Chapter 13, Problem 8PS

a.

To determine

To find: The probability that $3$ randomly selected US male adults are all more than $6$ feet tall.

Answer to Problem 8PS

Probability is near about $3.997\%$ .

Explanation of Solution

Given Information: Mean is $69$ , standard deviation is $3.1$ .

Calculation:

For $x=72inches$ ,

  $z=\frac{72-69}{3.1}\approx 1.0$

  $\begin{array}{l}P(x\ge 72)=P(z\ge 1.0)\\ \\ P(x\ge 72)=1-P(z\le 1.0)\\ \\ P(x\ge 72)=1-0.8413=0.1587\end{array}$

Looking value in the standard normal table,

Calculating probability that all three are less than $6$ feet tall,

  $\begin{array}{l}$ P={(0.1587)}^3$\\ P=0.003996969\\ \\ \Rightarrow P\approx 3.997\%\end{array}$

Thus, probability is near about $3.997\%$ .

b.

To determine

To find: The probability that $5$ randomly selected US male adults are all between $65-75$ inches tall.

Answer to Problem 8PS

Probability is near about $51.14\%$ .

Explanation of Solution

Given Information: Mean is $69$ , standard deviation is $3.1$ .

Calculation:

For $x=65$ ,

  $\begin{array}{l}z=\frac{65-69}{3.1}\\ \\ \approx -1.3\end{array}$

For $x=75$ ,

  $\begin{array}{l}z=\frac{75-69}{3.1}\\ \\ \approx 1.9\end{array}$

  $\begin{array}{l}P(65\le x\le 75)=P(-1.3\le z\le 1.9)\\ \\ P(65\le x\le 75)=P(z\le 1.9)-P(z\le -1.3)\\ \\ =0.9713-0.0968\\ \\ =0.8745\end{array}$

Calculating probability that all five are between $65-75$ inches tall,

  $\begin{array}{l}$ P={(0.8745)}^5$=0.51144\end{array}$

Thus, probability is near about $51.14\%$ .