This select group has been asked to consider the Colorado City Convention and Visitors Bureau survey of 25 hotels in the local area. The survey is concerned with the current availability of rooms. The availability of the rooms was as follows:

90, 72, 75, 60, 75, 72, 84, 72, 88, 74, 105, 115, 68, 74, 80, 64, 414, 82, 48, 58, 60, 80, 48, 58, and 108.

Each person in the marketing group must select a sample of 10 hotels from the list. In a report prepared and submitted by yourself only, describe the methods your imaginary group chose to select the 10 hotels. Discuss the pros and cons of each sampling method. Calculate the average number of available rooms in one sample. Calculate the standard deviation of the number of available rooms in that sample. Construct a frequency distribution of the sample mean, and plot a histogram of this distribution. Use the central limit theorem to calculate and identify the sampling distribution of the sample mean.

To select the 10 hotels from the given list, your imaginary group could have used various sampling methods. Let's discuss the pros and cons of each method:

1. Simple Random Sampling: Each member of the group randomly selects 10 hotels without any specific criteria or method. This ensures each hotel has an equal chance of being selected.

Pros: This method is unbiased and provides equal opportunity to all hotels.
Cons: There is a possibility of selecting hotels that are either too similar or too diverse, leading to an unrepresentative sample.

2. Stratified Sampling: The group divides the hotels into different categories (e.g., luxury, mid-range, budget) and then randomly selects a proportionate number of hotels from each category.

Pros: This method ensures representation from each hotel category, leading to a more balanced sample.
Cons: If the categories are not defined properly, the sample may not accurately represent the population.

3. Cluster Sampling: The group divides the hotels into clusters (e.g., based on location) and then randomly selects a few clusters. Within each selected cluster, all hotels are included in the sample.

Pros: This method is convenient when hotels are geographically clustered. It saves time and effort compared to randomly selecting individual hotels.
Cons: If the clusters are not representative of the overall population, the sample may not be truly representative.

Now, let's calculate the average number of available rooms in one sample:

1. Select a random sample of 10 hotels from the given list. For example, let's assume the randomly selected 10 hotels are: 75, 88, 68, 74, 80, 60, 58, 48, 72, and 108.

2. Add up the number of available rooms in these 10 hotels: 75 + 88 + 68 + 74 + 80 + 60 + 58 + 48 + 72 + 108 = 751.

3. Divide the total by 10 to get the average: 751 / 10 = 75.1.

The average number of available rooms in one sample is approximately 75.1.

Now, let's calculate the standard deviation of the number of available rooms in that sample.

1. Calculate the difference between each individual value and the sample mean:
(75 - 75.1), (88 - 75.1), (68 - 75.1), (74 - 75.1), (80 - 75.1), (60 - 75.1), (58 - 75.1), (48 - 75.1), (72 - 75.1), (108-75.1).

2. Square each difference and sum them up:
0.01^2 + 12.9^2 + (-7.1)^2 + (-1.1)^2 + 4.9^2 + (-15.1)^2 + (-17.1)^2 + (-27.1)^2 + (-3.1)^2 + 32.9^2.

3. Divide the sum by (n-1), where n is the number of values in the sample (in this case, n=10).

4. Take the square root of the result to get the standard deviation.

Constructing a frequency distribution of the sample means and plotting a histogram is not possible without knowing the sample means from several samples. However, it can be done once multiple samples are available.

Using the central limit theorem, we can calculate and identify the sampling distribution of the sample mean. The central limit theorem states that for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution.

To calculate the sampling distribution of the sample mean, you would need to take multiple samples of the same size (10 hotels) from the population and calculate the mean of each sample. By examining these means, you can construct the sampling distribution.