twenty percent of the employees of a large company are female.use the normal approximation of the binomial probabilities to answer the following question. what is the probability that in a random sample of 80 employeses
a)exactly 16 will be female?
b)14 or more will be female?
c)15 or fewer will be female?
To determine the probability using the normal approximation of binomial probabilities, we need to follow certain steps. Let's solve each question one by one:
a) To find the probability that exactly 16 employees will be female:
1. Calculate the mean (μ) of the binomial distribution: μ = n * p, where n is the sample size (80) and p is the probability of success (20% or 0.2).
μ = 80 * 0.2 = 16
2. Calculate the standard deviation (σ) of the binomial distribution: σ = sqrt(n * p * (1 - p)) = sqrt(80 * 0.2 * 0.8).
σ ≈ 3.58
3. We can now use the normal approximation.
Convert the problem to a standard normal distribution by subtracting the mean and dividing by the standard deviation: (16 - 16) / 3.58 = 0 / 3.58 = 0.
The probability of exactly 16 employees being female is the same as finding the probability of getting a Z-score of 0, which is approximately 0.5.
Therefore, the probability that exactly 16 employees will be female is 0.5.
b) To find the probability that 14 or more employees will be female:
1. Calculate the mean (μ) and standard deviation (σ) as done in part (a).
μ = 16
σ ≈ 3.58
2. Convert 14 to a standard normal distribution: (14 - 16) / 3.58 ≈ -0.56.
We need to find the probability of getting a Z-score greater than or equal to -0.56. Using a normal distribution table or calculator, we can find this probability.
c) To find the probability that 15 or fewer employees will be female:
1. Calculate the mean (μ) and standard deviation (σ) as done previously.
μ = 16
σ ≈ 3.58
2. Convert 15 to a standard normal distribution: (15 - 16) / 3.58 ≈ -0.28.
We need to find the probability of getting a Z-score less than or equal to -0.28. Using a normal distribution table or calculator, we can find this probability.
Using a normal distribution table or calculator, you can find the probabilities for parts (b) and (c) based on the Z-scores obtained.