Question

In our rental shop case, we know that on the busiest day we can expect 150 rentals, which forms the number of independent events or trials. We also know that, historically, 60% of our customers rent skis and 40% rent snowboards, which provides our probability. If we decide that we only need to have 65 snowboards in stock, what is the probability that we will run out of snowboard rentals on any specific day?

Answer 1

Solution :

Number of rentals on the busiest day = 150

Percentage of customers who rent snowboards = 40%

We have decided that we need to have 65 snowboards in stock and we have to obtain the probability that on any day we will run out of snowboard rentals.

We will run out out of snowboards rentals if number of customers who rent snowboards is greater than 65.

Now, population of customers who rent snowboard (p) = 40% = 0.40

Number of maximum customers on any specific day (n) = 150

If out of 150 customeers, 65 customers rent snowboard, then sample proportion of customers who rented snowboard is, p̂ = 65/150 = 0.4333

Hence, we need to obtain the probability that sample proportion of customers who will rent snowboards, will be more than 0.4333 on any day. This probability will be equal to the required probability of running out of snowboard rentals on any specific day.

i.e. We need to obtain P(p̂ > 0.4333).

Now if nP > 5 and nq > 5, then sampling distribution of sample proportions (p̂) follows approximately normal distribution with mean p and variance pq/n.

i.e. p̂ ~ N(p, pq/n)

(Where, p is population proportion, q = 1 - p, n is sample size.)

We have, p = 0.40, q = 1 - 0.40 = 0.60 and n = 150

np = 150×0.40 = 60 which is greater than 5.

nq = 150×0.60 = 90 which is greater than 5.

Hence, sampling distribution of sample proportions (p̂) of customers who rent snowboards will be approximately normal with mean p = 0.40 and variance pq/n = 0.40×0.60/150 = 0.0016.

We have to obtain P(p̂ > 0.4333).

We know that if p̂ ~ N(p, pq/n) then

Using "pnorm" function of R we get, P(Z > 0.8333) = 0.2023

Hence, the probability that we will run out of snowboard rentals on any specific day is 0.2023.

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