Question: Use Excel to illustrate the Central Limit Theorem when sampling from a binomial distribution.

1. Generate at least 1000 random samples (you can generate more if you wish), each with 4 observations, from a binomial random variable \(y\) where \(p=0.2\) (the probability of a success), and \(\mathrm{n}=10\) trials. That is, the random variable y represents the number of successes in 10 trials, where the probability of a success is 0.2. Calculate the sample mean for each sample.
   - Plot a frequency histogram of the sample means and comment on the plot.
   - Determine the range, mean and standard deviation of the 1000 sample means.
   - How do the calculated mean and standard deviation compare with what you would have expected from a theoretical analysis?
2. Generate at least 1000 random samples (you can generate more if you wish), each with 30 observations, from a binomial random variable \(y\) where \(p=0.2\) (the probability of a success), and \(\mathrm{n}=10\) trials. That is, the random variable y represents the number of successes in 10 trials, where the probability of a success is 0.2. Calculate the sample mean for each sample.

- Plot a frequency histogram of the sample means and comment on the plot. Determine the range, mean and standard deviation of the 1000 sample means.

- How do the calculated mean and standard deviation compare with what you would have expected from a theoretical analysis?

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