🎯⭐ INTERACTIVE LESSON

Confidence Intervals for Proportions

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Confidence Intervals for Proportions - Complete Interactive Lesson

Part 1: Inference for Proportions Basics

📊 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

Random: Data from a random sample or experiment
Normal: $np \\geq 10$ and $n(1-p) \\geq 10$ (use $\\hat{p}$ for CIs)

Standard Error

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

Key Distinction

Purpose	Formula for SD
Confidence interval	$SE = \\sqrt{\\hat{p}(1-\\hat{p})/n}$
Hypothesis test	(use value)

Concept Check U0001f3af

Conditions Check 🧮

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

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

2) $n (1 - h a t p) = ?$

Part 2: Confidence Intervals for Proportions

📏 Confidence Intervals for Proportions

Part 2 of 7 — One-Sample Z Interval

Formula

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

Part 3: Hypothesis Tests for Proportions

⚖️ Hypothesis Tests for Proportions

Part 3 of 7 — One-Sample Z Test

Steps

State hypotheses: $H_0: p = p_0$ vs. (or or )

Part 4: Two-Proportion Inference

📊 Two-Proportion Inference

Part 4 of 7 — Comparing Two Proportions

Confidence Interval for $p_1 - p_2$

$(h a t p_{1} - h a t p_{2})$

Part 5: Sample Size Determination

📐 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 $z^*$ :

$n = l e f t (f r a c z^{*} M E r i g h t)^{2} h a t p (1 - h a$

Part 6: Problem-Solving Workshop

🏆 Problem-Solving Workshop

Part 6 of 7 — AP-Style Practice

AP FRQ Template for Inference

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

Common Mistakes to Avoid

Using $\\hat{p}$ in the test statistic SE (should use $p_0$ )
Using in the CI SE (should use )

Part 7: Mixed Review

📝 Mixed Review

Part 7 of 7 — Comprehensive Review

Quick Reference

Procedure	SE Formula	When to Use
1-prop CI	$\\sqrt{\\hat{p}(1-\\hat{p})/n}$	Estimating

Independent: Sample

< 10\\%

of population (10% condition)

SE = \\sqrt{p_0(1-p_0)/n}

SE = s q r t p_{0} (1 - p_{0}) / n

n (1 - ha t p) = ?

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

Common Critical Values

Confidence Level	$z^*$
90%	1.645
95%	1.960
99%	2.576

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

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

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

CI: $(0.303, 0.397)$

Concept Check U0001f3af

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)

H_{a} : p n e q p_{0}

Check conditions (Random, Normal, Independent)

Calculate the test statistic:

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

Find the p-value
Conclude in context

P-Value Decision Rules

If p-value	Decision
$\\leq \\alpha$	Reject $H_0$
$> \\alpha$	Fail to reject $H_0$

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

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

Concept Check U0001f3af

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?

(hatp1​−hatp2​)pmz∗sqrtfrachatp1​(1−hatp1​)n1​+fra

Hypothesis Test for $p_1 - p_2$

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

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

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

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)

Concept Check U0001f3af

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

n=left(fracz∗MEright)2hatp(1−hatp)

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

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

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.

Concept Check U0001f3af

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)

Saying “accept

H_0

” instead of “fail to reject

H_0

”

Forgetting to check conditions

Not interpreting in context

Concept Check U0001f3af

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)

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

Concept Check U0001f3af

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?

Confidence Intervals for Proportions

Common Critical Values

Interpretation

Example

P-Value Decision Rules

Example

Hypothesis Test for $p_1 - p_2$

Key Difference

Example

AP Exam Tips

Confidence Intervals for Proportions

Confidence Intervals for Proportions - Complete Interactive Lesson

Part 1: Inference for Proportions Basics

📊 Inference for Proportions

The Setting

Conditions for Inference

Standard Error

Key Distinction

Part 2: Confidence Intervals for Proportions

📏 Confidence Intervals for Proportions

Formula

Part 3: Hypothesis Tests for Proportions

⚖️ Hypothesis Tests for Proportions

Steps

Part 4: Two-Proportion Inference

📊 Two-Proportion Inference

Confidence Interval for p1−p2p_1 - p_2p1​−p2​

Part 5: Sample Size Determination

📐 Sample Size Determination

Finding the Required Sample Size

Part 6: Problem-Solving Workshop

🏆 Problem-Solving Workshop

AP FRQ Template for Inference

Common Mistakes to Avoid

Part 7: Mixed Review

📝 Mixed Review

Quick Reference

Common Critical Values

Interpretation

Example

P-Value Decision Rules

Example

Hypothesis Test for p1−p2p_1 - p_2p1​−p2​

Key Difference

Example

AP Exam Tips

Confidence Interval for $p_1 - p_2$

Hypothesis Test for $p_1 - p_2$