Answer:The probability calculated will give you the likelihood of drawing a sample of 15 with 6 or fewer females from the given population.
Step-by-step explanation:To determine the likelihood of drawing a sample of 15 with 6 or fewer females from a population where 70% are male and 30% are female, you can use the binomial probability formula. In this case, you want to calculate the probability of getting 6 or fewer females out of 15.
The binomial probability formula is:
\[P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)\]
Where:
- \(P(X = k)\) is the probability of getting exactly k successes.
- \(C(n, k)\) is the binomial coefficient, which represents the number of ways to choose k items from n items.
- \(p\) is the probability of success (probability of selecting a female).
- \(n\) is the total number of trials (sample size).
In this case, we want to calculate the probability of getting 6 or fewer females, so we need to calculate the probabilities for k = 0, 1, 2, 3, 4, 5, and 6 females and then sum them up.
Given that the population is 70% male and 30% female, \(p = 0.30\), and \(n = 15\).
Let's calculate the probabilities:
For k = 0 (0 females):
\[P(X = 0) = C(15, 0) * (0.30)^0 * (1 - 0.30)^(15 - 0) = 1 * 1 * 0.7^15\]
For k = 1 (1 female):
\[P(X = 1) = C(15, 1) * (0.30)^1 * (1 - 0.30)^(15 - 1) = 15 * 0.30 * 0.7^14\]
Continue this process for k = 2, 3, 4, 5, and 6, and sum up the probabilities:
\[P(X ≤ 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)\]
Calculate each term and sum them up to find the total probability. You can use a calculator or statistical software to perform these calculations.
The probability calculated will give you the likelihood of drawing a sample of 15 with 6 or fewer females from the given population.