Question

Often, travelers who have no intention of showing up fail to cancel their hotel reservations in a timely manner. These travelers are known, in the parlance of the hospitality trade, as "no-shows." To protect against no-shows and late cancellations, hotels invariably overbook rooms. A study reported in the Journal of Travel Research examined the problems of overbooking rooms in the hotel industry. The data in the table, extracted from the study, represent daily numbers of late cancellations and no-shows for a random sample of 30 days at a large (500-room) hotel. Based on this sample, how many rooms, at a minimum, should the hotel overbook each day?

18 16 16 16 14 18 16 18 14 19 15 19 9 20 10 10 12 14 18 12 14 14 17 12 18 13 15 13 15 19
(a) Find the sample size, mean, mode, range, standard deviation, and variance for the first seven numbers from the data manually. Show relevant work

Answer 1

The sample size for the first seven numbers is 7. The mean is 16.29, the mode is 16, the range is 4, the standard deviation is approximately 1.38, and the variance is 1.9.

Step-by-step explanation:

To calculate the statistics for the first seven numbers in the data set (18, 16, 16, 16, 14, 18, 16), we will find the sample size, mean, mode, range, standard deviation, and variance.

• Sample Size (n): The number of observations in our sample is 7.
• Mean: The average is calculated as the sum of all numbers divided by the sample size. (18 + 16 + 16 + 16 + 14 + 18 + 16) / 7 = 16.29 (rounded to two decimal places).
• Mode: The mode is the number that appears most frequently in the data set, which is 16.
• Range: The range is the difference between the highest and lowest values. 18 (highest) - 14 (lowest) = 4.
• Standard Deviation: This is calculated using the formula for the sample standard deviation, which involves finding the square root of the sample variance. First, we find each number's deviation from the mean, square these deviations, average them, and then take the square root.
  Deviations (rounded to two decimal places): 1.71, -0.29, -0.29, -0.29, -2.29, 1.71, -0.29. Squaring each gives the values: 2.92, 0.08, 0.08, 0.08, 5.24, 2.92, 0.08. Summing these and dividing by n-1 (since it's a sample): (2.92 + 0.08 + 0.08 + 0.08 + 5.24 + 2.92 + 0.08) / 6 = 1.9. Taking the square root gives a standard deviation of approximately 1.38.
• Variance: The variance is the squared standard deviation, which is 1.9.