You are to imagine that you are part of a select marketing group. This group is to consider the Colorado City Convention and Vistitors Bureau survey of 25 hotels. The survey is concerned with the current availability of rooms. They are as follows: 90, 72,75,60,75,72,84,72,105,115,68,74,80,64,414,84 48,58,60,80,48,58 and108
Pick 10 hotels....
Use the central limit theorem to calculate and identify the sampling distribution of the sample mean.
To calculate and identify the sampling distribution of the sample mean using the central limit theorem, you first need to select a random sample of hotels from the given data. Here's how you can pick 10 hotels from the survey:
1. Create a list of the hotel values:
[90, 72, 75, 60, 75, 72, 84, 72, 105, 115, 68, 74, 80, 64, 414, 84, 48, 58, 60, 80, 48, 58, 108]
2. Randomly select 10 values from the list using a random number generator or any other method of your choice. For example, let's say you randomly selected the following 10 hotels:
[60, 84, 72, 80, 58, 72, 68, 64, 108, 80]
Now that you have your sample of 10 hotels, you can proceed to calculate and identify the sampling distribution of the sample mean:
3. Calculate the mean (average) of the selected hotels:
(60 + 84 + 72 + 80 + 58 + 72 + 68 + 64 + 108 + 80) / 10 = 726 / 10 = 72.6
The sample mean for this particular sample is 72.6.
4. According to the central limit theorem, the sampling distribution of the sample mean will follow a normal distribution, regardless of the shape of the population distribution, as long as the sample size is sufficiently large.
In this case, since you only sampled 10 hotels, which is a relatively small sample size, the shape of the population distribution may still have some influence on the sampling distribution of the sample mean. However, if you were to sample a larger number of hotels, the sampling distribution would eventually approach a normal distribution.
Note: To get a more accurate estimation of the sampling distribution, you would need to repeat steps 2-3 multiple times and calculate the sample mean for each random sample. By doing so, you can observe the distribution of these sample means and analyze it to further identify the sampling distribution of the sample mean.