where:

• $\hat{p}$ = sample proportion (point estimate)
• $z^*$ = critical z-value (depends on confidence level)
• $SE(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Common critical values:

• 90% CI: $z^* = 1.645$
• 95% CI: $z^* = 1.96$
• 99% CI: $z^* = 2.576$

Conditions for Validity

Before constructing CI for proportions, verify:

1. Random sample: Data collected randomly
2. Independence: Sampling without replacement; use 10% rule (n ≤ 0.10N)
3. Success/failure rule: Both $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$

If these fail, don't use the standard formula.

Worked Example

A poll of 400 likely voters finds 220 support Candidate A. Construct a 95% CI for the population proportion.

Step 1: $\hat{p} = \frac{220}{400} = 0.55$

Step 2: Check conditions:

• Random sample ✓
• n = 400 ≤ 0.10(population) ✓ (assume population is large)
• $n\hat{p} = 400(0.55) = 220 \geq 10$ ✓
• $n(1-\hat{p}) = 400(0.45) = 180 \geq 10$ ✓

Step 3: Calculate SE: $SE = \sqrt{\frac{0.55 \cdot 0.45}{400}} = \sqrt{\frac{0.2475}{400}} = \sqrt{0.00061875} \approx 0.0249$

Step 4: Calculate margin of error (ME): $ME = 1.96 \times 0.0249 \approx 0.0488$

Step 5: Confidence interval: $0.55 \pm 0.0488 = (0.5012, 0.5988)$

Interpretation: We are 95% confident that between 50.1% and 59.9% of likely voters support Candidate A.

Margin of Error

The margin of error (ME) represents the uncertainty in the point estimate:

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

Larger ME means wider CI (less precise). Smaller ME means narrower CI (more precise).

Sample Size for Desired Margin of Error

To find the sample size needed for margin ME:

$n = \frac{(z^*)^2 \cdot p(1-p)}{ME^2}$

If p is unknown, use p = 0.5 (gives maximum sample size; most conservative).

Example: You want 95% CI with ME = 0.03. How large must the sample be?

$n = \frac{(1.96)^2 \cdot 0.5(0.5)}{(0.03)^2} = \frac{3.8416 \cdot 0.25}{0.0009} = \frac{0.9604}{0.0009} \approx 1068$

Common Mistakes

1. Wrong formula: Using $z^*$ instead of $t^*$ (t-values are for means, not proportions)
2. Forgetting to check conditions: Missing the success/failure rule or 10% rule
3. Using p instead of $\hat{p}$: The SE formula uses the sample proportion, not the population proportion

AP Exam Tip

Know the margin of error formula cold. Many free-response questions ask you to find the sample size needed to achieve a specific margin. Always verify conditions before calculating; if conditions fail, state which ones and explain why the interval is not valid.

Or: (0.50, 0.60) rounded

Step 7: Interpret the interval We are 95% confident that the true proportion of voters who support the proposition is between 0.50 and 0.60 (or 50% and 60%).

This means:

• If we repeated this sampling process many times
• About 95% of intervals would contain true p
• This specific interval either contains p or doesn't
• But the process is reliable 95% of the time

Answer: 95% CI for p: (0.50, 0.60)

We are 95% confident that between 50% and 60% of all voters support the proposition.