For this assignment you are to imagine that you are part of a select marketing group. 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 list of 25, your imaginary group can consider multiple sampling methods. Let's discuss the pros and cons of two common sampling methods: simple random sampling and stratified random sampling.
1. Simple random sampling:
Pros:
- Fair representation: Every hotel in the list has an equal chance of being selected, ensuring a fair representation of the overall population.
- Simplicity: It is relatively easy to implement as you can randomly select hotels using a random number generator or by drawing lots.
Cons:
- Potential bias: Since the list of hotels is not categorized or stratified, it is possible to end up with a sample that does not adequately represent the different characteristics of the hotels (e.g., luxury, budget, location).
- Variability: The sample may not accurately reflect the true distribution of hotel availability in the population due to random chance.
2. Stratified random sampling:
Pros:
- Improved representation: By dividing the hotels into strata based on specific characteristics (e.g., luxury, location), you can ensure that each stratum of hotels is adequately represented in the sample.
- Enhanced precision: This method allows for targeted analysis within each stratum, leading to more precise insights regarding hotel availability.
Cons:
- Complexity: The process of categorizing hotels into strata and selecting samples from each stratum can be more time-consuming and require more effort.
- Potential bias: If the chosen strata do not accurately represent the population, the sample may not be truly representative.
Once you have selected your sample of 10 hotels, you can calculate the average number of available rooms in that sample by summing up the number of available rooms for each selected hotel and dividing it by 10.
To calculate the standard deviation of the number of available rooms in the sample, you would need to calculate the variance first. The variance measures the variability of data points around the mean. Subtract the mean of the sample from each data point, square the differences, sum them up, and then divide by the sample size minus one. Finally, take the square root of the variance to get the standard deviation.
To construct a frequency distribution of the sample mean, you would need to determine the range for the sample means and divide it into intervals. Count the number of sample means falling within each interval to create the frequency distribution.
To plot a histogram of this distribution, you would represent the intervals on the x-axis and the corresponding frequencies on the y-axis, creating rectangular bars for each interval. The heights of the bars represent the frequency of sample means falling within each interval.
Lastly, you can use the central limit theorem to calculate and identify the sampling distribution of the sample mean. According to the central limit theorem, as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution. This allows you to make inferences about the population based on the sample mean.
To calculate the sampling distribution of the sample mean, you would need to calculate the mean and standard deviation of the sample means. The mean of the sampling distribution of the sample mean is equal to the population mean, and the standard deviation is equal to the population standard deviation divided by the square root of the sample size.
I hope this explanation helps you understand the methods for selecting a sample, calculating average and standard deviation, constructing a frequency distribution and histogram, as well as using the central limit theorem to identify the sampling distribution of the sample mean.