a poll is given, showing 50% are in favor of a new building project. if 150 people are chosen at random, answer the following.
a) which is the correct wording for the random variable?
rv x = the number of 150 randomly selected people that are in favor of a new building project
b) pick the correct symbol: n = 150
c) pick the correct symbol: p = 0.5
d) what is the probability that exactly 72 of them are in favor of a new building project?
round final answer to 4 decimal places.
e) what is the probability that less than 72 of them are in favor of a new building project?
round final answer to 4 decimal places.
f) what is the probability that more than 72 of them are in favor of a new building project?
round final answer to 4 decimal places.
g) what is the probability that exactly 79 of them are in favor of a new building project?
round final answer to 4 decimal places.
h) what is the probability that at least 79 of them are in favor of a new building project?
round final answer to 4 decimal places.
i) what is the probability that at most 79 of them are in favor of a new building project
round final answer to 4 decimal places.

Question

Accepted Answer

# Explanation:
## Step1: Identify the distribution
Since we have a fixed number of trials ($n = 150$), two possible outcomes (in - favor or not in - favor), and a constant probability of success ($p=0.5$), $X$ follows a binomial distribution $X\sim B(n = 150,p = 0.5)$. The probability mass function of a binomial distribution is $P(X=k)=\binom{n}{k}p^{k}(1 - p)^{n - k}$, where $\binom{n}{k}=\frac{n!}{k!(n - k)!}$

## Step2: Calculate $P(X = 72)$
For $n = 150$, $k = 72$, $p = 0.5$
$$
\begin{align*}
\binom{150}{72}&=\frac{150!}{72!(150 - 72)!}=\frac{150!}{72!78!}\
P(X = 72)&=\binom{150}{72}(0.5)^{72}(0.5)^{150 - 72}\
&=\binom{150}{72}(0.5)^{150}\
\end{align*}
$$
Using a calculator or software (e.g., in R: `dbinom(72,150,0.5)`), $P(X = 72)\approx0.0639$

## Step3: Calculate $P(X\lt72)$
$P(X\lt72)=\sum_{k = 0}^{71}\binom{150}{k}(0.5)^{k}(0.5)^{150 - k}$
Using a calculator or software (e.g., in R: `pbinom(71,150,0.5)`), $P(X\lt72)\approx0.4407$

## Step4: Calculate $P(X\gt72)$
Since $P(X\gt72)=1 - P(X\leq72)$ and $P(X\leq72)=P(X\lt72)+P(X = 72)$
$P(X\gt72)=1-(P(X\lt72)+P(X = 72))$
$P(X\gt72)=1-(0.4407 + 0.0639)=0.4954$

## Step5: Calculate $P(X = 79)$
For $n = 150$, $k = 79$, $p = 0.5$
$$
\begin{align*}
\binom{150}{79}&=\frac{150!}{79!(150 - 79)!}=\frac{150!}{79!71!}\
P(X = 79)&=\binom{150}{79}(0.5)^{79}(0.5)^{150 - 79}\
&=\binom{150}{79}(0.5)^{150}
\end{align*}
$$
Using a calculator or software (e.g., in R: `dbinom(79,150,0.5)`), $P(X = 79)\approx0.0639$

## Step6: Calculate $P(X\geq79)$
$P(X\geq79)=\sum_{k = 79}^{150}\binom{150}{k}(0.5)^{k}(0.5)^{150 - k}$
Using a calculator or software (e.g., in R: `1 - pbinom(78,150,0.5)`), $P(X\geq79)\approx0.4954$

## Step7: Calculate $P(X\leq79)$
$P(X\leq79)=\sum_{k = 0}^{79}\binom{150}{k}(0.5)^{k}(0.5)^{150 - k}$
Using a calculator or software (e.g., in R: `pbinom(79,150,0.5)`), $P(X\leq79)\approx0.9361$

# Answer:
d) $0.0639$
e) $0.4407$
f) $0.4954$
g) $0.0639$
h) $0.4954$
i) $0.9361$

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QUESTION IMAGE

Question

Explanation:

Step1: Identify the distribution

Step2: Calculate \(P(X = 72)\)

Step3: Calculate \(P(X\lt72)\)

Step4: Calculate \(P(X\gt72)\)

Step5: Calculate \(P(X = 79)\)

Step6: Calculate \(P(X\geq79)\)

Step7: Calculate \(P(X\leq79)\)

Answer: