Question

In: Statistics and Probability

New York City is the most expensive city in the United States for lodging. The mean...

New York City is the most expensive city in the United States for lodging. The mean hotel room rate is $205 per night (USA Today, April 30, 2012). Assume that room rates are normally distributed with a standard deviation of $53. Use Table 1 in Appendix B.

a. What is the probability that a hotel room costs $224 or more per night (to 4 decimals)?

b. What is the probability that a hotel room costs less than $139 per night (to 4 decimals)?

c. What is the probability that a hotel room costs between $201 and $299 per night (to 4 decimals)?

d. What is the cost of the 20% most expensive hotel rooms in New York City? Round up to the next dollar.

Solutions

Expert Solution

Solution :

Given that ,

mean = = 205

standard deviation = = 53

P(x 224) = 1 - P(x < 224)

= 1 - P((x - ) / (224 - 205) / 53)

= 1 -  P(z 0.36)  

= 1 - 0.6406   

= 0.3594

P(x 224) = 0.3594

Probability = 0.3594

b)

P(x < 139) = P((x - ) / < ( 139 - 205) / 53) = P(z < -1.25)

Using standard normal table,

P(x < 139) = 0.1056

Probability = 0.1056

(c)

P(201 < x < 299) = P((201 - 205 / 53) < (x - ) / < (299 - 205) / 53) )

P(201 < x < 299) =  P(-0.08 < z < 1.77)

P(201 < x < 299) = P(z < 1.77) - P(z < -0.08)

P(201 < x < 299) = 0.9616 - 0.4681 = 0.4935

Probability = 0.4935

(d)

P(Z > z) = 20%

1 - P(Z < z) = 0.20

P(Z < z) = 1 - 0.20 = 0.80

P(Z < 0.84) = 0.80

z = 0.84

Using z-score formula,

x = z * +

x = 0.84 * 53 + 205 = 249.52 = 250

Cosr = $250


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