Relationship between Confidence Intervals and Hypothesis Tests

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Chapter: Biostatistics for the Health Sciences: Tests of Hypotheses

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 = θ0, versus the alternative hypothesis that θ θ0 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 = θ0 as follows: If θ0 is contained in the 100(1 – α)% confidence interval for θ, then do not reject H0; if θ0 lies outside the region, then reject H0. 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 = θ0, against a two-sided alternative at the 100α% significance level. Then, the set of all values of θ0 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 H0, 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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