advertisement

Review of Student’s t-test Comparison of means between two populations Assumptions These assumptions allow the sampling distribution of the test statistic to be calculated. normal distributions equal variances random sampling Hypotheses H0: B A 0 HA: B A 0 Test statistic based on sample estimators of population means. The sampling distributions are: 2 2 xB x A ~ N B A , A B nA nB z x B x A B A A2 nA B2 nB xB x A 1 1 2 n A nB ~ N 0,1 under the null hypothesis, and assumption of equal valiances. Substituting the sample estimator of the population variance: xB x A t 1 1 s 2 n A nB ~ tn A nB 2 Decision rule The decision on whether or not to reject the null hypothesis in favor of the alternative is based on the observed value of the test statistic being far from its central value. The critical points are determined such that the ex ante risk of false rejection (probability of ‘Type I’ error, denoted ‘’) is limited, traditionally to levels such as 0.05 and 0.01. Thus the decision rule: Reject H0 if t t1 2 with degrees of freedom (‘dof’) For example, if = 0.05, then the critical points are n A nB 2 . t0.975 . The null hypothesis is traditionally viewed as the status quo ante, the state of knowledge (or ignorance) before new evidence is evaluated. Thus, limiting the risk of Type I error makes the hypothesis test a conservative procedure. At the same time, the failure of some particular sample evidence to lead to the rejection of the null hypothesis is not to be viewed as a confirmation of the null hypothesis. The test may simply lack sufficient power to detect a meaningful violation. P-value Since the willingness to take risks is a subjective, but necessary, component of statistical testing, a transformation of the test statistic is used that allows other subjective determinations of risk. The p-value (‘probability value’) of the test is the probability of observing a value of the test statistic more damaging to the null hypothesis than the value actually observed, or the ex post risk of rejecting the null hypothesis. p - value prob t tobs With this transformation of the test statistic, the rejection rule can be restated as: Reject H0 if p