# 45: Central Limit Theorem Activity

### This is an activity to verify the Central Limit Theorem.  Be sure to enter in your answers as decimals rather than fractions when necessary.

A population consists of the numbers  Find the population mean $$\mu$$.  Do this by hand and show your work.

Assuming samples of size $$n = 2$$ are drawn with replacement between each selection.  Let $$\bar{x}$$ represent the mean of each of these samples.  Find $$\mu_{\bar{x}}$$, the mean of the sample means.  First, complete the table of means.  The first row has been completed for you.

Now fill in the table below of the possible sample means, the probabilities of those means occurring, and the product of the mean and its probability.
 $$\bar{x}$$ $$p(\bar{x})$$ $$\bar{x} p(\bar{x})$$

Now add all of the entries of the last column to arrive at $$\mu_{\bar{x}}$$.

$$\mu_{\bar{x}} =$$

Next decide how $$\mu_{\bar{x}}$$ compares to $$\mu$$.
$$\mu_{\bar{x}}$$  $$\mu$$

Next, find the population standard deviation, $$\sigma_{x}$$.  First, complete the table below.
 $$x$$ $$x - \mu$$ $$(x - \mu )^2$$

Now, add up the last column:

$$\sum_{i=1}^4 (x - \mu)^2 =$$

Now, find $$\sigma_{x}$$ using the formula $$\sqrt{\frac{\sum_{i=1}^N (x_{i} - \mu)^2}{N}}$$ Round your answer to four decimal places.

$$\sigma_{x} =$$

Now let's work on $$\sigma_{\bar{x}}$$.  Complete the table below:
 $$\bar{x}$$ $$\bar{x}^2$$ $$p(\bar{x})$$ $$\bar{x}^2p(\bar{x})$$

Now, add up the last column:

$$\sum \bar{x}^2 p(\bar{x}^2) =$$

Now, find $$\sigma_{\bar{x}}$$ using the formula $$\sqrt{\sum_{i=1}^n [\bar{x_i}^2 p(\bar{x_i})] - \mu^2}$$ Round your answer to four decimal places.

$$\sigma_{\bar{x}} =$$

Next decide how $$\sigma_{\bar{x}}$$ compares to $$\sigma$$.

$$\sigma_{\bar{x}}$$    $$\sigma$$

Now, find $$\frac{\sigma_{x}}{\sqrt{n}}$$.  Round your answer to four decimal places.

$$\frac{\sigma_{x}}{\sqrt{n}} =$$

Finally, compare  $$\sigma_{\bar{x}}$$ to $$\frac{\sigma_{x}}{\sqrt{n}}$$.
$$\sigma_{\bar{x}}$$     $$\frac{\sigma_{x}}{\sqrt{n}}$$

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