To solve this problem, we need to calculate the probability that at least 5 out of 8 randomly selected human resource (HR) managers say that job applicants should follow up within two weeks. Let's break this problem down step-by-step.
1. Identify the Parameters:
- Probability of success (p) = 0.54 (The percentage of HR managers who say job applicants should follow up within two weeks)
- Number of trials (n) = 8 (The number of HR managers selected)
- Number of successes (k) = 5 or more
2. Understand the Binomial Distribution:
This type of problem involves a binomial distribution because each HR manager has two possible outcomes: either they say job applicants should follow up within two weeks (success) or they do not (failure). The probability of observing a specific number of successes in a fixed number of trials can be modelled with the binomial distribution formula. The formula for the binomial distribution is:
[tex]\[
P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}
\][/tex]
where [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient calculated as [tex]\(\frac{n!}{k!(n-k)!}\)[/tex].
3. Calculate the Required Probability:
To find the probability of at least 5 successes, we need to sum the probabilities of obtaining 5, 6, 7, and 8 successes out of 8 trials. Mathematically, it can be represented as:
[tex]\[
P(X \geq 5) = P(X=5) + P(X=6) + P(X=7) + P(X=8)
\][/tex]
4. Summing the Probabilities:
We sum the individual binomial probabilities:
[tex]\[
P(X \geq 5) = \sum_{i=5}^{8} \binom{8}{i} (0.54)^i (0.46)^{8-i}
\][/tex]
where [tex]\(0.54\)[/tex] is the probability of success and [tex]\(0.46\)[/tex] is the probability of failure (1-p).
5. Compute the Cumulative Probability:
Adding each of these probabilities together can be computationally intensive, but we use known methods or computational tools to find:
[tex]\[
P(X \geq 5) = 0.4537
\][/tex]
Hence, the probability that at least 5 out of 8 human resource managers say that job applicants should follow up within two weeks is 0.4537 when rounded to four decimal places.
