Question

your body mass index (bmi) is your weight in kilograms divided by the square of your height in meters. online bmi calculators allow you to enter weight in pounds and height in inches. high bmi is a common but controversial indicator of being overweight or obese. a study by the national center for health statistics found that the bmi of american young women (ages 20 to 29) is approximately normally distributed with mean 26.8 and standard deviation 7.4.
(a) people with bmi less than 18.5 are often classified as \underweight.\
what percent of american young women are underweight by this criterion? (round to 4 decimal places. then convert your answer to a percentage.) %
(b) suppose we select two american young women in this age group at random.
find the probability that at least one of them is classified as underweight. (round to 4 decimal places. leave your answer in decimal form.)

Explanation:

Step1: Calculate the z - score for part (a)

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 18.5\), \(\mu=26.8\), and \(\sigma = 7.4\).

$$z=\frac{18.5 - 26.8}{7.4}=\frac{- 8.3}{7.4}\approx - 1.1216$$

Step2: Find the probability for part (a)

Using the standard normal distribution table or a calculator with a normal - distribution function (\(P(Z<z)\)), for \(z=-1.1216\), \(P(Z < - 1.1216)\approx0.1314\) (using a calculator: \(P(X<18.5)=\Phi(-1.1216)\), where \(\Phi(z)\) is the cumulative distribution function of the standard normal distribution).

Step3: Calculate the probability for part (b)

Let \(p = 0.1314\) (the probability of a single woman being underweight from part (a)). The probability that a woman is not underweight is \(q=1 - p=1 - 0.1314 = 0.8686\).
The probability that neither of the two women is underweight is \(q\times q=(0.8686)^{2}=0.7545\).
The probability that at least one is underweight is \(P(X\geq1)=1 - P(X = 0)\).

$$P(X\geq1)=1-(0.8686)^{2}=1 - 0.7545=0.2455$$

Answer:

(a) \(13.14\%\)
(b) \(0.2455\)