Question

A common characterization of obese individuals is that their body mass index is at least 30 [BMI = weight/(height)², where height is in meters and weight is in kilograms]. An article reported that in a sample of female workers, 265 had BMIs of less than 25, 159 had BMIs that were at least 25 but less than 30, and 122 had BMIs exceeding 30. Is there compelling evidence for concluding that more than 20% of the individuals in the sampled population are obese?

Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.)

What is the probability of not concluding that more than 20% of the population is obese when the actual percentage of obese individuals is 24%? (Round your answer to four decimal places.)

Answer 1

Ans:

sample proportion of women who are obese=122/(265+159+122)=0.2234

n=265+159+122=546

Test statistic:

z=(0.2234-0.20)/sqrt(0.20*(1-0.20)/546)

z=1.37

p-value=P(z>1.37)=0.0854

probability of not concluding that more than 20% of the population is obese when the actual percentage of obese individuals is 24%=P(type II error)=P(fail to reject Ho/when H0 is false)

assume alpha=0.05

critical z value=1.645

p-hat=0.20+1.645*sqrt(0.2*(1-0.2)/546)

p-hat=0.2+1.645*0.01712

p-hat=0.2+0.02816=0.22816

Now,

z=(0.22816-0.24)/sqrt(0.24*(1-0.24)/546)

z=-0.648

P(type II error)=P(z<=-0.648)=0.2585