A sample of 125 is drawn form population with proportion equal to .065 determine the probability of success
A 80 or fewer successes
B 82 or fewer successes
C 75 or fewer successes
To determine the probability of success, we can use the binomial probability formula:
P(X = k) = (n C k) * p^k * (1-p)^(n-k)
Where:
n = sample size
k = number of successes
p = probability of success
A) 80 or fewer successes:
P(X ≤ 80) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 80)
n = 125
p = 0.065
Using the formula, we calculate the probability of each individual success and sum them up:
P(X ≤ 80) = sum[(n C k) * p^k * (1-p)^(n-k)], where k ranges from 0 to 80
B) 82 or fewer successes:
P(X ≤ 82) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 82)
We use the same formula, but this time we sum up the probabilities for k ranging from 0 to 82.
C) 75 or fewer successes:
P(X ≤ 75) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 75)
Again, we use the formula to sum up the probabilities for k ranging from 0 to 75.
To find the probability of success in a binomial distribution, we need to use the binomial probability formula.
The formula for calculating the probability of getting exactly x successes in n trials, given a probability of success p, is:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
- nCx represents the number of combinations of choosing x items from a set of n items (nCx = n! / (x! * (n-x)!))
- p is the probability of success in one trial
- x is the number of successes
- n is the total number of trials
For the given problem, the population proportion is 0.065, which means the probability of success in one trial (p) is 0.065. The sample size is 125 (n = 125).
Now, let's calculate the probabilities:
A) 80 or fewer successes:
P(80 or fewer successes) = P(x <= 80)
= P(0) + P(1) + P(2) + ... + P(80)
= Σ (125Ck * 0.065^k * (1 - 0.065)^(125 - k)) for k = 0 to 80
B) 82 or fewer successes:
P(82 or fewer successes) = P(x <= 82)
= P(0) + P(1) + P(2) + ... + P(82)
= Σ (125Ck * 0.065^k * (1 - 0.065)^(125 - k)) for k = 0 to 82
C) 75 or fewer successes:
P(75 or fewer successes) = P(x <= 75)
= P(0) + P(1) + P(2) + ... + P(75)
= Σ (125Ck * 0.065^k * (1 - 0.065)^(125 - k)) for k = 0 to 75
To calculate these probabilities, you need to substitute the values of nCx, p, x, and n in the formula for each case and sum up the individual probabilities using the given ranges.
Note: These calculations require advanced statistical software or a binomial probability table.