7.18 Performance of airlines In 2015, the on-time arrival rate of all major domestic and regional airlines operating between Australian airports has a bell-shaped distribution roughly with mean 0.86 and standard deviation 0.1.

a. Let X denote the number of flights arriving on time when you observe one flight. State the probability distribution of X. (This also represents the population distribution you would get if you could observe an infinite number of flights.)

b. You decide to observe the airport arrival tables for one day. At the end of the day, you were able to check the arrival times of 100 flights. Show that the sampling distribution of your sample mean number of flights on time has mean = 0.86 and standard deviation = 0.01.

c. Refer to part

b. Using the central limit theorem, find the probability that the mean number of flights on time is at least 0.88, so that you have a gain of at least 2% with regard to the rate of on-time flights in the population.

(Hint: Find the probability that a normal random variable with mean 0.86 and standard deviation 0.01 exceeds 0.88.)