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7. Product and Process Comparisons
7.1. Introduction
7.1.3. What are statistical tests?

Critical values and \(p\) values

Determination of critical values Critical values for a test of hypothesis depend upon a test statistic, which is specific to the type of test, and the significance level, \(\alpha\), which defines the sensitivity of the test. A value of \(\alpha\) = 0.05 implies that the null hypothesis is rejected 5 % of the time when it is in fact true. The choice of \(\alpha\) is somewhat arbitrary, although in practice values of 0.1, 0.05, and 0.01 are common. Critical values are essentially cut-off values that define regions where the test statistic is unlikely to lie; for example, a region where the critical value is exceeded with probability \(\alpha\) if the null hypothesis is true. The null hypothesis is rejected if the test statistic lies within this region which is often referred to as the rejection region(s). Critical values for specific tests of hypothesis are tabled in chapter 1.
Information in this chapter This chapter gives formulas for the test statistics and points to the appropriate tables of critical values for tests of hypothesis regarding means, standard deviations, and proportion defectives.
\(p\)-values Another quantitative measure for reporting the result of a test of hypothesis is the \(p\)-value. The \(p\)-value is the probability of the test statistic being at least as extreme as the one observed given that the null hypothesis is true. A small \(p\)-value is an indication that the null hypothesis is false.
Good practice It is good practice to decide in advance of the test how small a \(p\)-value is required to reject the test. This is exactly analagous to choosing a significance level, \(\alpha\), for test. For example, we decide either to reject the null hypothesis if the test statistic exceeds the critical value (for \(\alpha\) = 0.05) or analagously to reject the null hypothesis if the \(p\)-value is smaller than 0.05. It is important to understand the relationship between the two concepts because some statistical software packages report \(p\)-values rather than critical values.
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