A common characterization of obese individuals is that their body mass index is

Question

A common characterization of obese individuals is that their body mass index is at least 30. An article reported that in a sample of 546 female workers, 426 had BMIs of less than 30, and 120 had BMIs at least 30. (a) Is there compelling evidence for concluding that more than 20% of the individuals in the female working population are obese? Use alpha = 0.05: be sure to state the appropriate hypotheses and determine the rejection region. (b) What is the probability of NOT concluding that more than 20% of the female working population is obese assuming the actual percentage is 24%

Explanation / Answer

Q.2 (a) H0 : p = 0.20

Ha : p > 0.20

standard error of sampling se0 = sqrt [p(1-p)/N] = sqrt [0.20 * 0.80/ 546] = 0.017

p = 120/546 = 0.2198

so test statistic

Z = (p - 0.20)/se0 = (0.2197 - 0.20)/ 0.017 = 1.16

critical value of Z = 1.645

so critical region is p > 0.20 + 1.645 * 0.017 or p > 0.228

here proportion of obese people in sample is less than the critical or rejection region.

(b) Here it is talking about calculation type II probability.

so Pr (Type II error) = Pr (p < 0.228; 0.24 ; 0.017)

so Z - value = ( 0.228 - 0.24)/ 0.017 = -0.706

so probaility of type II error = 0.239

so there are 0.239 probability that we will not reject the null even if actual female working percentage is 24%.

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