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 \(C(n,k)=\frac{n!}{k!(n - k)!}\). To find \(P(x\geq14)\), we use \(P(x\geq14)=\sum_{k = 14}^{25}P(X = k)\). Another way is \(P(x\geq14)=1-\sum_{k = 0}^{13}P(X = k)\).

Step2: Calculate using binomial cumulative distribution function

Using a binomial cumulative distribution function (either a statistical software, calculator with binomial CDF function, or formula - based calculation). For \(n = 25\), \(p=0.5\)

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Using a calculator or software (e.g., in R: 1 - pbinom(13,25,0.5)), we get the result.

Answer:

\(P(x\geq14)=0.387\) (rounded to three decimal places)