Grades Pre-Calculus & Calculus Advanced StatisticsConfidence Intervals & Data Collection

Confidence Intervals & Data Collection

📊 Advanced Statistics

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Confidence Intervals & Data Collection

Construct confidence intervals for population parameters and evaluate study design.

A confidence interval (CI) provides a range of plausible values for an unknown population parameter. A 95% CI means that if we repeated the study many times, 95% of the resulting intervals would contain the true parameter.

\text{CI for } \mu: \quad \bar{x} \pm z^* \cdot \frac{\sigma}{\sqrt{n}} \quad \text{(z-interval, } \sigma \text{ known)}

\text{CI for } \mu: \quad \bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}} \quad \text{(t-interval, } \sigma \text{ unknown)}

Common Critical Values (z*)

90% CI: z* = 1.64595% CI: z* = 1.96099% CI: z* = 2.576

Higher confidence → wider interval. Larger n → narrower interval (more precision).

\text{Margin of Error } = z^* \cdot \frac{\sigma}{\sqrt{n}} \implies n = \left(\frac{z^* \cdot \sigma}{E}\right)^2

📊Constructing a 95% CI

Sample: n = 36, x̄ = 52, σ = 12. 95% CI: 52 ± 1.96·(12/6) = 52 ± 3.92 = (48.08, 55.92). We are 95% confident the true mean is between 48.08 and 55.92.

Good Study Design

Random sampling from populationControl and treatment groupsBlinding to reduce biasLarge enough sample size

Sources of Bias

Convenience sampling (non-random)Voluntary response biasUndercoverage of groupsConfounding variables

Statistical analysis is only as valid as the data collection process — "garbage in, garbage out."

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Remember This!

To halve the margin of error, quadruple the sample size (since n appears under a square root: E ∝ 1/√n). Reducing the confidence level also narrows the interval but at the cost of certainty.

✏️ Try It!

A 95% confidence interval for a population mean is (42, 58). What is the sample mean x̄?

Practice with CalcVerse

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