HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 606

Procedure for testing hypotheses The procedure for hypothesis testing involves the following six steps: 1. Declare a null hypothesis, H0. This is the hypothesis to be tested. For example, H0: µ1-µ2 = 0, i.e., we hypothesize that the mean value of population 1 and the mean value of population 2 are the same. If H0 is true, any observed difference in means is attributed to errors in random sampling. 2. Declare an alternate hypothesis, H1. For the example under consideration, it could be H1: µ1-µ2 ≠ 0 [Note: this is what we really want to test.] 3. Determine or specify a test statistic, T. In the example under consideration, T will be based on the difference of observed means, X1-X2. 4. Use the known (or assumed) distribution of the test statistic, T. 5. Define a rejection region (the critical region, R) for the test statistic based on a pre-assigned significance level α. 6. Use observed data to determine whether the computed value of the test statistic is within or outside the critical region. If the test statistic is within the critical region, then we say that the quantity we are testing is significant at the 100α percent level. Notes: 1. For the example under consideration, the alternate hypothesis H1: µ1-µ2 ≠ 0 produces what is called a two-tailed test. If the alternate hypothesis is H1: µ1-µ2 > 0 or H1: µ1-µ2 < 0, then we have a one-tailed test. 2. The probability of rejecting the null hypothesis is equal to the level of significance, i.e., Pr[T∈R|H0]=α. The notation Pr[A|B] represents the conditional probability of event A given that event B occurs. Errors in hypothesis testing In hypothesis testing we use the terms errors of Type I and Type II to define the cases in which a true hypothesis is rejected or a false hypothesis is accepted (not rejected), respectively. Let T = value of test statistic, R = rejection region, A = acceptance region, thus, R∩A = ∅, and R∪A = Ω, where Ω = the parameter space for T, and ∅ = the empty set. The probabilities of making an error of Type I or of Type II are as follows: Page 18-35