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An insurance company offers a discount to homeowners who
Chapter 4, Problem 20E(choose chapter or problem)
An insurance company offers a discount to home-owners who install smoke detectors in their homes. A company representative claims that 80% or more of policyholders have smoke detectors. You draw a random sample of eight policyholders. Let X be the number of policyholders in the sample who have smoke detectors.
a. If exactly 80% of the policyholders have smoke detectors (so the representative’s claim is true, but just barely), what is \(P(X \leq 1)\)?
b. Based on the answer to part (a), if 80% of the policyholders have smoke detectors, would one policyholder with a smoke detector in a sample of size 8 be an unusually small number?
c. If you found that one of the eight sample policyholders had a smoke detector, would this be convincing evidence that the claim is false? Explain.
d. If exactly 80% of the policyholders have smoke detectors, what is \(P(X \leq 6)\)?
e. Based on the answer to part (d), if 80% of the policyholders have smoke detectors, would six policyholders with smoke detectors in a sample of size 8 be an unusually small number?
f. If you found that six of the eight sample policyholders had smoke detectors, would this be convincing evidence that the claim is false? Explain.
Questions & Answers
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QUESTION:
An insurance company offers a discount to home-owners who install smoke detectors in their homes. A company representative claims that 80% or more of policyholders have smoke detectors. You draw a random sample of eight policyholders. Let X be the number of policyholders in the sample who have smoke detectors.
a. If exactly 80% of the policyholders have smoke detectors (so the representative’s claim is true, but just barely), what is \(P(X \leq 1)\)?
b. Based on the answer to part (a), if 80% of the policyholders have smoke detectors, would one policyholder with a smoke detector in a sample of size 8 be an unusually small number?
c. If you found that one of the eight sample policyholders had a smoke detector, would this be convincing evidence that the claim is false? Explain.
d. If exactly 80% of the policyholders have smoke detectors, what is \(P(X \leq 6)\)?
e. Based on the answer to part (d), if 80% of the policyholders have smoke detectors, would six policyholders with smoke detectors in a sample of size 8 be an unusually small number?
f. If you found that six of the eight sample policyholders had smoke detectors, would this be convincing evidence that the claim is false? Explain.
ANSWER:Step 1 of 7
Let us consider a random variable X it presents a number of policyholders in the sample who have smoke detectors. Then X follows a binomial distribution with parameters “n and p” that is
\(\mathrm{X}\sim\mathrm{B}(\mathrm{n},\mathrm{\ p})\),
The probability mass function of the binomial distribution is given by
\(P(X)=\left(\begin{array}{l}n\\ x\end{array}\right)p^x(1-p)^{n-x},\mathrm{\ x}=0,\ 1,\ 2,\ldots,\mathrm{n}\)
Where,
n = sample size
= 8
x = random variable
p = probability of success
= 0.80
q = 1 - p (probability of failure)
= 1 - 0.80
= 0.20
Here our goal is:
a). We need to find \(P(X \leq 1)\) when exactly 80% of the policyholders have smoke detectors.
b). We need to check whether one policyholder with a smoke detector in a sample of size 8 is an unusually small number.
c). We need to check whether this would be convincing evidence that the claim is false if yes Explain.
d). We need to find \(P(X \leq 6)\) when exactly 80% of the policyholders have smoke detectors.
e). We need to check whether six policyholders with smoke detectors in a sample of size 8 is an unusually small number.
f). We need to check whether this would be convincing evidence that the claim is false if yes Explain.
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