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.
To find the expected number of flights that will arrive on time, we can multiply the probability of each flight arriving on time by the total number of flights in the sample.
The probability of a flight arriving on time is 80%, which can be written as 0.80 or 0.8.
Therefore, the expected number of flights that will arrive on time is:
Expected number = Probability of arriving on time × Total number of flights
Expected number = 0.80 × 200
Expected number = 160 flights
So, the expected number of flights that will arrive on time is 160.
To calculate the standard deviation, we can use the formula for the standard deviation of a binomial distribution:
Standard deviation = √(n × p × (1 - p))
Where:
n = Total number of flights in the sample (200)
p = Probability of a flight arriving on time (0.80)
Standard deviation = √(200 × 0.80 × (1 - 0.80))
Standard deviation = √(200 × 0.80 × 0.20)
Standard deviation = √(32)
Standard deviation ≈ 5.66
So, the standard deviation of this distribution is approximately 5.66.