1. 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

2. 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

3. 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

4. Example
STA248 week 10

5. 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

6. Example
STA248 week 10