Confidence Intervals — Complete Mastery Workbook

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What Is a Confidence Interval?

Core Definition

A confidence interval is a range of plausible values for an unknown population parameter, constructed from sample data. A C% confidence interval means: if we repeated this procedure many times, C% of the resulting intervals would capture the true parameter.

General Structure

$$\text{CI} = \hat{\theta} \pm z^* \cdot \text{SE}(\hat{\theta})$$

Estimate ± (Critical value × Standard Error)

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The 4 Key Formulas to Memorize

① One-Sample z-Interval for Mean (σ known)

$$\bar{x} \pm z^* \cdot \frac{\sigma}{\sqrt{n}}$$

② One-Sample t-Interval for Mean (σ unknown)

$$\bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}}, \quad df = n-1$$

③ One-Sample z-Interval for Proportion

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

④ Margin of Error (ME)

$$ME = z^* \cdot \frac{\sigma}{\sqrt{n}} \Rightarrow n = \left(\frac{z^* \cdot \sigma}{ME}\right)^2$$

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Critical Values — Memorize These!

Confidence Level	Two-tail α	z* (or t*→∞)
90%	0.10	1.645
95%	0.05	1.960
98%	0.02	2.326
99%	0.01	2.576

★ Must-Know Memory Points

Conditions for z-interval: SRS, known σ, Normal population or n ≥ 30

Conditions for t-interval: SRS, unknown σ, Normal pop. or n ≥ 30 (CLT)

Conditions for p-interval: SRS, $n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$, pop ≥ 10n

Wider CI ← higher confidence level OR smaller n OR larger σ/s

To halve ME, you must quadruple sample size n

df for one-sample t: df = n − 1

"Confident" ≠ "probability" — the parameter is fixed, not random!

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Correct vs. Incorrect Interpretation

❌ WRONG — Never say this

"There is a 95% probability that the true mean lies between 48.2 and 53.8."

✅ CORRECT — Say this

"We are 95% confident that the true population mean lies between 48.2 and 53.8. If this procedure were repeated many times, 95% of the resulting intervals would capture the true mean."

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Worked Example 1 — z-Interval for Mean

z-Interval

A quality control engineer measures the fill volume of 64 randomly selected bottles. The population standard deviation is known to be $\sigma = 2.4$ mL. The sample mean is $\bar{x} = 500.3$ mL. Construct a 95% confidence interval for the true mean fill volume.

Step 1 — Identify: One-sample z-interval (σ known), $\bar{x}=500.3$, $\sigma=2.4$, $n=64$, $C=95\%$

Step 2 — Conditions: (1) SRS ✓ (2) σ known ✓ (3) n=64 ≥ 30, CLT applies ✓

Step 3 — Critical value: $z^* = 1.960$ for 95%

Step 4 — Standard Error: $$SE = \frac{\sigma}{\sqrt{n}} = \frac{2.4}{\sqrt{64}} = \frac{2.4}{8} = 0.30$$

Step 5 — Margin of Error: $$ME = z^* \cdot SE = 1.960 \times 0.30 = 0.588$$

Step 6 — Interval: $$500.3 \pm 0.588 \Rightarrow (499.712,\ 500.888)$$

95% CI: (499.71 mL, 500.89 mL)

Interpretation: We are 95% confident the true mean fill volume is between 499.71 mL and 500.89 mL.

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Worked Example 2 — Proportion Interval

p-Interval

In a survey of 400 registered voters, 220 said they support a new education policy. Construct a 90% confidence interval for the true proportion of all voters who support the policy.

Step 1 — Identify: One-sample z-interval for proportion, $n=400$, $x=220$

Step 2 — Sample proportion: $$\hat{p} = \frac{220}{400} = 0.55$$

Step 3 — Conditions: $n\hat{p}=220 \ge 10$ ✓, $n(1-\hat{p})=180 \ge 10$ ✓, SRS ✓

Step 4 — Critical value: $z^* = 1.645$ for 90%

Step 5 — SE: $$SE = \sqrt{\frac{0.55 \times 0.45}{400}} = \sqrt{\frac{0.2475}{400}} = \sqrt{0.00061875} \approx 0.02488$$

Step 6 — ME: $ME = 1.645 \times 0.02488 \approx 0.0409$

Step 7 — Interval: $0.55 \pm 0.041 \Rightarrow (0.509,\ 0.591)$

90% CI for p: (0.509, 0.591) or (50.9%, 59.1%)

Core Definition

Solutions & Explanations