Question

Conde Nast Traveler publishes a Gold List of the top hotels all over the world. The Broadmoor Hotel in Colorado Springs contains 700 rooms and is on the 2004 Gold List (Conde Nast Traveler, January 2004). Suppose Broadmoor's marketing group forecasts a demand of 670 rooms for the coming weekend. Assume that demand for the upcoming weekend is normally distributed with a standard deviation of 30.
a.What is the probability all the hotel's rooms will be rented (to 4 decimals)?

b. What is the probability 50 or more rooms will not be rented (to 4 decimals)?

Answer 1

Answer:

(a) The probability that all the hotel's rooms will be rented is 0.1587.

(b) The probability that 50 or more rooms will not be rented is 0.2514.

Step-by-step explanation:

We are given that the Broadmoor Hotel in Colorado Springs contains 700 rooms and is on the 2004 Gold List.

Suppose Broadmoor's marketing group forecasts a mean demand of 670 rooms for the coming weekend. Assume that demand for the upcoming weekend is normally distributed with a standard deviation of 30.

Let X = demand for rooms in the hotel

So, X ~ Normal($\mu=670,\sigma^{2} =30^{2}$)

The z-score probability distribution for the normal distribution is given by;

                           Z  =  $\frac{X-\mu}{\sigma}$  ~ N(0,1)

where, $\mu$ = mean demand for the rooms = 670

            $\sigma$ = standard deviation = 30

(a) The probability that all the hotel's rooms will be rented means that the demand is at least 700 = P(X $\geq$ 700)

          P(X $\geq$ 700) = P( $\frac{X-\mu}{\sigma}$ $\geq$ $\frac{700-670}{30}$ ) = P(Z $\geq$ 1) = 1 - P(Z < 1)

                                                             = 1 - 0.8413 = 0.1587

The above probability is calculated by looking at the value of x = 1 in the z table which has an area of 0.8413.

(b) The probability that 50 or more rooms will not be rented is given by = P(X $\leq$ 650)

         P(X $\leq$ 650) = P( $\frac{X-\mu}{\sigma}$ $\leq$ $\frac{650-670}{30}$ ) = P(Z $\leq$ -0.67) = 1 - P(Z < 0.67)

                                                             = 1 - 0.7486 = 0.2514

The above probability is calculated by looking at the value of x = 0.67 in the z table which has an area of 0.7486.