assume that when human resource managers are randomly selected, 50% say job applicants should follow up within two weeks. if 14 human resource managers are randomly selected, find the probability that fewer than 3 of them say job applicants should follow up within two weeks. the probability is □ (round to four decimal places as needed.)

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Accepted Answer

# Explanation:
## Step1: Identify the binomial distribution parameters
This is a binomial - distribution problem with $n = 14$ (number of trials, i.e., number of managers selected), $p=0.5$ (probability of success, i.e., a manager saying job - applicants should follow up within two weeks), and we want to find $P(X < 3)=P(X = 0)+P(X = 1)+P(X = 2)$. 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 $P(X = 0)$
$$
\begin{align*}
C(14,0)&=\frac{14!}{0!(14 - 0)!}=1\
P(X = 0)&=C(14,0)\times(0.5)^{0}\times(1 - 0.5)^{14-0}\
&=1\times1\times(0.5)^{14}\
&=\frac{1}{16384}\approx0.000061
\end{align*}
$$
## Step3: Calculate $P(X = 1)$
$$
\begin{align*}
C(14,1)&=\frac{14!}{1!(14 - 1)!}=\frac{14!}{1!×13!}=14\
P(X = 1)&=C(14,1)\times(0.5)^{1}\times(1 - 0.5)^{14 - 1}\
&=14\times0.5\times(0.5)^{13}\
&=14\times\frac{1}{2}\times\frac{1}{8192}\
&=\frac{14}{16384}\approx0.000854
\end{align*}
$$
## Step4: Calculate $P(X = 2)$
$$
\begin{align*}
C(14,2)&=\frac{14!}{2!(14 - 2)!}=\frac{14\times13\times12!}{2\times1\times12!}=91\
P(X = 2)&=C(14,2)\times(0.5)^{2}\times(1 - 0.5)^{14 - 2}\
&=91\times(0.5)^{2}\times(0.5)^{12}\
&=91\times\frac{1}{4}\times\frac{1}{4096}\
&=\frac{91}{16384}\approx0.005554
\end{align*}
$$
## Step5: Calculate $P(X < 3)$
$$
\begin{align*}
P(X < 3)&=P(X = 0)+P(X = 1)+P(X = 2)\
&=\frac{1 + 14+91}{16384}\
&=\frac{106}{16384}\
&\approx0.0065
\end{align*}
$$
# Answer:
0.0065

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QUESTION IMAGE

Question

Explanation:

Step1: Identify the binomial distribution parameters

Step2: Calculate \(P(X = 0)\)

Step3: Calculate \(P(X = 1)\)

Step4: Calculate \(P(X = 2)\)

Step5: Calculate \(P(X < 3)\)

Answer: