7.5: Minimum Sample Size
- Page ID
- 64946
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Minimum Sample Size for Mean & Proportion Estimates
When planning a study, it is critical to ask how much data do we need? Too small a sample may lead to inaccurate conclusions due to natural variability. On the other hand, time and resources may limit how large of a sample can be gathered. We will look at formulas in this section that give minimum estimates for sample sizes to ensure a given margin of error at a specified confidence level.
Estimating a Population Mean
For estimating a mean, the minimum sample size depends on:
- The desired margin of error (E): the maximum amount of error you're willing to accept in the estimate.
- The confidence level (determines the critical value, z*)
- An estimate of the population standard deviation (σ)
The formula to calculate the minimum sample size (\( n \)) is:
\[ n = \left( \frac{z^* \cdot \sigma}{E} \right)^2 \]
Example: Calculating Sample Size for a Mean
Suppose a researcher wants to determine the average height of sunflowers with a margin of error of 2 cm at a 95% confidence level. If a preliminary study suggests that the standard deviation of sunflower heights is 10 cm, then:
- Identify: \( z^* \approx 1.96 \) for 95% confidence
- Calculate: \[ n \approx \left( \frac{1.96 \times 10}{2} \right)^2 \approx 96.04 \]
Therefore, the researcher needs a sample of at least 97 sunflowers (always round up!) to achieve the desired confidence and precision.
Unknown Population Deviation
In most applications of statistics, we do not know the population standard deviation. It's then up to us to estimate them as best we can. One method is to do a smaller preliminary study to gain some insight into the required sample size. If the sampling is occurring over time, this result can be revised multiple times as the data is collected. It is feasible to also utilize standard deviation data from prior studies or comparable populations as a proxy. These methods may not produce perfect results, but it's critical to remember that this calculated value is a minimum estimate and there is nothing wrong with collecting more data as long as the sampling methods are sound and as unbiased as possible.
Estimating a Population Proportion
For estimating a proportion, the sample size formula also incorporates:
- The desired margin of error (E)
- The confidence level (determines the critical value, z*)
- Estimated proportion (\( \hat{p} \)): a guessed proportion based on past research or preliminary data
The minimum sample size formula for a proportion is:
\[ n = \left( \frac{z^*}{E} \right)^2 \cdot \hat{p}(1 - \hat{p}) \]
Example: Calculating Sample Size for a Proportion
Let’s say a school administrator wants to know the proportion of students who participate in after-school activities with a margin of error of 5% at a 90% confidence level. If a pilot survey showed that about 40% have participated, then:
- Identify: \( z^* \approx 1.645 \) for 90% confidence
- Calculate: \[ n = \left( \frac{1.645}{0.05} \right)^2 \cdot 0.40(1 - 0.40) = 259.78 \]
Thus, the administrator needs a sample of at least 260 students to estimate the proportion accurately and precisely.
Unknown Proportion Estimates
In this scenario, we have a much easier time when our estimates are unknown. The value \( \hat{p}(1 - \hat{p})\) achieves its maximum when \(\hat{p} = 0.5\). By using this value, we will simply calculate a higher sample size which can only help our overall accuracy.