46: Central Limit Theorem Activity
- Page ID
- 8634
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
\(\frac{\sigma_{x}}{\sqrt{n}} = \)