Hypothesis tests and confidence intervals have a one-to-one correspondence.

**RELATIONSHIP BETWEEN CONFIDENCE INTERVALS AND HYPOTHESIS TESTS**

Hypothesis tests and confidence intervals have a
one-to-one correspondence. This correspondence allows us to use a confidence
interval to form a hypothesis test or to use the critical regions defined for a
hypothesis test to construct a confidence inter-val. Up to this point, we have
not needed this relationship, as we have constructed hypothesis tests and
confidence intervals independently. However, in the next sec-tion we will
exploit this relationship for bootstrap tests. With the bootstrap, it is
natural to construct confidence intervals for parameters. We will use the
one-to-one correspondence between hypothesis tests and confidence intervals to
determine a bootstrap hypothesis test based on a bootstrap confidence interval
(refer to Section 9.12).

The correspondence works as follows: Suppose we
want to test the null hypothe-sis that a parameter θ = θ_{0}, versus the alternative hypothesis that θ ≠ θ_{0} at the 100 % significance level; we have a method
to obtain a 100(1 – α)% confidence interval for θ. Then we test the null hypothesis θ = θ_{0} as follows: If θ_{0} is contained in the 100(1 – α)%
confidence interval for θ, then do not reject *H*_{0}; if θ_{0} lies outside the region, then reject *H*_{0}. Such a test will have a
significance level of 100α%. By 100α% significance we mean the same thing as an α level but express as a percentage.

On the other hand, suppose we have a critical
region defined for the test of a null hypothesis that θ = θ_{0}, against a two-sided alternative at the 100α% significance level. Then, the set of all values of θ_{0} that would lead to not rejecting the null hypothesis
form a 100(1 – α)% confidence region for θ.

As an example let us consider the one sample test
of a mean with the variance known. Suppose we have a sample of size 25 with a
standard deviation of 5. The sample mean is 0.5, and we wish to test μ = 0
versus the alternative that μ ≠ 0. A 95% confidence interval for μ is then [ – 1.96 /√n, + 1.96 σ/√n] = [0.5 – 1.96, 0.5 + 1.96] = [–1.46, 2.46], since σ= 5 and √*n* = 5. Thus, values of the sample mean that fall into this interval are
in the nonrejection region for the 5% significance level test based on the
one-to-one correspondence between hypothesis tests and confidence intervals. In
our case with = 0.5, we do
not reject *H*_{0}, because 0
is contained in the interval The same would be true for any value in the
interval. The nonrejection region for the 5% level two-sided test contains the
values of such that 0 lies
inside the interval, and the rejection region is the set of values such that 0 lies outside the
interval, which is formed by +
1.96 < 0 or – 1.96 >
0 or < –1.96 or > 1.96 or || > 1.96.

Note that had we constructed the 5% two-sided test
directly, using the procedure we developed in Section 9.3, we would have
obtained the same result.

Also, by taking the critical region defined by || > 1.96 that we obtain
directly in Section 9.3, the one-to-one correspondence gives us a 95%
confidence interval [0.5 - 1.96, 0.5 + 1.96] = [–1.46, 2.46], exactly the confidence
interval we would get directly using the method of Section 8.4. In the formula
for the two-sided test, we replace with 0.5 and σ/√n with 1.0.

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