Confidence Intervals for Proportions

📊 Inference for Proportions

Part 1 of 7 — Inference for Proportions Basics

The Setting

We have a sample proportion $\\hat{p}$ and want to make inferences about the population proportion $p$.

Conditions for Inference

1. Random: Data from a random sample or experiment
2. Normal: $np \\geq 10$ and $n(1-p) \\geq 10$ (use $\\hat{p}$ for CIs)
3. Independent: Sample $< 10\\%$ of population (10% condition)

Standard Error

$SE(\\hat{p}) = \\sqrt{\\frac{\\hat{p}(1-\\hat{p})}{n}}$

Key Distinction

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Conditions Check 🧮

In a random sample of 200 voters, 120 support a candidate. $\\hat{p} = 0.60$.

1) $n\\hat{p} = ?$

2) $n(1-\\hat{p}) = ?$

3) Is the Normal condition met? (yes/no)

📏 Confidence Intervals for Proportions

Part 2 of 7 — One-Sample Z Interval

Formula

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

Common Critical Values

Interpretation

“We are [C]% confident that the true proportion of [context] is between [lower] and [upper].”

Example

$n = 400$, $\\hat{p} = 0.35$, 95% CI:

$0.35 \\pm 1.96\\sqrt{\\frac{0.35 \\times 0.65}{400}} = 0.35 \\pm 0.0467$

CI: $(0.303, 0.397)$

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Confidence Interval 🧮

$n = 500$, $\\hat{p} = 0.40$, 95% CI.

1) $SE = \\sqrt{0.40 \\times 0.60 / 500}$ = ? (round to 4 decimal places)

2) Margin of error = $1.96 \\times SE$ = ? (round to 4 places)

3) Lower bound of CI? (round to 3 places)

⚖️ Hypothesis Tests for Proportions

Part 3 of 7 — One-Sample Z Test

Steps

1. State hypotheses: $H_0: p = p_0$ vs. $H_a: p \\neq p_0$ (or $<$ or $>$)
2. Check conditions (Random, Normal, Independent)
3. Calculate the test statistic:

$z = \\frac{\\hat{p} - p_0}{\\sqrt{p_0(1-p_0)/n}}$

1. Find the p-value
2. Conclude in context

P-Value Decision Rules

Example

Claim: $p = 0.5$. Sample: $\\hat{p} = 0.56$, $n = 200$.

$z = \\frac{0.56 - 0.50}{\\sqrt{0.50 \\times 0.50 / 200}} = \\frac{0.06}{0.0354} = 1.70$

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Hypothesis Test 🧮

$H_0: p = 0.30$, $H_a: p > 0.30$. $n = 150$, $\\hat{p} = 0.36$.

1) $SE = \\sqrt{0.30 \\times 0.70 / 150}$ = ? (round to 4 places)

2) $z = (0.36 - 0.30) / SE$ = ? (round to 2 places)

3) Is this a one-tailed or two-tailed test?

📊 Two-Proportion Inference

Part 4 of 7 — Comparing Two Proportions

Confidence Interval for $p_1 - p_2$

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

Hypothesis Test for $p_1 - p_2$

$H_0: p_1 = p_2$ (or $p_1 - p_2 = 0$)

Use the pooled proportion: $\\hat{p}_c = \\frac{x_1 + x_2}{n_1 + n_2}$

$z = \\frac{(\\hat{p}_1 - \\hat{p}_2) - 0}{\\sqrt{\\hat{p}_c(1-\\hat{p}_c)\\left(\\frac{1}{n_1} + \\frac{1}{n_2}\\right)}}$

Key Difference

• CI: Use individual $\\hat{p}_1$ and $\\hat{p}_2$ in the SE
• Test: Use the pooled $\\hat{p}_c$ (assuming $H_0: p_1 = p_2$ is true)

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Two-Proportion Test 🧮

Group 1: $x_1 = 45$, $n_1 = 100$. Group 2: $x_2 = 30$, $n_2 = 100$.

1) $\\hat{p}_1 = ?$

2) Pooled $\\hat{p}_c = (45 + 30)/(100 + 100) = ?$

3) $\\hat{p}_1 - \\hat{p}_2 = ?$

📐 Sample Size Determination

Part 5 of 7 — Planning a Study

Finding the Required Sample Size

For a desired margin of error $ME$ at confidence level $z^*$:

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

If no prior estimate of $p$ exists, use $\\hat{p} = 0.5$ (maximizes $n$, conservative).

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

Example

Want a 95% CI with margin of error $\\leq 0.03$:

$n = \\left(\\frac{1.96}{0.03}\\right)^2 (0.25) = (65.33)^2(0.25) = 4268.4(0.25) = 1067.1$

Round up: $n = 1068$

🔑 Always round UP to the next whole number when computing sample size.

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Sample Size Calculation 🧮

Desired: 95% CI, margin of error $\\leq 0.04$, no prior estimate of $p$.

1) What value of $\\hat{p}$ should you use?

2) $n = (1.96/0.04)^2 \\times 0.25 = ?$ (round to nearest integer)

3) What $n$ do you report? (remember rounding rule)

🏆 Problem-Solving Workshop

Part 6 of 7 — AP-Style Practice

AP FRQ Template for Inference

1. State: Name the procedure and define parameters
2. Plan: Check conditions (Random, Normal, Independent)
3. Do: Show calculations
4. Conclude: Interpret in context

Common Mistakes to Avoid

• Using $\\hat{p}$ in the test statistic SE (should use $p_0$)
• Using $p_0$ in the CI SE (should use $\\hat{p}$)
• Saying “accept $H_0$” instead of “fail to reject $H_0$”
• Forgetting to check conditions
• Not interpreting in context

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AP Practice 🧮

A poll finds 52% of 1000 voters favor a candidate. Test $H_0: p = 0.50$ vs. $H_a: p > 0.50$ at $\\alpha = 0.05$.

1) $z = (0.52 - 0.50)/\\sqrt{0.25/1000}$ = ? (round to 2 places)

2) p-value $\\approx P(Z > z) \\approx ?$ (round to 3 places)

3) Decision at $\\alpha = 0.05$? (reject/fail to reject)

📝 Mixed Review

Part 7 of 7 — Comprehensive Review

Quick Reference

AP Exam Tips

• Always state your hypotheses using the parameter $p$, not $\\hat{p}$
• Check all three conditions: Random, Normal, Independent
• Give a conclusion IN CONTEXT

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Final Challenge 🧮

$n = 800$, $\\hat{p} = 0.65$, 95% CI.

1) Margin of error $= 1.96 \\times \\sqrt{0.65 \\times 0.35/800} \\approx ?$ (round to 3 places)

2) Lower bound of CI?

3) Upper bound of CI?