Question

Obesity is defined as a body mass index (BMI) of 30 /kgm2 or more. A 95% confidence interval for the percentage of U.S. women aged 30 to 39 who were obese was found to be 25.3% to 27.1%. What was the sample size? Round the intermediate calculations to four decimal places and round up your final answer to the next whole number.

Answer 1

Given the 95% confidence interval for the percentage of U.S. women aged 30 to 39 who were obese was found to be 25.3% to 27.1%

Upper limit of confidence interval = + Z_/2 * * (1 - ) / n

lower limit limit of confidence interval = - Z_/2 * * (1 - ) / n

Where is the sample proportion

n is the sample size

Given confidence level = 95% = 0.95

= 1 - confidence level

= 1 - 0.95

= 0.05

/2 = 0.035

Z_/2 will be z-score that has an area of 0.025 to its right or 0,975 which is 1.96 from online calculator

Z_/2 = 1.96

Upper limit of confidence interval = + Z_/2 * * (1 - ) / n

0.271 = + 1.96 * * (1 - ) / n ------------- (1)

lower limit limit of confidence interval = - Z_/2 * * (1 - ) / n

0.253 = - 1.96 * * (1 - ) / n --------------- (2)

Adding 1 and 2 we have

0.524 = 2 *

= 0.262

Substituting = 0.262 in (1) we have

0.271 = 0.262 + 1.96 * 0.262* (1 - 0.262) / n

0.009 = 1,96 * 0.193356 / n

0.193356 / n = 0.009 / 1.96

0.193356 / n = 0.004592

Squaring on both sides we have

(0.193356 / n) = 0.004592²

n = 0.193356 / 0.0000211

n = 9170.32604

n = 9171 rounded up to next whole number

So Sample size = 9171