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 pick 10 hotels from the survey, you can randomly select any 10 numbers from the given list. Here are 10 randomly selected hotels from the survey: 72, 75, 60, 68, 74, 80, 48, 58, 60, and 108.
Now let's calculate and identify the sampling distribution of the sample mean using the central limit theorem. The central limit theorem states that for a large enough 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, we need to find the mean and standard deviation of the sample.
1. Calculate the mean of the sample:
Sum up the values of the 10 hotels you selected: 72 + 75 + 60 + 68 + 74 + 80 + 48 + 58 + 60 + 108 = 703
Divide the sum by the sample size (10): 703 / 10 = 70.3
2. Calculate the standard deviation of the sample:
Subtract the mean from each value and square the difference:
(72 - 70.3)^2, (75 - 70.3)^2, (60 - 70.3)^2, (68 - 70.3)^2, (74 - 70.3)^2, (80 - 70.3)^2, (48 - 70.3)^2, (58 - 70.3)^2, (60 - 70.3)^2, (108 - 70.3)^2
Sum up the squared differences: 4586.6
Divide the sum by (n - 1), where n is the sample size (10 - 1): 4586.6 / 9 = 509.62
Take the square root of the result: √509.62 ≈ 22.59
Now, with the mean of the sample (70.3) and the standard deviation of the sample (22.59), we have the parameters to describe the sampling distribution of the sample mean.
The sampling distribution of the sample mean is approximately normal with a mean equal to the population mean (which is unknown in this case) and a standard deviation equal to the population standard deviation divided by the square root of the sample size. However, since we don't have the population standard deviation, we can use the sample standard deviation as an estimate.
So, the estimated standard deviation of the sampling distribution of the sample mean can be calculated by dividing the sample standard deviation (22.59) by the square root of the sample size (10):
Estimated standard deviation = 22.59 / √10 ≈ 7.14
Therefore, the sampling distribution of the sample mean has an estimated mean of 70.3 and an estimated standard deviation of 7.14.