Confidence Intervals for Proportions

What is a Confidence Interval?

Confidence Interval (CI): Range of plausible values for population parameter

Form: statistic ± margin of error

Interpretation: We are C% confident the interval contains the true parameter

Example: 95% CI for p: (0.52, 0.58)
We are 95% confident true population proportion is between 0.52 and 0.58

One-Sample CI for Proportion

Formula:

$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Where:

• $\hat{p}$ = sample proportion
• z* = critical value (from confidence level)
• n = sample size

Critical Values

Common confidence levels:

| Confidence Level | z* | |------------------|-----| | 90% | 1.645 | | 95% | 1.96 | | 99% | 2.576 |

Higher confidence → wider interval

Example 1: Simple CI

Survey: 400 voters, 220 support candidate

$\hat{p} = \frac{220}{400} = 0.55$

95% CI:

$SE = \sqrt{\frac{0.55(0.45)}{400}} = \sqrt{0.0006188} \approx 0.0249$

$CI = 0.55 \pm 1.96(0.0249) = 0.55 \pm 0.049$

$(0.501, 0.599)$

Interpretation: We are 95% confident between 50.1% and 59.9% of voters support the candidate.

Conditions for CI

Random: Random sample
Normal: np̂ ≥ 10 and n(1-p̂) ≥ 10
Independent: n ≤ 10% of population

Check ALL before proceeding!

Margin of Error

Margin of Error (ME):

$ME = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Factors affecting ME:

• Larger z* (higher confidence) → larger ME
• Larger n → smaller ME
• p̂ near 0.5 → larger ME (maximum variability)

Sample Size for Desired ME

To achieve margin of error m:

$n = \left(\frac{z^*}{m}\right)^2 \hat{p}(1-\hat{p})$

Conservative approach (if no estimate): Use p̂ = 0.5

$n = \left(\frac{z^*}{m}\right)^2 (0.25)$

Example: Want ME = 0.03 with 95% confidence

$n = \left(\frac{1.96}{0.03}\right)^2 (0.25) = (65.33)^2(0.25) \approx 1068$

Need at least 1068 people!

Interpreting Confidence Level

95% confidence means:

• If we repeated sampling many times and built 95% CI each time
• About 95% of intervals would contain true p
• About 5% would miss true p

NOT:

• "95% chance p is in our interval" (p is fixed!)
• "95% of data is in interval"

Our specific interval either contains p or it doesn't (we just don't know which)

Increasing Confidence

Want higher confidence (say 99% instead of 95%):

• Use larger z* (2.576 instead of 1.96)
• Interval becomes wider
• Trade-off: More confidence but less precision

Example 2: With Interpretation

Survey of 500 students: 285 have jobs

$\hat{p} = \frac{285}{500} = 0.57$

Conditions:

• Random: Assume random sample ✓
• Normal: 500(0.57) = 285 ≥ 10, 500(0.43) = 215 ≥ 10 ✓
• Independent: 500 < 10% of all students (assume) ✓

90% CI:

$SE = \sqrt{\frac{0.57(0.43)}{500}} \approx 0.0221$

$CI = 0.57 \pm 1.645(0.0221) = 0.57 \pm 0.036$

$(0.534, 0.606)$

Interpretation: We are 90% confident that between 53.4% and 60.6% of all students have jobs.

Common Mistakes

❌ Saying "95% of data in interval"
❌ Saying "95% chance p in interval"
❌ Not checking conditions
❌ Using t* instead of z* for proportions
❌ Rounding p̂ too early

Two-Sample CI for Difference in Proportions

Comparing two groups:

$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

Conditions: Each group meets conditions separately

Interpretation: If interval contains 0, no significant difference

Calculator Commands (TI-83/84)

STAT → TESTS → A:1-PropZInt

Enter:

• x (count of successes)
• n (sample size)
• C-Level (confidence level as decimal)

Calculate → gives interval

Relationship to Hypothesis Testing

If testing H₀: p = p₀ at significance level α:

Equivalent: Check if (1-α)% CI contains p₀

• If p₀ in CI → fail to reject H₀
• If p₀ not in CI → reject H₀

Quick Reference

Formula: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Conditions: Random, np̂ ≥ 10 and n(1-p̂) ≥ 10, n < 10%N

Common z:* 1.645 (90%), 1.96 (95%), 2.576 (99%)

Sample size: $n = \left(\frac{z^*}{m}\right)^2 p(1-p)$

Remember: Higher confidence → wider interval. Larger sample → narrower interval. Always check conditions and interpret in context!