7.4: Effect of Sample Size and Confidence Level
- Page ID
- 58918
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Now that we’ve built and interpreted confidence intervals, it’s time to reflect on something subtle but important: what affects how wide or narrow those intervals are?
In this section, we’ll focus on two powerful factors:
- The confidence level (like 90%, 95%, 99%)
- The sample size used to build the interval
Higher Confidence Comes with a Cost
When we want to be more confident (say, 99% instead of 95%), we’ll need to include a wider range of values. This makes sense to be more sure we’ve captured the true value, we must cast a wider net.
Here’s how it looks with the same sample data:
- Confidence level = 90% → narrower interval
- Confidence level = 95% → standard choice
- Confidence level = 99% → wider interval
The critical value (z* or t*) increases as confidence increases and so does the margin of error.
Larger Samples Give Greater Precision
Sample size also plays a critical role. As the sample size increases:
- The standard error decreases
- The margin of error gets smaller
- The confidence interval becomes narrower
Reminder: Standard error is affected by sample size:
\( \text{SE} = \frac{s}{\sqrt{n}} \) for a mean or \( \text{SE} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \) for a proportion
Example 1: Different Confidence Levels
You survey 100 students about weekly screen time. The average is 23.4 hours/week with a standard deviation of 5.1 hours.
Compare the width of the following intervals:
- 90% confidence: \( t^* \approx 1.660 \), ME = 0.85 → Interval = (22.55, 24.25)
- 95% confidence: \( t^* \approx 1.984 \), ME = 1.02 → Interval = (22.38, 24.42)
- 99% confidence: \( t^* \approx 2.626 \), ME = 1.35 → Interval = (22.05, 24.75)
Note that the standard error (SE) is the same in all these calculations, only the critical value changes.
Conclusion: As confidence increases, the interval gets wider.
Example 2: Changing the Sample Size
The same data is collected, but from only 25 students. Now compare with:
- Mean = 23.4, SD = 5.1, n = 25 → SE = 1.02
Notice that while the sample standard deviation is the same, the smaller n means there is a larger standard error (SE). This makes the margin of error (ME) bigger as well.
- 95% CI: ME = \( 2.064 \cdot 1.02 \approx 2.11 \), Interval = (21.29, 25.51)
If n=100 (the previous example). 95% CI: (22.38, 24.42)
Conclusion: A larger sample size gives a more precise or narrow interval with the same confidence level.
Interactive Demo: Confidence Interval Width
Adjust the sliders below to change the sample size (n) and confidence level. Watch how the margin of error and confidence interval width respond.
100Sample standard deviation: 15
Standard error:
Margin of error:
Confidence interval:
Check Your Understanding: Confidence Intervals
Question 1: Compared to a 90% confidence interval, a 99% confidence interval for the same data is...
Question 2: Compared to a 90% confidence interval, a 99% confidence interval for the same data is...
narrowwiden
stay the same
Why This Matters
Different studies require different levels of precision and certainty. These tools help researchers design better surveys and polls. You’ll use these relationships to:
- Plan studies with a target margin of error
- Decide how trustworthy an estimate may be
- Compare results responsibly
Now that we’ve built our understanding of confidence intervals, we’re ready to apply them in new ways including hypothesis testing. But first, let’s reflect on how we interpret and communicate these results effectively.