Answer:
Given data: One flight with total seats = 100
Full fare passengers, cost per ticket=$150, mean=56 passengers, SD=23
Discount fare passengers, cost per ticket=$100, mean=88 passengers, SD=44
(a) Here, though there is a hint to use the CDF, since the confidence interval is not given we will make some simplying assumptions that will reduce the complexity of the question, of course keeping the question statistically correct.
this question wants us to maximize total revenue per flight (one way), we can do that by taking only full fare passengers or total revenue will be 150*100=$15,000, but since historical probability shows a mean of 56 with a standard deviation of 23, we can assume in best case scenario total full fare ticket passengers will be 56+23=79, leaving 21 tickets for discount passenger, in this case the total revenues will be 79*150+21*100=$13,950
(b) Now, the new constrained policy is giving a clear cut number of seats to each category of pasengers, 44 for discount (total revenues 44*100) and 56 for full fare (total revenues 56*150) both of which are within the probabilities given earlier (full fare mean=56, discount mean=88). Total revenues in case will be 44*100+56*150=$12,800.
(c) Gain is the difference of the excess revenues in both cases of optimal total revenues and limited seats policy or answer (a) - answer (b) = $13,950- $12,800=$1,150
(d) Realistically speaking, there is no answer for this question without a clear cut confidence interval. Another simplifying assumption we can make here is taking the mean passengers as expected bookings (can be tweaked once confidence interval or degree of significance is given). so total revenues in this case will be 44*100 from discount and 56*150 from full fare passengers. That is still similar to answer (c) due to our assumption/lack of constraints, so our optimal booking will be 54 full fare tickets and 44 discount passenger tickets. You can also take worst case scenario by subtracting SD of each passenger type from the mean or go the best case scenario in which SD of full fare will be added to the mean while the pending seats (left over from 100) will be the total to discount fare for optimal revenue collection.
