Simple versus Composite Hypothesis

1. Simple versus Composite Hypothesis • Recall, a simple hypothesis completely specifies the distribution. A composite does not. • When testing a simple null hypothesis versus a composite alternative, the power of the test is a function of the parameter of interest. • In addition, the power is also affected by the sample size. week 9

2. Example week 9

3. Test for Mean of Normal Population σ2 is known • Suppose X1, …, Xn is a random sample from a N(μ, σ2) distribution where σ2 is known. We are interested in testing hypotheses about μ. • The test statistics is the standardized version of the sample mean . • We could test three sets of hypotheses… week 9

4. Test for Mean of Normal Population σ2 is unknown • Suppose X1, …, Xn is a random sample from a N(μ, σ2) distribution where σ2 is unknown, n is small and we are interested in testing hypotheses about μ. • The test statistics is... week 9

5. Example • In a metropolitan area, the concentration of cadmium (Cd) in leaf lettuce was measured in 6 representative gardens where sewage sludge was used as fertilizer. The following measurements (in mg/kg of dry weight) were obtained. Cd: 21 38 12 15 14 8 • Is there evidence that the mean concentration of Cd is higher than 12. week 9

6. Test for Mean of a Non-Normal Population • Suppose X1, …, Xn are iid from some distribution with E(Xi)=μ and Var(Xi)= σ2. Further suppose that n is large and we are interested in testing hypotheses about μ. • Since n is large the CLT applies to the sample mean and the test statistics is again the standardized version of the sample mean . week 9

7. Example –Binomial Distribution • Suppose X1,…,Xn are random sample from Bernoulli(θ) distribution. • We are interested in testing hypotheses about θ… week 9