Test for Mean of a Non-Normal Population – small n Suppose X1, …, Xn are iid from some distribution with E(Xi)=μ and Var(Xi)= σ2. Further suppose that n is small and we are interested in testing hypotheses about μ. Can use the t-test since it is robust as long as there are no extreme outliers and skewness. Alternatively, we can use bootstrap hypothesis testing. STA248 week 10

Bootstrap Hypothesis Testing - Introduction Suppose we have a small sample from some population and we wish to test vs . As a test statistics we will use the sample mean . We reject the H0 in favor of Ha if is large. The P-values will be . We want the bootstrap estimate of this P-value. STA248 week 10

Bootstrap Hypothesis Testing - Procedure To obtain the bootstrap estimate of the P-value we need to generate samples with H0 true. Instead of re-sampling from original data, we resample from . Draw B bootstrap samples (sampling with replacement for non-parametric bootstrap) from and for each bootstrap sample calculate , j =1,…,B. The bootstrap estimate of the P-value is ….   For bootstrap testing, B is typically at least 3000. Similarly, can calculate the P-value for a lower-tailed test and a two-tailed test… STA248 week 10

Example STA248 week 10

Test for a Single Variance Suppose X1, …, Xn is a random sample from a N(μ, σ2) distribution. We are interested in testing versus a one sided or a two sided alternative… Then… STA248 week 10

Example STA248 week 10