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 number that will arrive on time: 160
Standard deviation of this distribution: 28.28
To find the expected number of flights that will arrive on time, we can multiply the probability of a flight arriving on time by the total number of flights in the sample.
Given that 80% of flights arrived on time, the probability is 0.8. Assuming the on-time rate is representative of the entire industry, we can multiply this probability by the total number of flights in the sample, which is 200.
Expected Number of On-Time Flights = Probability of On-Time * Total Number of Flights
Expected Number of On-Time Flights = 0.8 * 200
Expected Number of On-Time Flights = 160
Therefore, the expected number of flights that will arrive on time is 160.
To find the standard deviation, we need to calculate the square root of the product of the probability of an event occurring (p) and the probability of it not occurring (q), multiplied by the sample size (n).
Standard Deviation = √(p * q * n)
Where p is the probability of success (arrival on time), q is the probability of failure (not arrival on time), and n is the sample size.
In this case, p = 0.8 (probability of flight arriving on time), q = 1 - p = 0.2 (probability of flight not arriving on time), and n = 200 (sample size).
Standard Deviation = √(0.8 * 0.2 * 200)
Standard Deviation = √(0.16 * 200)
Standard Deviation = √32
Standard Deviation ≈ 5.656
Therefore, the standard deviation of this distribution is approximately 5.656.