Confidence Interval Formula

A confidence interval is a range of values, calculated from sample data, constructed so that the procedure captures the true population parameter a.

The Formula

estimate±margin of error\text{estimate} \pm \text{margin of error}

When to use: Instead of saying 'the average is 50,' you say 'I'm 95% confident the average is between 47 and 53.' The interval acknowledges uncertainty from sampling.

Quick Example

Poll: 52% support candidate, margin of error ±3%\pm 3\%. 95% CI: 49%-55%. True support is probably in this range.

Notation

CI is the confidence interval. zα/2z_{\alpha/2} is the critical z-value (1.96 for 95%). α\alpha is the significance level. The margin of error is E=zα/2⋅SEE = z_{\alpha/2} \cdot SE.

What This Formula Means

A confidence interval is a range of values, calculated from sample data, constructed so that the procedure captures the true population parameter a specified percentage of the time (e.g., 95%). It quantifies the uncertainty inherent in using a sample to estimate a population value.

Instead of saying 'the average is 50,' you say 'I'm 95% confident the average is between 47 and 53.' The interval acknowledges uncertainty from sampling.

Formal View

A (1−α)100%(1-\alpha)100\% confidence interval for μ\mu is xˉ±zα/2⋅σn\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}. When σ\sigma is unknown, use xˉ±tα/2,n−1⋅sn\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}.

Worked Examples

Example 1

medium
From n=64n=64 samples, xˉ=120\bar{x}=120, σ=16\sigma=16 known. Find the 99% CI for μ\mu.

Answer

[114.85, 125.15][114.85,\ 125.15]

First step

1
SE=16/8=2SE=16/8=2; z∗=2.576z^*=2.576.

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Example 2

medium
How large should nn be so that a 95% z-CI for μ\mu (with σ=12\sigma=12) has margin ≤2\le 2?

Example 3

hard
xˉ=68\bar{x}=68, s=12s=12, n=9n=9. Construct a 95% t-CI for μ\mu. Use t∗=2.306t^*=2.306 (df =8=8).

Common Mistakes

  • Thinking 95% CI means 95% of data falls there - The safer move is to ask "Am I using sample-to-sample variation to make a population claim with uncertainty stated clearly?" and then state the data source, denominator, or variable before interpreting the result.
  • Interpreting as probability for one interval - The safer move is to ask "Am I using sample-to-sample variation to make a population claim with uncertainty stated clearly?" and then state the data source, denominator, or variable before interpreting the result.
  • Confusing confidence with probability - The safer move is to ask "Am I using sample-to-sample variation to make a population claim with uncertainty stated clearly?" and then state the data source, denominator, or variable before interpreting the result.
  • Choosing confidence interval from a keyword alone - Keywords like estimate, confidence, sample are only clues; the data structure must match the concept.

Why This Formula Matters

Confidence Interval is the bridge from sample data to population reasoning. It matters because real data are incomplete, so students must learn to state uncertainty, check conditions, and avoid claiming more than the sample design supports.

Frequently Asked Questions

What is the Confidence Interval formula?

A confidence interval is a range of values, calculated from sample data, constructed so that the procedure captures the true population parameter a specified percentage of the time (e.g., 95%). It quantifies the uncertainty inherent in using a sample to estimate a population value.

How do you use the Confidence Interval formula?

Instead of saying 'the average is 50,' you say 'I'm 95% confident the average is between 47 and 53.' The interval acknowledges uncertainty from sampling.

What do the symbols mean in the Confidence Interval formula?

CI is the confidence interval. zα/2z_{\alpha/2} is the critical z-value (1.96 for 95%). α\alpha is the significance level. The margin of error is E=zα/2⋅SEE = z_{\alpha/2} \cdot SE.

Why is the Confidence Interval formula important in Statistics?

Confidence Interval is the bridge from sample data to population reasoning. It matters because real data are incomplete, so students must learn to state uncertainty, check conditions, and avoid claiming more than the sample design supports.

What do students get wrong about Confidence Interval?

Students often know a procedure related to confidence interval but skip the recognition step: Am I using sample-to-sample variation to make a population claim with uncertainty stated clearly? That leads to a calculation or graph that looks reasonable but answers a different question.

What should I learn before the Confidence Interval formula?

Before studying the Confidence Interval formula, you should understand: standard error, sampling distribution.

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