Question

You may need to use the appropriate appendix table to answer this question. New York City is the most expensive city in the United States for lodging. The mean hotel room rate is $204 per night. T 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? (Round your answer to four decimal places. ) (b) What is the probability that a hotel room costs less than $110 per night? (Round your answer to four decimal places. ) (c) What is the probability that a hotel room costs between $200 and $280 per night? (Round your answer to four decimal places. ) (d) what is the cost in dollars of the 20% most expensive hotel rooms in New York City? (Round your answer to the nearest cent. )

Answer 1

Answer: (a) To find the probability that a hotel room costs $225 or more per night, we need to calculate the z-score first:

z = (225 - 204) / 55 = 0.3818

Using the standard normal distribution table or calculator, we find the probability that a z-score is greater than or equal to 0.3818 is 0.3524. Therefore, the probability that a hotel room costs $225 or more per night is 0.3524.

(b) To find the probability that a hotel room costs less than $110 per night, we need to calculate the z-score:

z = (110 - 204) / 55 = -1.7091

Using the standard normal distribution table or calculator, we find the probability that a z-score is less than or equal to -1.7091 is 0.0446. Therefore, the probability that a hotel room costs less than $110 per night is 0.0446.

(c) To find the probability that a hotel room costs between $200 and $280 per night, we need to calculate the z-scores for both values:

z1 = (200 - 204) / 55 = -0.0727

z2 = (280 - 204) / 55 = 1.3818

Using the standard normal distribution table or calculator, we find the probability that a z-score is between -0.0727 and 1.3818 is 0.5244. Therefore, the probability that a hotel room costs between $200 and $280 per night is 0.5244.

(d) To find the cost in dollars of the 20% most expensive hotel rooms in New York City, we need to find the z-score that corresponds to the 80th percentile:

z = invNorm(0.8) ≈ 0.8416

Using the z-score formula, we can find the corresponding hotel room rate:

z = (x - 204) / 55

0.8416 = (x - 204) / 55

x - 204 = 0.8416 * 55

x ≈ 250.29

Therefore, the cost in dollars of the 20% most expensive hotel rooms in New York City is approximately $250.29 per night.