Statistical Estimation and Confidence Intervals

What Is Statistical Estimation?

Estimation: The process of using sample data to infer population parameters (μ, σ, p).

Point Estimation: Estimating a parameter with a single number (e.g., using sample mean x̄ to estimate population mean μ)
Interval Estimation: Estimating a range likely to contain the parameter (e.g., 95% confidence interval)

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Properties of Good Estimators

Property	Meaning
Unbiasedness	E(estimator) = parameter (no systematic bias)
Efficiency	Minimum variance among all unbiased estimators for the same n
Consistency	Converges to the true parameter as n → ∞
Sufficiency	Uses all available information in the sample

Why sample variance uses n−1: s² = Σ(xᵢ−x̄)²/(n−1) — dividing by n−1 instead of n produces an unbiased estimator of the population variance (Bessel’s correction).

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Confidence Interval

Confidence Interval: A range expected to contain the true parameter

Incorrect interpretation: “There is a 95% probability that the population mean lies in this interval”
Correct interpretation: “If we repeated this procedure 100 times, approximately 95 of those intervals would contain the true population mean”

Confidence Interval for the Population Mean

When σ (population SD) is known (Z distribution):

$$\begin{aligned} CI &= \bar{x} \pm Z_{\alpha/2} \times (\frac{\sigma}{\sqrt{n}}) \\ 95\% CI: Z_{0.025} &= 1.96 \\ 99\% CI: Z_{0.005} &= 2.576 \end{aligned}$$

When σ is unknown (t distribution, small samples):

$$CI = \bar{x} \pm t(\alpha /2, n-1) \times (\frac{s}{\sqrt{n}})$$

Confidence Interval for a Population Proportion

$$\hat{p} \pm Z_{\alpha/2} \times \sqrt{\hat{p}(1-\hat{p})/n}$$

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Confidence Level vs Interval Width

Confidence Level	Z value	Interval Width
90%	1.645	Narrower
95%	1.960	Moderate
99%	2.576	Wider

Higher confidence level: Wider interval (more accurate, less precise)
Larger sample size: Narrower interval (more precise)

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Determining Sample Size

Minimum sample size to achieve a desired margin of error (E):

$$\begin{aligned} &n \geq (Z_{\alpha/2} \times \sigma / E)^2 \\ &For proportion estimation: \\ &n \geq (Z_{\alpha/2})^2 \times \hat{p}(1-\hat{p}) / E^2 \quad \text{(When p̂ is unknown, use p̂ = 0.5 → maximum sample size)} \end{aligned}$$

Example: 95% confidence, σ = 10, margin of error E = 2
n ≥ (1.96 × 10 / 2)² = 9.8² = 96.04 → n = 97

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Key Concept Cards

Correct Interpretation of a 95% CI ★★★★★ : If we repeated this sampling procedure 100 times, approximately 95 of the resulting intervals would contain the true population parameter. It does NOT mean there is a 95% probability the parameter is in this specific interval. Memory tip: CI = long-run frequency interpretation

Sample Variance Denominator n−1 ★★★★☆ : Dividing by n underestimates the population variance (bias). Dividing by n−1 produces an unbiased estimator. Related to degrees of freedom. Memory tip: sample variance = divide by n−1 (unbiasedness)

Higher Confidence Level → Wider Interval ★★★★☆ : A 99% CI is wider than a 95% CI. To be more certain of capturing the parameter, you need a wider net. Memory tip: confidence level ↑ → interval ↑ → precision ↓

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Practice Questions

Q. With n = 100, x̄ = 50, σ = 20, what is the 95% confidence interval?

SE = 20/√100 = 2. 95% CI = 50 ± 1.96×2 = [46.08, 53.92].

Q. To cut the margin of error in half at 95% confidence, how must the sample size change?

Since E ∝ 1/√n, halving the margin of error requires multiplying n by 4.