In yesterday’s post<\/a> I began about hypothesis testing, type I, and type II errors. Specifically, a type II error would be continuing to believe the mean loaf mass is 454g, when in fact it’s drifted to 470g.<\/p>\n Let’s imagine that, indeed, the mean loaf mass is 470g. Furthermore, let’s imagine the standard deviation is calculated, from the sample, as 25g.<\/p>\n Imagine a sample size of 31 loaves, so the degrees of freedom is 30. Since the standard deviation is calculated from the sample, we use a t statistic. With null hypothesis H0<\/sub>=454g, at significance level of 5%, and with a two-sided test, the rejection region is x<\/span>>454+2.042*25\/(31)1\/2<\/sup> or x<\/span><454-2.042*25\/(31)1\/2<\/sup>.<\/p>\n Rejection region:<\/p>\n x<\/span> > 463g, or x<\/span> < 445g <\/p>\n If the mean mass truly is 470g, the probability that x<\/span> > 463g is 93.5%, while the probability it’s less than 445g is virtually 0. Therefore, the probability of realizing that the mean loaf mass is no longer 454g (the power of the test in this context) is 93.5%. Conversely, the probability of continuing to believe the mean loaf mass is 454g (a type II error), is 6.5%.<\/p>\n In stats texts, β is sometimes used to refer to the probability of a type II error, while 1-β is used to refer to the power of the test, aka its ability to determine the falsity of H0<\/sub>. In this case, β=6.5%, while 1-β=93.5%.<\/p>\n Source:<\/p>\n Harnett, Donald L. and James L. Murphy. Statistical Analysis for Business and Economics<\/u>. Don Mills: Addison-Wesley, 1986.<\/p>\n