Interpreting Confidence Intervals

What Does "95% Confident" Mean?

Confidence level describes the method, not a specific interval

Correct interpretation: "If we repeated this sampling process many times and constructed a 95% CI each time, about 95% of those intervals would contain the true parameter."

NOT:

• "95% chance the parameter is in this interval" (parameter is fixed!)
• "95% of the data falls in this interval"
• "We are 95% sure this interval contains the parameter"

Visualizing Confidence Level

Imagine 100 different samples:

• Each produces different CI
• About 95 capture true parameter (green)
• About 5 miss true parameter (red)

Our interval is one of these – we don't know if it's green or red!

Example: Correct vs Incorrect

95% CI for mean: (45, 55)

✓ Correct: "We are 95% confident the true mean is between 45 and 55."

✓ Correct: "If we repeated sampling many times, 95% of intervals would capture the true mean."

✗ Incorrect: "There is a 95% probability the mean is between 45 and 55."

✗ Incorrect: "95% of data values are between 45 and 55."

✗ Incorrect: "The sample mean has a 95% chance of being in this interval."

Components of Interpretation

Good interpretation includes:

1. Confidence level: "We are 95% confident..."
2. Parameter (not statistic): "...the true mean (or proportion)..."
3. Context: "...test score for all students..."
4. Interval: "...is between 73 and 82."

Template: "We are [C]% confident that the true [parameter in context] is between [lower bound] and [upper bound]."

Context Matters

Generic: "We are 95% confident μ is between 45 and 55."

Better: "We are 95% confident the mean height of adult males is between 45 and 55 inches."

Even better: "We are 95% confident the mean height of adult males in California is between 45 and 55 inches."

Always state parameter in context of the problem!

Margin of Error Interpretation

CI = statistic ± ME

Interpretation of ME: "We estimate the parameter is within [ME] of [statistic] with [C]% confidence."

Example: ME = 3, $\bar{x}$ = 50, 95% confidence

"We estimate the true mean is within 3 of our sample mean of 50 with 95% confidence."

Width of Interval

Narrower interval:

• More precise estimate
• But requires larger sample or lower confidence

Wider interval:

• Less precise
• But higher confidence or smaller sample

Trade-off: Precision vs confidence

Factors affecting width:

1. Confidence level: Higher → wider
2. Sample size: Larger → narrower
3. Population variability: More variable → wider

Comparing Intervals

Two non-overlapping intervals suggests difference

Example:

• Group 1: (52, 58)
• Group 2: (65, 71)

No overlap → strong evidence of difference

Two overlapping intervals:

• May or may not be significant difference
• Need formal hypothesis test to determine

Using CI for Decisions

Testing H₀: μ = μ₀ at α significance level

Equivalent to: Check if μ₀ is in (1-α) CI

Example: H₀: μ = 50, α = 0.05, 95% CI: (52, 58)

50 not in interval → Reject H₀

But: CI gives MORE information than test (plausible range of values)

Two-Sided vs One-Sided

Two-sided CI: Interval (L, U)

• Most common
• Symmetric around estimate

One-sided CI:

• Upper bound: (-∞, U)
• Lower bound: (L, ∞)
• Less common
• For directional questions

Practical vs Statistical Significance

Statistically significant: Interval doesn't contain null value

Practically significant: Interval contains values that matter in practice

Example: CI for improvement: (0.5, 2.5) points on 100-point test

• Statistically significant (doesn't contain 0)
• But practically? Is 0.5-2.5 point improvement meaningful?

Always consider both statistical AND practical significance!

Common Misinterpretations

❌ "95% of the data is in the interval"

• No! Interval is for parameter (mean/proportion), not individual values
• Prediction interval for individuals (different calculation)

❌ "There's a 95% probability μ is in the interval"

• No! μ is fixed (not random). Interval is random.
• Either μ is in it (probability 1) or not (probability 0)

❌ "We are 95% confident the sample mean is in the interval"

• No! We KNOW sample mean (it's the center of the interval!)
• Confident about population mean, not sample mean

❌ "95% of all samples will give this interval"

• No! Different samples give different intervals
• 95% of intervals (not samples) capture μ

Confidence vs Probability

Probability: Long-run frequency (objective)

• Coin has 50% probability of heads

Confidence: Measure of method reliability

• Method produces correct intervals 95% of the time
• But specific interval either right or wrong

Subtle but important distinction!

Reporting Confidence Intervals

In writing:

• State interval with confidence level
• Interpret in context
• Include units

Example report: "Based on a random sample of 100 students, the 95% confidence interval for mean study time is (8.2, 10.8) hours per week. We are 95% confident that the true mean study time for all students is between 8.2 and 10.8 hours per week."

Limitations of Confidence Intervals

CI only valid if:

• Conditions met (random, normal, independent)
• No bias in data collection
• No measurement errors
• Proper statistical procedure used

CI doesn't account for:

• Sampling bias
• Response bias
• Measurement error
• Non-random sampling

Garbage in, garbage out! CI from biased sample is meaningless.

Choosing Confidence Level

Common choices:

• 90% (less stringent, narrower)
• 95% (standard in many fields)
• 99% (very stringent, wider)

Higher confidence:

• Safer (more likely to capture parameter)
• But less precise (wider interval)

Choice depends on:

• Consequences of being wrong
• Field conventions
• Desired precision

Quick Reference

Correct interpretation template: "We are [C]% confident that the true [parameter in context] is between [L] and [U]."

Common mistakes to avoid:

• Probability statements about parameter
• Statements about data/sample
• Forgetting context
• Confusing confidence with probability

Remember: Confidence describes the method's reliability, not probability that this specific interval is correct. Always interpret in context with proper terminology!