Question

in a survey of a group of men, the heights in the 20 - 29 age group were normally distributed, with a mean of 69.3 inches and a standard deviation of 2.0 inches. a study participant is randomly selected. complete parts (a) through (d) below. (a) find the probability that a study participant has a height that is less than 65 inches. the probability that the study participant selected at random is less than 65 inches tall is 0.0158. (round to four decimal places as needed.) (b) find the probability that a study participant has a height that is between 65 and 70 inches. the probability that the study participant selected at random is between 65 and 70 inches tall is . (round to four decimal places as needed.)

Explanation:

Step1: Calculate z - scores

For \(x = 65\), \(z_1=\frac{65 - 69.3}{2.0}=\frac{- 4.3}{2.0}=-2.15\). For \(x = 70\), \(z_2=\frac{70 - 69.3}{2.0}=\frac{0.7}{2.0}=0.35\).

Step2: Find probabilities from z - table

The probability corresponding to \(z_1=-2.15\) is \(P(Z < - 2.15)=0.0158\) (from part (a)). The probability corresponding to \(z_2 = 0.35\) is \(P(Z<0.35)=0.6368\) from the standard - normal table.

Step3: Calculate the required probability

\(P(-2.15<Z<0.35)=P(Z < 0.35)-P(Z < - 2.15)=0.6368 - 0.0158=0.6210\)

Answer:

0.6210