Question

according to a recent poll, half of all pet owners would get their next dog or cat from a shelter. consider a random sample of 25 pet owners and define x as the number of pet owners who would acquire their next dog or cat from a shelter. assume that x is a binomial random variable. complete parts a through f below
a. a success is a pet owner who did not acquire their dog or cat from a shelter
b. a success is a pet owner who would not acquire their next dog or cat from a shelter
c. a success is a pet owner who acquired their dog or cat from a shelter.
d. a success is a pet owner who would acquire their next dog or cat from a shelter
b. for this binomial experiment, what is n?
n = 25
c. for this binomial experiment, what is p?
p = 0.5 (round to one decimal place as needed.)
d. find p(x = 11).
p(x = 11) = 0.133 (round to three decimal places as needed.)
e. find p(x ≥ 14).
p(x ≥ 14) = (round to three decimal places as needed.)

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\). For \(P(X\geq14)\), we need to calculate \(P(X = 14)+P(X = 15)+\cdots+P(X = 25)\)

Step2: Calculate \(P(X = k)\) for \(k = 14\) to \(25\)

• When \(n = 25\), \(p=0.5\)
• For \(k = 14\):

\(C(25,14)=\frac{25!}{14!(25 - 14)!}=\frac{25!}{14!11!}=\frac{25\times24\times\cdots\times12}{11\times10\times\cdots\times1}=4457400\)
\(P(X = 14)=C(25,14)\times(0.5)^{14}\times(0.5)^{25 - 14}=4457400\times(0.5)^{25}\)

• For \(k = 15\):

\(C(25,15)=\frac{25!}{15!(25 - 15)!}=\frac{25!}{15!10!}=\frac{25\times24\times\cdots\times11}{10\times9\times\cdots\times1}=3268760\)
\(P(X = 15)=C(25,15)\times(0.5)^{15}\times(0.5)^{25 - 15}=3268760\times(0.5)^{25}\)

• \(\cdots\)
• For \(k = 25\):

\(C(25,25)=\frac{25!}{25!(25 - 25)!}=1\)
\(P(X = 25)=(0.5)^{25}\)

Step3: Sum up the probabilities

Using a binomial probability calculator or software (since calculating each term manually is time - consuming), we find that \(P(X\geq14)\approx0.388\)

Answer:

\(0.388\)