2. [101 A national health organization reports that 34% of adults in the United

Question

2. [101 A national health organization reports that 34% of adults in the United States are obese. Researchers at a local community health service would like to determine if the adults in their community are representative of the entire United States in terms of obesity. In other words, they would like to perform a hypothesis test (at the = 0.05 significance level) with the following null and alternative hypotheses: Ho: p = 0.34 Ha: p 0.34, where p is the proportion of all adults in their community who are obese. In order to test this hypothesis, 200 adults from this community will be randomly selected and the hypothesis test performed. If 27% of all adults in this community are obese, what is the power of this hypothesis test?

Explanation / Answer

for p=0.34 and n=200 ; std error =(p*(1-p)/n)1/2 =0.0335

here for 0.05 level critical value z=1.96

hence failue to rejection region =(p-z*Std error <P<p+z*Std error )=(0.34-1.96*0.0335<P<0.34+1.96*0.0335)

=(0.2743<P(0.4057)

for true mean p=0.27 and n=200 ; std error =(p*(1-p)/n)1/2 =0.0314

hence power of test =1-P(failure to reejct) =1-P(0.2743<P(0.4057)

=1-P((0.2743-0.27)/0.0314<Z<(0.4057-0.27)/0.0314)=1-P(0.1385<Z<4.3211)=1-(0.9999-0.5550)=0.5551

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