7.1.5. What is the relationship between a test and a confidence interval?

There is a correspondence between hypothesis testing and confidence intervals

In general, for every test of hypothesis there is an equivalent statement about whether the hypothesized parameter value is included in a confidence interval. For example, consider the previous example of linewidths where photomasks are tested to ensure that their linewidths have a mean of 500 micrometers. The null and alternative hypotheses are:

$H_0$:   mean linewidth = 500 micrometers

$H_a$:   mean linewidth $\ne$ 500 micrometers

Hypothesis test for the mean

For the test, the sample mean, $\overline{Y}$, is calculated from $N$ linewidths chosen at random positions on each photomask. For the purpose of the test, it is assumed that the standard deviation, $\sigma$, is known from a long history of this process. A test statistic is calculated from these sample statistics, and the null hypothesis is rejected if: $$\begin{array}{r}\frac{\overline{Y}-500}{\sigma /\sqrt{N}}\le z_{\alpha /2}\text{ or }\frac{\overline{Y}-500}{\sigma /\sqrt{N}}\ge z_{1-\alpha /2},\end{array}$$ where $z_{\alpha /2}$ and $z_{1-\alpha /2}$ are tabled values from the normal distribution.

Equivalent confidence interval

With some algebra, it can be seen that the null hypothesis is rejected if and only if the value 500 micrometers is not in the confidence interval $$\overline{Y}\pm \frac{z_{1-\alpha /2}\sigma }{\sqrt{N}}.$$

Equivalent confidence interval

In fact, all values bracketed by this interval would be accepted as null values for a given set of test data.