2. 3 points Suppose a new movie called Mootopia just released and seems to be do

Question

2. 3 points Suppose a new movie called Mootopia just released and seems to be doing amaz- ingly well at the box office, beloved amongst kids and adults alike. Suppose at a big theater, the number of tickets bought per customer is a random variable with mean 2.5 and standard deviations ox - 2. Suppose for a noon showing on a weekend, the size of the theater is 275 seats. Further suppose 100 customers have lined up to buy tickets. Using the Central Limit theorem find the probability that there will be no unsatisfied customers.

Explanation / Answer

Solution

Let X = average number of tickets bought per customer. Then, we are given that

Mean of X, µX = 2.5 and Standard deviation of X, X = 2.0.

So, by Central Limit Theorem, as n is large,

Z = [{X – E(X)}/SD(X)] ~ N(0, 1).

i.e., Z = [{X – 2.5)}/2] ~ N(0, 1)………………………………………………………..(1)

Now, with a capacity of 275 seats in the theatre, and given 100 customers in the line, there will be no unsatisfied customer, if the average number of tickets bought per customer is less than 2.75.

So, the required probability

= P(X 2.75)

= P[Z {(2.75 – 2.5)/2]   

= P(Z 0.375)

= 0.64617 ANSWER

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