Question

Assume that the heights of men are normally distributed with a mean of 69.0 inches and a standard deviation of 2.8 inches. The U.S. Marine Corps requires that the heights of men be between 64 and 78 inches. If 500 men want to enlist in the U.S. Marine Corps, how many would you not expect to meet the height requirements?

Answer 1

19

Explanation:

If a continuous random variable X is normally distributed with mean μ and variance σ², it is written as:

`\boxed{X \sim\text{N}(\mu,\sigma^2)}`

Given:

• Mean μ = 69.0 inches
• Standard deviation σ = 2.8 inches

Therefore, if the heights of men are normally distributed:

`\boxed{X \sim\text{N} \left(69.0, 2.8^2 \right)}`

where X is height in inches.

Given:

• The U.S. Marine Corps requires that the heights of men be between 64 and 78 inches.

Find the probability that the heights of men is between 64 and 78 inches by calculating P(64 ≤ X ≤ 78).

Calculator input for "normal cumulative distribution function (cdf)":

• Lower bound: x = 64
• Upper bound: x = 78
• μ = 69.0
• σ = 2.8

⇒ P(64 ≤ X ≤ 78) = 0.9622733869...

To calculate how many men who want to enlist in the U.S. Marine Corps would not expect to meet the height requirements from a group of 500, multiply 500 by 1 less than the probability found above:

⇒ 500 × [1 - P(64 ≤ X ≤ 78)]

⇒ 500 × [1 - 0.9622733869...]

⇒ 500 × 0.03772661307...

⇒ 18.86330654...

Therefore, we would expect for 19 men to not meet the height requirements.