Confirmation of the central limit theorem by simulation.

Central Limit Theorem (CLT) is: The theoretical basis for this approach happens when its conditions are met. This theorem provides powerful ways of answering a variety of statistical questions.

Suppose there are 30 students in a class and each draws an independent sample of n=100 observations from the same normally distributed population with a known and variance ². Suppose we know the mean of the population is, = 0.5 and ² = 0.0833. The results of these observations are summarized in the following table.

Sample	Mean, y	Variance, S2	Sample	Mean, y	Variance, S2
1	0.4809	0.0912	16	0.5094	0.0675
2	0.4861	0.0867	17	0.4637	0.0901
3	0.4916	0.0835	18	0.5033	0.0835
4	0.5373	0.0859	19	0.5227	0.0803
5	0.4840	0.0883	20	0.5207	0.0882
6	0.5947	0.0719	21	0.5060	0.0821
7	0.5026	0.0901	22	0.4858	0.0830
8	0.4969	0.0860	23	0.5511	0.0851
9	0.4837	0.0842	24	0.4715	0.0852
10	0.4936	0.0790	25	0.5381	0.0881
11	0.4990	0.0786	26	0.4931	0.0933
12	0.4999	0.0771	27	0.5510	0.0763
13	0.4804	0.0754	28	0.5102	0.0869
14	0.5071	0.0864	29	0.4700	0.0887
15	0.4838	0.0719	30	0.4745	0.0769

Instructions:

1. Please discuss the ways you would confirm that this scenario follows the Central Limit Theorem (CLT). Please support your decisions through calculating the sample statistics and compare them with population parameters. You may also utilize Excels statistical functions to build your formulations for this decision model. Hints: We want to check if:
   • The average of all means [Mean of Means] agrees with population Mean.
   • The sample variance S² is consistent with ²/n.
   • The distribution of sample means is Approximately Normal, i.e., the means appear to follow the normal distribution.