6.4: CLT for Means and Proportions
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- 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. The Central Limit Theorem (CLT) states:
Even if the original population isn't normally distributed, the sampling distribution of the sample mean will become approximately normal as sample size increases.
This idea doesn’t just apply to means, but also works for sample proportions.
Two Types of Statistics
For now, we will discuss the two different statistics:
- Sample Means, \( \bar{x} \)
- Sample Proportions, \( \hat{p} \)
When we take a sample, there is an element of randomness. If we were to take repeated samples, we would see a sampling distribution for both mean and proportions start to emerge. This sampling distribution has the following properties:
- Centers around the population value (mean or proportion)
- Gets narrower as sample size increases
- Approximates a normal distribution given the below conditions
CLT Summary: Means vs. Proportions
| Statistic | Population Parameter | Mean of Sampling Distribution | Standard Error | Normality Conditions |
|---|---|---|---|---|
| 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’s different:
- The formulas for standard error
- The conditions required for “approximately normal”
- Notation!
What’s the same:
- The general shape of the sampling distribution is normal (under the right conditions)
- The center equals the population parameter
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
Try to see why this is a silly scenario! If we actually knew the population mean, there would be no need to sample. This simply illustrates that when we sample there is a quantifiable uncertainty.
Example 2: Sample Proportion – Product Support
To consider a more realistic scenario, suppose that we already expect a proportion of 60% from past studies. We can utilize the CLT to compare with new data. If we resample with a sample size \(n = 100\) we would have:
- 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 \( n(1 - p) = 40 \))
Consider if we end up with a value such as 0.63 from our sample. This is within our sample error, and not too surprising. Instead, if we had a sample proportion of 0.7, this is a significantly different value than our original assumption that it may warrent a review!
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.