Question

two local hotels, x and y, rent out rooms nightly. let x represent the number of rooms hotel x rents out nightly; and let y represent the number of rooms hotel y rents out nightly. the mean of x is 49, and the mean of y is 58. which answer choice correctly calculates and interprets the mean of the sum, s = x + y?
μ_s = 53.5; the hotels can expect to rent out 53.5 rooms, on average, each night.
μ_s = 53.5; on one randomly selected night, the total number of rooms rented out will be 53.5.
μ_s = 107; the hotels can expect to rent out 107 rooms, on average, each night.
μ_s = 107; on one randomly selected night, the total number of rooms rented out will be 107.

Explanation:

Step1: Recall the formula for the mean of the sum of two random variables

The formula for the mean of the sum of two random variables \(S = X + Y\) is \(\mu_S=\mu_X+\mu_Y\).

Step2: Substitute the given means

Given \(\mu_X = 49\) and \(\mu_Y=58\), then \(\mu_S=49 + 58\).

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The mean of a random variable represents the expected value. So, for \(S = X + Y\), \(\mu_S = 107\) means that the hotels can expect to rent out 107 rooms, on average, each night.

Answer:

\(\mu_S = 107\); the hotels can expect to rent out 107 rooms, on average, each night. So the correct answer is: \(\mu_S=107\); the hotels can expect to rent out 107 rooms, on average, each night.