|
7. Product and Process Comparisons 7.1. Introduction 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:
\(H_a\): mean linewidth \(\ne\) 500 micrometers |
| Hypothesis test for the mean | For the test, the sample mean, \(\bar{Y}\), 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: $$ \begin{eqnarray} \frac{\bar{Y}-500}{\sigma / \sqrt{N}} \le z_{\alpha/2} \,\, \mbox{ or } \,\, \frac{\bar{Y}-500}{\sigma / \sqrt{N}} \ge z_{1-\alpha/2} \, , \end{eqnarray} $$ 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 $$ \bar{Y} \pm \frac{z_{1-\alpha/2} \, \sigma}{\sqrt{N}} \, . $$ |
| Equivalent confidence interval | In fact, all values bracketed by this interval would be accepted as null values for a given set of test data. |