Question

New York City is the most expensive city in the United States for lodging. The mean hotel room rate is $204 per night. Assume that room rates are normally distributed with a standard deviation of $55.

a. What is the probability that a hotel room costs $225 or more per night?

b. What is the probability that a hotel room costs less than $140 per night?

c. What is the probability that a hotel room costs between $200 and $300 per night?

d. What is the cost of the 20% most expensive hotel rooms in the New York City?

Answer 1

Solution:

Given: The room rates are normally distributed with mean hotel room rate is $204 per night and a standard deviation of $55.

That is:   X ~ Normal

Part a. What is the probability that a hotel room costs $225 or more per night?

Find z score:

Thus we get:

Look in z table for z= 0.3 and 0.08 and find area.

Thus from z table we get: P( Z < 0.38 ) = 0.6480

Thus

Part b. What is the probability that a hotel room costs less than $140 per night?

That is: P( X < 140) =..........?

Find z score:

Thus we get:

P( X < 140) =P( Z < -1.16)

Look in z table for z= -1.1 and 0.06 and find area.

Thus from z table we get: P( Z < -1.16 ) = 0.1230

Thus

P( X < 140) =P( Z < -1.16)

P( X < 140) =0.1230

Part c. What is the probability that a hotel room costs between $200 and $300 per night?

That is: P( 200 < X < 300) =.........?

Find z score:

and

Thus we get:

P( 200 < X < 300) = P( -0.07 < Z < 1.75)

P( 200 < X < 300) = P(Z < 1.75) - P( Z < -0.07 )

Look in z table for z = -0.00 and 0.07 as well as for z = 1.7 and 0.05 and find corresponding area.

Thus from z table we get:

P( Z < -0.07) = 0.4721

and

P( Z < 1.75) = 0.9599

Thus:

P( 200 < X < 300) = P(Z < 1.75) - P( Z < -0.07 )

P( 200 < X < 300) =   0.9599 - 0.4721

P( 200 < X < 300) = 0.4878

Part d. What is the cost of the 20% most expensive hotel rooms in the New York City?

That is find x value such that :

P( X > x) =20%

P( X > x) =0.2000

Thus find z value such that:

P( Z > z) =0.2000

then

P( Z < z) =1 - P( Z > z )

P( Z < z) =1 - 0.2000

P( Z < z) = 0.8000

Look in z table for Area = 0.8000 and find corresponding z value.

From above z table , we can see that area 0.7995 is most closet to 0.8000 and it corresponds to 0.8 and 0.04

Thus z = 0.84

Now use following formula to find x value:

thus the cost of the 20% most expensive hotel rooms in the New York City is $ 250.2