Download Statistics 571 Discussion 5: Hypothesis Testing and p-values and more Exams Data Analysis & Statistical Methods in PDF only on Docsity! STATISTICS 571 TA: Perla Reyes DISCUSSION 5 Review 1. The notion “MEAN”: (a) Population mean µ. is the mean of entire population, usually unknown. We use sample mean x̄ to estimate it. (b) Sample mean. is a certain number. After you get a set of observations –sample–, the number 1 n ∑n i=1 xi = x̄ is the sample mean. (c) Random variable X̄.: Suppose you decide to get a sample of size n from the population. Before your experiment, you know you will get n random variables from the population, and their average X̄ = 1n ∑n i=1 Xi is still a random variable. When you get different samples, X̄ may change. Any certain sample mean is a realization of the random variable X̄. 2. Steps for a significant test: (a) Parameter of interest Aspect of the population that it is of interest: µ, p. (b) Formulate the null(H0) and alternative(HA) hypothesis. i. H0 is the position that we wish to support unless there is strong evidence against it. Standard, known from before, established value. H0 : µ = µ0. ii. HA is challenging assertion or new idea, that one wishes to be able to check. Usually two- sided (HA : µ 6= µ0) is prefered unless there is a strong reason to use one-sided (HA : µ < µ0 or HA : µ > µ0). (c) Find test statistic and null distribution. The test statistic and the distribution of the test statistic under the null hypothesis depend on: i. The parameter we are testing. ii. The distribution of our population. iii. The information we have about the distribution of our population. The following table resume all the possible options that we have until now. Parameter of Interest Population Distribution Information Test statistic Distribution statistic µ N(µ, σ2) σ2 known Z = X̄−µ0 σ/ √ n Exactly N(0, 1) µ N(µ, σ2) σ2 unknown T = X̄−µ0 S/ √ n Exactly Tn−1 µ Unknown σ2 known and n large Z = X̄−µ0 σ/ √ n Approximately N(0, 1) µ Unknown σ2 unknown and n large T = X̄−µ0 S/ √ n Approximately N(0, 1) p Bi(n, p) np0 ≥ 5 and nq0 ≥ 5 Z = P Xi−np0√ np0q0 Approximately N(0, 1) p Bi(n, p) np0 < 5 or nq0 < 5 Y = ∑ Xi Exactly Bi(n, p0) (d) Calculate p-value. The p-value is the probability of observing an event as extreme or more extreme than what we observed, if H0 is true and using HA to determine what kinds of data constitute ”extreme” data. The are the possible options, for two cases. All other cases have a similar construction. email: reyes@stat.wisc.edu 1 Office: 248 MSC M2:30-3:30 R3:30-4:30
