45: Central Limit Theorem Activity
- Page ID
- 8634
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.
\(\bar{x} \) | \(p(\bar{x}) \) | \(\bar{x} p(\bar{x}) \) |
Now add all of the entries of the last column to arrive at \(\mu_{\bar{x}} \).
\(\mu_{\bar{x}} = \)
\(x \) | \(x - \mu \) | \((x - \mu )^2 \) | |
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.
\(\sigma_{x} = \)
\(\bar{x} \) | \(\bar{x}^2 \) | \(p(\bar{x}) \) | \(\bar{x}^2p(\bar{x}) \) |
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 \(\sigma \).
Now, find \(\frac{\sigma_{x}}{\sqrt{n}} \). Round your answer to four decimal places.
\(\frac{\sigma_{x}}{\sqrt{n}} = \)