Question

The heights of 18-year-old men are approximately normally distributed with mean 68 inches and standard deviation 3 inches. What is the probability that an 18-year-old man selected at random is greater than 65 inches tall? Round your answer to four decimal places.

Answer 1

Answer:

The probability that an 18-year-old man selected at random is greater than 65 inches tall is 0.8413.

Step-by-step explanation:

We are given that the heights of 18-year-old men are approximately normally distributed with mean 68 inches and a standard deviation of 3 inches.

Let X = heights of 18-year-old men.

So, X ~ Normal($\mu=68,\sigma^{2} =3^{2}$)

The z-score probability distribution for the normal distribution is given by;

                              Z  =  $\frac{X-\mu}{\sigma}$  ~ N(0,1)

where, $\mu$ = mean height = 68 inches

           $\sigma$ = standard deviation = 3 inches

Now, the probability that an 18-year-old man selected at random is greater than 65 inches tall is given by = P(X > 65 inches)

       P(X > 65 inches) = P( $\frac{X-\mu}{\sigma}$ > $\frac{65-68}{3}$ ) = P(Z > -1) = P(Z < 1)

                                                                = 0.8413

The above probability is calculated by looking at the value of x = 1 in the z table which has an area of 0.8413.