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
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
To determine the probability of observing a certain number of successes in a given sample, we can use the binomial probability formula:
P(x) = C(n, x) * p^x * q^(n-x)
Where:
P(x) represents the probability of getting exactly x successes.
n represents the sample size.
x represents the number of successes.
p represents the probability of success.
q represents the probability of failure, which is equal to 1 - p.
C(n, x) represents the number of combinations of n objects taken x at a time.
Given:
Sample size (n) = 125
Proportion of success (p) = 0.065
A) Probability of 80 or fewer successes (x <= 80):
P(80 or fewer successes) = P(0) + P(1) + P(2) + ... + P(80)
To calculate this probability, we need to calculate the probability for each individual value of x from 0 to 80, and then sum them up.
P(80 or fewer successes) = ∑[x = 0 to 80](C(125, x) * p^x * q^(n-x))
B) Probability of 82 or fewer successes (x <= 82):
P(82 or fewer successes) = P(0) + P(1) + P(2) + ... + P(82)
To calculate this probability, we need to calculate the probability for each individual value of x from 0 to 82, and then sum them up.
P(82 or fewer successes) = ∑[x = 0 to 82](C(125, x) * p^x * q^(n-x))
C) Probability of 75 or more successes (x >= 75):
P(75 or more successes) = P(75) + P(76) + P(77) + ... + P(125)
To calculate this probability, we need to calculate the probability for each individual value of x from 75 to 125, and then sum them up.
P(75 or more successes) = ∑[x = 75 to 125](C(125, x) * p^x * q^(n-x))
The calculations for these probabilities involve evaluating the binomial coefficient (C(n, x)), raising p to the power of x, raising q to the power of (n-x), and summing up all the terms.