Concept explainers
An airplane has 100 seats for passengers. Assume that the
Find the probability that everyone who appears for the flight will get a seat, if the airline sells 105 tickets.
Answer to Problem 1SE
The probability that everyone who appears for the flight will get a seat is 0.9774.
Explanation of Solution
Given info:
There are 100 seats for passengers in an airplane. The probability that a person holding a ticket appears for the flight is 0.90.
Calculation:
The random variable X is defined as the number of people out of 105 who appear for the flight. Then the random variable X follows binomial distribution with parameters
The condition for the normal approximation to the binomial probability distribution is
That is,
Thus, the normal distribution can be used to approximate the binomial distribution.
Mean:
Substitute 105 for n and 0.90 for p.
Standard deviation:
Substitute n as 60 and p as 0.40.
The approximate binomial probability using the normal distribution is,
By using continuity correction, the value 0.5 is added from the value of 100.
Substitute 100 for a,
The formula to convert X values into z score is,
Substitute 94.5 for np and 3.0741 for
The above probability can be obtained by finding the areas to the left of 1.95.
Use Table A.2: Cumulative Normal Distribution to find the area.
Procedure:
For z at 1.95,
- Locate 1.9 in the left column of the Table A.2.
- Obtain the value in the corresponding row below 0.05.
That is,
Thus, the probability that everyone who appears for the flight will get a seat, if the airline sells 105 tickets is 0.9774.
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Chapter 4 Solutions
Statistics for Engineers and Scientists
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