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, 266 had BMIs of less than 25, 158 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.

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 is the probability of not concluding that more than 20% of the population is obese when the actual percentage of obese individuals is 25%? (Round your answer to four decimal places.)

Answer 1

SOLUTION:

From given data,

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, 266 had BMIs of less than 25, 158 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

The hypothesis are :

H₀ : =0.2

H_a : > 0.2

(a) State the appropriate hypotheses with a significance level of 0.05.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.)

This is right tailed test.

Out of

= 266+158+123 = 547

females in the sample =   = 123 had BMIs exceeding 30

the sample proportion is:

= / = 123 / 547

= 0.2248

The test statistic is :

= - /

= 0.2248 - 0.2 /

= 0.0248 / 0.0171027649

=1.45

P-Value is

P-Value = P(>1.45)

P-Value = 1 - P(<1.45)

P-Value = 1 - 0.9265

P-Value = 0.0735

P-Value = 0.07

What is the probability of not concluding that more than 20% of the population is obese when the actual percentage of obese individuals is 25%? (Round your answer to four decimal places.)

Let = 0.05 = >

Critical value = z = 1.645

The probability of not concluding that more than 20% of the population is obese when the actual percentage of obese individuals is 25% is

P = 25/100 = 0.25

= ( - + z / )

= (0.2 - 0.25 + 1.645 / )

= (-0.0218659 / 0.01851428)

= (-1.18) = 0.1190

= 0.1190