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.
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Bootstrap Hypothesis Testing - Introduction
• Suppose we have a small sample from some population and we wish
to test H 0 :   0 vs H a :   0 .
• As a test statistics we will use the sample mean X .
• We reject the H0 in favor of Ha if X is large.
• The P-values will be PX  x .| H 0 is true 
• We want the bootstrap estimate of this P-value.
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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
yi  xi  x  0 .
• Draw B bootstrap samples (sampling with replacement for nonparametric bootstrap) from y1 ,..., yn and for each bootstrap sample
*
calculate y j , 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 twotailed test…
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Example
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Test for a Single Variance
• Suppose X1, …, Xn is a random sample from a N(μ, σ2) distribution.
• We are interested in testing H 0 :  2   02 versus a one sided or a
two sided alternative…
• Then…
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Example
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