7.1.3.1. Critical values and p values (2024)

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

7.1.3.1.

Critical values and \(p\) values

Determination of critical valuesCritical values for a test of hypothesis depend upon a teststatistic, which is specific to the type of test, and thesignificance level, \(\alpha\),which defines the sensitivity of the test. A value of \(\alpha\)= 0.05implies that the null hypothesis is rejected 5 % of the time whenit is in fact true. The choice of \(\alpha\)is somewhat arbitrary, although in practice values of 0.1, 0.05, and0.01 are common. Critical values are essentially cut-off valuesthat define regions where the test statistic is unlikely to lie;for example, a region where the critical value is exceeded withprobability \(\alpha\)if the null hypothesis is true. The null hypothesis is rejectedif the test statistic lies within this region which is oftenreferred to as the rejection region(s).Critical values forspecific tests of hypothesis are tabled in chapter 1. Information in this chapterThis chapter gives formulas for the test statistics and points tothe appropriate tables of critical values for tests of hypothesisregarding means, standard deviations, and proportion defectives. \(p\)-valuesAnother quantitative measure for reporting the result of a testof hypothesis is the \(p\)-value.The \(p\)-valueis theprobability of the test statistic being at least as extreme as the oneobserved given that the null hypothesis is true. A small \(p\)-valueis an indication that the null hypothesis is false.Good practiceIt is good practice to decide in advance of the test how smalla \(p\)-valueis required to reject the test. This is exactlyanalagous to choosing a significance level, \(\alpha\),for test.For example, we decide either to reject the null hypothesisif the test statistic exceeds the critical value (for\(\alpha\) = 0.05)oranalagously to reject the null hypothesis if the \(p\)-valueis smaller than 0.05. It is important to understand therelationship between the two concepts because some statisticalsoftware packages report \(p\)-valuesrather than critical values.