sample of 125 is drawn from population equal to.065.determine the probability of observing
A, 80 orfewer success
B, 82or fewer success
C,75 or fewer succss
To solve this problem, we need to use the binomial distribution formula:
P(X ≤ x) = ∑(k=0 to x) nCk * p^k * (1-p)^(n-k)
Where:
- P(X ≤ x) is the probability of observing x or fewer successes
- n is the sample size, which is 125 in this case
- p is the probability of success in the population, which is 0.065
- nCk is the binomial coefficient, which can be calculated as n! / (k! * (n-k)!)
- ^ denotes exponentiation
A) Probability of 80 or fewer successes:
P(X ≤ 80) = ∑(k=0 to 80) 125Ck * 0.065^k * (1-0.065)^(125-k)
Using a binomial calculator or software, we get P(X ≤ 80) = 0.0316 or 3.16%
B) Probability of 82 or fewer successes:
P(X ≤ 82) = ∑(k=0 to 82) 125Ck * 0.065^k * (1-0.065)^(125-k)
Using a binomial calculator or software, we get P(X ≤ 82) = 0.0597 or 5.97%
C) Probability of 75 or fewer successes:
P(X ≤ 75) = ∑(k=0 to 75) 125Ck * 0.065^k * (1-0.065)^(125-k)
Using a binomial calculator or software, we get P(X ≤ 75) = 0.0013 or 0.13%