7.5: Interpreting Confidence Intervals
- Page ID
- 58919
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Now that we've learned how to build confidence intervals for means and proportions, it's time to focus on the most important part: interpreting what they tell us and what they don't.
Let’s work through a real example where a confidence interval helps us answer a bigger question.
Example: Is This Coin Fair?
You’re handed a coin that someone claims is completely fair. That is, the probability of heads is exactly 0.50.
You want to test this claim by flipping the coin 100 times and calculating the proportion of heads.
You do the test and count 57 heads out of 100 flips. That gives a sample proportion:
\( \hat{p} = \frac{57}{100} = 0.57 \)
Step 1: Build a Confidence Interval
The sample proportion is 0.57. We’ll use a 95% confidence level to see which values are plausible for the true proportion of heads, \( p \).
First, calculate the standard error:
\( SE = \sqrt{ \frac{0.57(1 - 0.57)}{100} } \approx 0.049 \)
Find margin of error using \( z^* = 1.96 \):
\( ME = 1.96 \cdot 0.049 \approx 0.096 \)
Confidence interval:
\( (0.57 - 0.096,\ 0.57 + 0.096) = (0.474,\ 0.666) \)
Step 2: Interpret the Interval
We are 95% confident that the true proportion of heads is between 0.474 and 0.666.
Now we ask: Is 0.50 inside this interval?
Yes! So we cannot rule out the possibility that the coin is fair, even though we saw 57 heads. The result is not strong enough to say the coin is biased.
The interval leaves open the possibility that \( p = 0.5 \), since that value is inside the range of likely population values.
Make a Reasoned Conclusion
- The sample proportion was 0.57 (more heads than expected)
- But sampling variability explains small surprises
- Since 0.50 is within the confidence interval → this could happen by chance
- Conclusion: No strong evidence the coin is unfair
This is the power of confidence intervals: they don’t give us absolute truths, but they help us understand what values are reasonable based on our data.
- Suppose instead we had seen 62 heads out of 100. Would 0.50 still be inside the 95% confidence interval?
- If you needed strong evidence to prove the coin wasn’t fair, would you increase your sample size or your confidence level?
- How would this interval change if we had flipped the coin 1,000 times?