A snack bar sells hot dogs at the rate of 3 per every 10 minutes. Consider model

Question

A snack bar sells hot dogs at the rate of 3 per every 10 minutes. Consider modeling the number of hot dogs sold during any given minute as a Poisson random variable having = 3/10. Please note that a Poisson random variable has expected value and variance equal to as well. That is, the expected value and variance on the number of hot dogs sold in any particular minute are both equal to = 3/10.

(a) Use a normal distribution approach to determine how many minutes the snack bar needs to remain open so that there is a 77% chance the snack bar will sell at least 130 hot dogs.

(b) For any single hour of operation, the snack bar donates $1.00 to charity whenever they sell between 15 to 22 hot dogs and they donate $2.00 to charity whenever they sell 23 or more hot dogs during the hour. (They donate nothing to charity during an hour in which they sell 14 or less hot dogs.)

(i) Derive the distribution on the amount of money the snack bar donates to charity during any single hour of operation.

(ii) Determine the probability the snack bar donates at least $190 to charity over the next 200 hours of operation.

Explanation / Answer

If we do normal approximation of the given poisson distribution
mean = 3/10 = 0.3
std. dev. = sqrt(0.3) = 0.5477

(A)
P(X>130) = 0.77
z-value = 0.74

(130-0.3*n)/0.5477 = 0.74
n = (130 - 0.74*0.5477)/0.3 = 431.98 = 432

Hence snacks bar should remain open for 432 mins.

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