Consumer Reports rated airlines and found that 80% of flights involved in the study arrived on time (that is, within 15 minutes of scheduled arrival time). Assuming that the on-time rate is representive of the entire commercial airline industry, consider a random sample of 200 flights. What is the expected number that will arrive on time? What is the standard deviation of this distribution.
Expected Numer = 160
Consumer Reports rated airlines and found that 78% of the flights involved in the study arrived on time (that is, within 15 minutes of scheduled arrival time). Assuming that the on-time arrival rate is representative of the entire commercial airline industry, consider a random sample of 190 flights. (Round your answers to two decimal places.)
To solve this problem, we can use the binomial distribution formula:
P(x) = (nCx) * p^x * q^(n-x)
Where:
P(x) is the probability of getting exactly x successes.
n is the number of trials (sample size).
x is the number of successes.
p is the probability of success (80% or 0.8).
q is the probability of failure (1 - p or 20% or 0.2).
nCx is the number of combinations of n items taken x at a time.
Step 1: Calculate the expected number of flights that will arrive on time.
The expected number (mean) is calculated by multiplying the sample size (n) by the probability of success (p).
Expected number = n * p = 200 * 0.8 = 160 flights
Therefore, the expected number of flights that will arrive on time is 160.
Step 2: Calculate the standard deviation of this distribution.
The standard deviation (σ) is calculated as the square root of (n * p * q).
Standard deviation = √(n * p * q) = √(200 * 0.8 * 0.2) ≈ √32 ≈ 5.65685425
Therefore, the standard deviation of this distribution is approximately 5.65685425.
To find the expected number of flights that will arrive on time, we need to multiply the proportion of flights that arrived on time by the total number of flights in the sample.
Given that 80% of flights arrived on time, we can represent this as a probability of success, p = 0.8. The sample size is 200 flights.
The expected number of flights that will arrive on time is calculated by multiplying the probability of success by the sample size:
Expected number = p * sample size
Expected number = 0.8 * 200
Expected number = 160
Therefore, we expect that 160 out of the 200 flights in the sample will arrive on time.
To calculate the standard deviation of this distribution, we need to use the formula for calculating the standard deviation of a binomial distribution, which is √(n * p * (1-p)).
Standard deviation = √(sample size * probability of success * probability of failure)
Standard deviation = √(200 * 0.8 * (1-0.8))
Standard deviation = √(200 * 0.8 * 0.2)
Standard deviation = √(32)
Standard deviation ≈ 5.66
Therefore, the standard deviation of the distribution is approximately 5.66.