6.4: CLT for Means and Proportions
- Page ID
- 58913
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In the last two sections, we explored what sampling distributions are and how they behave especially for means. The Central Limit Theorem (CLT) showed us a remarkable truth:
Even if the original population isn't normally distributed, the sampling distribution of the sample mean will become approximately normal as sample size increases.
That's powerful. And the good news is: this idea doesn’t just apply to means. It also works for sample proportions.
Two Types of Statistics, Same Big Idea
We often work with two kinds of sample statistics:
- Sample Means, \( \bar{x} \)
- Sample Proportions, \( \hat{p} \)
If we repeat sampling and compute these statistics over and over, we get a distribution of values — a sampling distribution.
For both types of statistics, the sampling distribution:
- Centers around the population value (mean or proportion)
- Gets narrower as sample size increases
- Approximates a normal distribution (if conditions are met)
CLT Summary: Means vs. Proportions
| Statistic | Population Parameter | Mean of Sampling Distribution | Standard Error | Shape |
|---|---|---|---|---|
| Sample Mean \( \bar{x} \) | \( \mu \) | \( \mu \) | \( \displaystyle \frac{\sigma}{\sqrt{n}} \) | Approximately normal if \( n \geq 30 \), or if population normal |
| Sample Proportion \( \hat{p} \) | \( p \) | \( p \) | \( \displaystyle \sqrt{ \frac{p(1 - p)}{n} } \) | Approximately normal if \( np \geq 10 \) and \( n(1 - p) \geq 10 \) |
What Changes? What Stays the Same?
What’s different:
- The formulas for standard error
- The conditions required for “approximately normal”
What’s the same:
- The general shape of the sampling distribution is normal (under the right conditions)
- The center equals the population parameter
- They follow the same logic: take many samples → compute statistic → observe distribution
Conceptual Comparisons
Example 1: Sample Mean – Commute Times
The population average commute time is 28 minutes with a standard deviation of 6 minutes. You take a sample of 40 commuters.
The sampling distribution of \( \bar{x} \) will be approximately normal with:
- Center: 28 minutes
- Standard error: \( \displaystyle \frac{6}{\sqrt{40}} \approx 0.95 \) minutes
Example 2: Sample Proportion – Product Support
You survey customers to estimate what proportion support a new product. Based on past data, you expect about 60% approval.
If \( n = 100 \), then:
- Center: 0.60
- Standard error: \( \displaystyle \sqrt{ \frac{0.6(0.4)}{100} } = 0.049 \)
- The sampling distribution of \( \hat{p} \) will be normal (since \( np = 60 \) and \( n(1 - p) = 40 \))
Why This Matters
The Central Limit Theorem tells us that sampling distributions behave in a predictable, normal way even when the underlying population does not. This allows us to:
- Estimate margin of error
- Build confidence intervals
- Run hypothesis tests
Related Video
In the next chapter, we use this knowledge to create confidence intervals a way to estimate population values using sample statistics and error margins.