Statistics (Statistics Project Sample)

This is a statistics Question
1. Assume that a population is normally distributed with a mean of 100 and a standard deviation of 15. Would it be unusual for the mean of a sample of 3 to be 115 or more? Why or why not?
Mean=100
Standard deviation =15
Sample size (n) = 3
We use 95% significance level where 1-0.95=0.05/2=0.025 from tables =1.96
Formula =
Replacing the value in the formula
It would not be unusual for the mean to be 115 since it lies in between the range of 116.97 and 83.03.
a. What if the size for each sample was increased to 20? Would a sample mean of 115 or more be considered unusual? Why or why not?
Mean= 100
Standard deviation= 15
Sample Size= 20
With the same significance level of 95%
The sample mean of 115 would be considered unusual since the sample mean of 115 exceeds the limit in place of 106.57 and 93.43.
b. Why is the Central Limit Theorem used?
Central Limit Theorem is the main stay of statistics and probability. The theorem shows that as the size of a sample increases, the mean distribution between multiple samples becomes like a Gaussian distribution (Illowsky and Dean 2017). CLT shows a constant distribution for estimates. You can use this to inquire about the probabilities of the estimates we make.