIONSHIP BETWEEN CONFIDENCE
INTERVALS AND HYPOTHESIS TESTS
Suppose we want to test the null hypothesis that μ 1 − μ 2 = 0 versus the
two - sided alternative that μ 1 − μ 2 ≠ 0. We wish to test at the 0.05 sig-
nifi cance level. Construct a 95% confi dence interval for the mean dif-
ference. For the hypothesis test, we reject the null hypothesis if and
only if the confi dence interval does not contain 0. The resulting hypoth-
esis test has signifi cance level 0.05. Conversely, suppose we have a
hypothesis test with the null hypothesis μ 1 − μ 2 = 0 versus the alterna-
tive that μ 1 − μ 2 ≠ 0. Look at the region of values for the test statistic
where the null hypothesis is rejected. This region determines a set of
values for μ 1 − μ 2 that defi nes a 95% confi dence region for μ 1 − μ 2.
The same type of argument can be used to equate one - sided confi -
dence intervals with one - sided tests. So what we have shown is that for
every hypothesis test about a parameter with a given test statistic, there
corresponds a confi dence interval whose confi dence level = 1 − signifi -
cance level of the test. On the other hand, if we can construct a confi -
dence interval (one or two - sided) for a parameter θ , we can defi ne a
test of hypothesis about θ based on the confi dence interval, and the
hypothesis test will have signifi cance level α if the confi dence level is