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, 262 had BMIs of less than 25, 156 had BMIs that were at least 25 but less than 30, and 123 had BMIs exceeding 30. Is there compelling evidence for concluding that more than 20% of the individuals in the sampled population are obese?

(a)

State the appropriate hypotheses with a significance level of 0.05.

H₀: p = 0.20
H_a: p < 0.20H₀: p > 0.20
H_a: p = 0.20    H₀: p = 0.20
H_a: p ≠ 0.20H₀: p = 0.20
H_a: p > 0.20

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.)

z=P-value=

What can you conclude?

Reject the null hypothesis. There is sufficient evidence that more than 20% of the population of female workers is obese.Do not reject the null hypothesis. There is not sufficient evidence that more than 20% of the population of female workers is obese.    Do not reject the null hypothesis. There is sufficient evidence that more than 20% of the population of female workers is obese.Reject the null hypothesis. There is not sufficient evidence that more than 20% of the population of female workers is obese.

(b)

Explain in the context of this scenario what constitutes type I error.

A type I error would be declaring that 20% or less of the population of female workers is obese, when in fact more than 20% are actually obese.A type I error would be declaring that 20% or more of the population of female workers is obese, when in fact less than 20% are actually obese.    A type I error would be declaring that less than 20% of the population of female workers is obese, when in fact 20% or more are actually obese.A type I error would be declaring that more than 20% of the population of female workers is obese, when in fact 20% or less are actually obese.

Explain in the context of this scenario what constitutes type II error.

A type II error would be declaring that 20% or less of the population of female workers is obese, when in fact more than 20% are actually obese.A type II error would be declaring that 20% or more of the population of female workers is obese, when in fact less than 20% are actually obese.    A type II error would be declaring that less than 20% of the population of female workers is obese, when in fact 20% or more are actually obese.A type II error would be declaring that more than 20% of the population of female workers is obese, when in fact 20% or less are actually obese.

(c)

What is the probability of not concluding that more than 20% of the population is obese when the actual percentage of obese individuals is 26%? (Round your answer to four decimal places.)

Answer 1

We are testing,

H0: p= 0.20 vs H1: p>0.20

Under H0, Test statistic: -0.2/(√0.2*0.8/n) ~N(0,1) distribution

Now, = 123/(123+156+262)= 0.2274

And n= 123+156+262= 541

So, Test statistic: (0.2274-0.2)/(√0.2*0.8/541) = 1.59

p-value of this one sided z test is: P(z>1.59) = 0.0559

(From the standard normal distribution tables)

Since the p-value of our test > significance level of 0.05, we have insufficient evidence to Reject H0 at the 5% level of significance. So option b is correct here.

b) Type I error is the rejection of a true null hypothesis. It means rejecting H0 when its infact true. Here it means concluding that p>0.20 when it's false. So it means declaring that more than 20% of the population is obese when it's infact less.

Type II error is the non rejection of a false null hypothesis. It means concluding that p<=0.2 when it's actually false. So it means concluding that less than or equal to 20% of the population is obese when it's infact more.