This is a binomial distribution problem, where:
n = 125 (sample size)
p = 0.065 (proportion of success in population)
To solve this problem, we need to use the binomial probability formula:
P(x) = nCx * p^x * (1-p)^(n-x)
where
nCx = combination of n things taken x at a time
p^x = probability of x successes
(1-p)^(n-x) = probability of (n-x) failures
A) To find the probability of observing 80 or fewer successes:
P(x ≤ 80) = Σ P(x=0) + P(x=1) + P(x=2) + ... + P(x=80)
= Σ 125C0 * 0.065^0 * (1-0.065)^(125-0) + 125C1 * 0.065^1 * (1-0.065)^(125-1) + 125C2 * 0.065^2 * (1-0.065)^(125-2) + ... + 125C80 * 0.065^80 * (1-0.065)^(125-80)
Using a binomial calculator or software, we can find that
P(x ≤ 80) = 0.163
B) To find the probability of observing 82 or fewer successes:
P(x ≤ 82) = Σ P(x=0) + P(x=1) + P(x=2) + ... + P(x=82)
= Σ 125C0 * 0.065^0 * (1-0.065)^(125-0) + 125C1 * 0.065^1 * (1-0.065)^(125-1) + 125C2 * 0.065^2 * (1-0.065)^(125-2) + ... + 125C82 * 0.065^82 * (1-0.065)^(125-82)
Using a binomial calculator or software, we can find that
P(x ≤ 82) = 0.215
C) To find the probability of observing 75 or more successes:
P(x ≥ 75) = Σ P(x=75) + P(x=76) + ... + P(x=125)
= Σ 125C75 * 0.065^75 * (1-0.065)^(125-75) + 125C76 * 0.065^76 * (1-0.065)^(125-76) + ... + 125C125 * 0.065^125 * (1-0.065)^(125-125)
Using a binomial calculator or software, we can find that
P(x ≥ 75) = 0.824
A sample of 125 is drown population with proportion equal to.065. determine the probability of observing
A 80 or fewer successes
B 82 or fewer successes
C 75 or more sucesses
1 answer