Question

The distribution of heights for adult men in a certain population is approximately normal with mean 70 inches and standard deviation 4 inches. Which of the following represents the middle 80 percent of the heights?

A) 50 inches to 73.37 inches

B) 62 inches to 78 inches

C) 64.87 inches to 75.13 inches

D) 66 inches to 74 inches

E) 66.63 inches to 90 inches

Answer 1

Answer:

The correct option is (C).

Step-by-step explanation:

Let X = heights for adult men in a certain population.

The random variable X follows a Normal distribution with mean μ = 70 inches and standard deviation, σ = 4 inches.

Compute the heights that represents the middle 80 percent of the heights as follows:

$P(x_{1}<X<x_{2})=0.80$

$P(\frac{x_{1}-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{x_{2}-\mu}{\sigma})=0.80$

$P(-z<Z<z)=0.80\\P(Z<z)-P(Z<-z)=0.80\\P(Z<z)-[1-P(Z<z)]=0.80\\2P(Z<z)=1.80\\P(Z<z)=0.90$

The value of z for the probability P (Z < z) = 0.90 is,

z = 1.282

*Use a z-table.

Compute the two heights as follows:

$-1.28=\frac{x_{1}-70}{4}\\x_{1}=70-(1.282\times 4)\\x_{1}=64.872\\x_{1}\approx 64.87$                $1.28=\frac{x_{2}-70}{4}\\x_{2}=70+(1.282\times 4)\\x_{2}=75.128\\x_{2}\approx 75.13$

Thus, the two heights which represents the middle 80 percent of the heights are 64.87 inches and 75.13 inches.

The correct option is (C).

Answer 2

The height that represents the middle 80 percent of the heights is 64.87 inches and 74.13 inches and this can be determined by using the given data.

Given :

The distribution of heights for adult men in a certain population is approximately normal with a mean of 70 inches and a standard deviation of 4 inches.

The heights that represent the middle 80 percent of the heights are:

$\rm P(x_1<X<x_2) = 0.80$

$\rm P(\dfrac{x_1-\mu}{\sigma}<\dfrac{X-\mu}{\sigma}<\dfrac{x_2-\mu}{\sigma}) = 0.80$

P(-z < Z < z) = 0.80

P(Z < z) - P(Z < -z) = 0.80

P(Z < z) - (1 - P(Z < z)) = 0.80

2P(Z < z) = 1.80

P(Z < z) = 0.90

So, the value of z is 1.282.

Now, put the values of height as follows:

$-1.28=\dfrac{x_1-70}{4}$

$x_1 = 70- 5.128$

$x_1 = 64.872$

$1.28=\dfrac{x_2-70}{4}$

$x_2 = 1.28\times 4+70$

$x_2 = 75.128$

$x_2 \approx 75.13$

Therefore, the height that represents the middle 80 percent of the heights is 64.87 inches and 74.13 inches.

For more information, refer to the information given below:

https://brainly.com/question/23044118