An airline experiences a no-show rate of 6%. What is the maximum number of reservations that it could accept for a flight with a capacity of 160, if it wants the probability of accommodating all reservation holders to be at least 95%.
Use the normal approximation to the binomial distribution.
mean = np
standard deviation = √npq
Therefore:
mean = .94n
standard deviation = √(n)(.94)(.06) = √(.0564n)
Using z-scores:
z = (x - mean)/sd
With the data:
1.645 = (160 - .94n)/√(.0564n)
Solve for n.
(Hint: Round answer to 165.)
Wouldn't you have to use the continuity correction factor if using the normal approximation to the binomial?
Create one week schedule. You have 5 members of management. Manager 1 is the Store Manager Isabela, Manager 2 is the Assistant Manager Madian, Manager 3 is another Assistant Manager Bryan, Manager 4 is Area Supervisor Violeta, Manager 5 is Front End Supervisor Virginia. Store Manager, and both Assistant Mangers can work 9 hours 30 minutes per day. Area Supervisor and Front End Supervisor can only work 8 hours per day. Each manager needs to be off 2 days per week. There needs to be at least 3 OR 4 (your choice) managers schedule per day. Morning shifts starts at 8:00am. Closing shift ends at 10:30pm with one manager closing per day. The store opens seven days per week.
Here is a possible schedule:
Monday:
- 8:00am-5:30pm: Isabela, Madian, Bryan
- 11:30am-8:00pm: Violeta, Virginia
Tuesday:
- 8:00am-5:30pm: Isabela, Madian, Virginia
- 11:30am-8:00pm: Bryan, Violeta
Wednesday:
- 8:00am-5:30pm: Isabela, Bryan, Virginia
- 11:30am-8:00pm: Madian, Violeta (closing)
Thursday:
- 8:00am-5:30pm: Isabela, Bryan, Violeta
- 11:30am-8:00pm: Madian, Virginia
Friday:
- 8:00am-5:30pm: Isabela, Madian, Virginia
- 11:30am-10:30pm: Bryan (closing), Violeta
Saturday:
- 8:00am-5:30pm: Isabela, Bryan, Virginia
- 11:30am-10:30pm: Madian (closing), Violeta
Sunday:
- 10:00am-7:30pm: Isabela, Madian, Bryan
- 10:30am-7:00pm: Violeta, Virginia (off)
To determine the maximum number of reservations that the airline could accept, we need to consider the probability of exceeding the flight's capacity due to no-shows. Let's break down the steps to solve this problem:
Step 1: Calculate the complement of the desired probability.
The complement of probability is equal to 1 minus the desired probability. In this case, since the desired probability is 95% or 0.95, the complement is 1 - 0.95 = 0.05.
Step 2: Determine the maximum number of no-shows.
The no-show rate is given as 6%. To calculate the maximum number of no-shows, we multiply the no-show rate by the flight's capacity:
0.06 * 160 = 9.6.
Since we cannot have a fraction of a person, we round this number up to the nearest whole number, which gives us a maximum of 10 no-shows.
Step 3: Subtract the maximum number of no-shows from the flight's capacity.
To determine the maximum number of reservations, subtract the maximum number of no-shows from the flight's capacity:
160 - 10 = 150.
Therefore, the airline could accept a maximum of 150 reservations for the flight to ensure a probability of at least 95% of accommodating all reservation holders.