Question

in the general american population, about 3.9 percent of adult men are 62\ or taller. out of 30 adult men find the probability that 1. exactly half are s are 62\ or taller. 2. at most a third i are 62\ or taller. 3. at least 10 are 62\ or taller. leave your answers upto 3 decimal places after trounding. example: 0.88999 should be 0.890 0.9998 should be 1.000

Explanation:

Step1: Identify binomial distribution parameters

Let \(n = 30\) (number of trials, i.e., number of adult - men), \(p=0.039\) (probability of an adult man being 6'2" or taller). The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\).

Step2: Calculate probability for exactly half

If exactly half (\(k = 15\)), then \(C(30,15)=\frac{30!}{15!(30 - 15)!}\), \(p = 0.039\), \(1-p=0.961\).
\[

$$\begin{align*} P(X = 15)&=\frac{30!}{15!15!}\times(0.039)^{15}\times(0.961)^{15}\\ &\approx0 \end{align*}$$

\]

Step3: Calculate probability for at most a third

A third of 30 is \(k = 10\). \(P(X\leq10)=\sum_{k = 0}^{10}C(30,k)\times(0.039)^{k}\times(0.961)^{30 - k}\).
Using a binomial - probability calculator or software, we find \(P(X\leq10)\approx1.000\).

Step4: Calculate probability for at least 10

\(P(X\geq10)=1 - P(X\lt10)=1-\sum_{k = 0}^{9}C(30,k)\times(0.039)^{k}\times(0.961)^{30 - k}\approx0\)

Answer:

1. 0.000
2. 1.000
3. 0.000