Question

New York City is the most expensive city in the United States for lodging. The mean hotel room rate is $204 per night (USA Today, April 30, 2012). 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 (to 4 decimals)?

b. What is the probability that a hotel room costs less than $140 per night (to 4 decimals)?

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

Answer 1

Concepts and reason

Normal distribution: Normal distribution is a continuous distribution of data that has the bell-shaped curve. The normally distributed random variable x has mean and standard deviation .

Also, the standard normal distribution represents a normal curve with mean 0 and standard deviation 1. Thus, the parameters involved in a normal distribution are mean and standard deviation.

Standardized z-score: The standardized z-score represents the number of standard deviations the data point is away from the mean.

?If the z-score takes positive value when it is above the mean (0).

?If the z-score takes negative value when it is below the mean (0).

Central limit theorem:

If the size of the samples taken from the population are sufficiently large (> 30) and also has the finite variance; then the mean of all the samples taken would be approximately same as the population mean. And the variance would be the ratio of population variance and sample size. Also the sampling distribution would be approximately normal.

Fundamentals

If the sample size is considered then, the standard z-score is defined as,

Formula for calculating the probability,

Where, Z is a standard normal random variable. The probability can be calculated using Z-table.

Formula for calculating the probability is,

Formula for calculating the probability, is,

Suppose, consider the case,

Then,

Based on the value c, the value of t is obtained. However, the procedure is done with MS Excel.

Procedure to obtain the value of t using MegaStat procedure:

1.In EXCEL, Select Add-Ins > MegaStat > Probability.

2.Choose Continuous probability distributions.

3.Select Normal distribution and select calculate z given P and enter P .

4.Enter mean and standard deviation

5.Click Ok.

(a)

The probability that the room costs $225 or more is obtained as shown below:

From the information, the mean hotel room rate is $204, that is and standard deviation is $o=55 $.

The required probability is,

From the “standard normal table”, the area to the left of is 0.64803.

(b)

The probability that the room rate is less than $140 is obtained as shown below:

From the “standard normal table”, the area to the left of is 0.03515.

(c)

The probability that room rate will fall between $200 and $300 is obtained below:

From the information, . The required probability is,

From the “standard normal table”, the area to the left of is 0. 47210 and the area to the left of is 0.95818.

(d)

The cost of room rent is obtained as shown below:

From the information, and $o=55 $. The cost of the top 20% hotel rooms is to be calculated. That is,

Excel add-in (MegaStat) procedure for obtaining z value:

1.In EXCEL, Select Add-Ins > MegaStat > Probability.

2.Choose Continuous probability distributions.

3.Select Normal distribution and select calculate z given P and enter P as 0.01

4.Enter mean as 204 and standard deviation as 55.

5.Click Ok.

Follow the above instruction to obtain the below output:

From MegaStat output, the value z is –0.84

The cost of top 20% hotel rooms is obtained as shown below:

From MegaStat, the value z is –0.84

Ans: Part a

Thus, the probability that the that the room costs $225 or more is