week 9 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…

X

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 . X

week 9 7

Example –Binomial Distribution

• Suppose X1,…,Xn are random sample from Bernoulli(θ) distribution.

• We are interested in testing hypotheses about θ…