Point and Interval Estimation
AdvancedInference moves from sample to population: the sampling distribution of the sample mean is centred at μ with standard error σ/√n and is approximately normal for large n; estimate μ by x̄, then build the 95% confidence interval x̄ ± 1.96 standard errors, with the width controlled by the sample size.
- Inference moves from sample to population: x̄ estimates μ and s² estimates σ²; statistics are random, parameters fixed.
- Sampling distribution of x̄: E(x̄) = μ, Var(x̄) = σ²/n, standard error σ/√n; approximately normal for large n by the central limit theorem.
- Sample variance s² = Σ(xᵢ − x̄)²/(n−1); the divisor n−1 guarantees unbiasedness.
文體小知識：數學概念說明文——由總體與樣本講起，說明樣本平均值的抽樣分佈、標準誤、中心極限定理、點估計與無偏性，再構造和解讀總體平均值的置信區間，是統計推斷的核心課。
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1The core problem of statistical inference: the population is too large to measure exhaustively, so we sample and infer from the sample. Constants describing the population, such as the mean μ and variance σ², are population parameters — fixed but usually unknown; quantities computed from the sample, such as the sample mean x̄ and sample variance s², are sample statistics and vary from sample to sample. To find the mean height of all S5 students in Hong Kong, we cannot measure everyone — we take a random sample of one hundred.
2The sample mean is itself a random variable with its own distribution, the sampling distribution. The curriculum states: if the population mean is μ and variance σ², then E(x̄) = μ and Var(x̄) = σ²/n; the standard deviation of x̄, called the standard error, is σ divided by the square root of n. The larger the sample, the more tightly x̄ clusters around μ. If the population is itself normal, then regardless of n, x̄ follows N(μ, σ²/n).
3What if the population is not normal? The central limit theorem answers: for n large enough, the distribution of x̄ is approximately normal whatever the population distribution. For example, one fair die gives a uniform distribution with expectation 3.5; but throw sixty dice at once and average them, and the distribution of the average is already bell-shaped about 3.5. This theorem is the engine of inference, making normal methods valid for almost all large samples.
4Point estimation uses a single number to estimate a parameter: x̄ estimates μ and s² estimates σ². One criterion for a good estimator is unbiasedness: its expectation equals the parameter it estimates. Since E(x̄) = μ, x̄ is an unbiased estimator of μ; the sample variance is defined as the sum of squared deviations divided by n minus one — the denominator n−1 is chosen exactly so that the expectation of s² equals σ². Dividing by n instead would make s² systematically underestimate the population variance.
5A point estimate says nothing about error; interval estimation fills the gap: build an interval centred at x̄ and declare a confidence level. The correct reading of a 95% confidence interval: repeat the sampling and the interval construction many times, and in the long run about 95% of the intervals cover μ; μ is a fixed constant, and it is the interval that is random. Sampling from a normal population with known variance, the 95% confidence interval for μ is x̄ plus or minus 1.96 standard errors; when n is large and σ unknown, the curriculum allows replacing σ by the sample standard deviation s.
1Summary 重點摘要
Inference moves from sample to population: the sampling distribution of the sample mean is centred at μ with standard error σ/√n and is approximately normal for large n; estimate μ by x̄, then build the 95% confidence interval x̄ ± 1.96 standard errors, with the width controlled by the sample size.
2Vocabulary cards Vocabulary
A fixed constant describing the population, such as the mean μ or variance σ²; usually unknown and estimated from samples.
A quantity computed from a sample, such as x̄ or s²; it varies between samples and estimates a parameter.
The probability distribution of a statistic over all possible samples, e.g. the distribution of x̄.
The standard deviation of x̄, equal to σ/√n; it measures the precision of estimating μ by x̄.
For n large enough, the distribution of x̄ is approximately normal regardless of the population distribution.
Estimating a parameter by a single value: x̄ for μ and s² for σ².
An estimator whose expectation equals the parameter it estimates; E(x̄) = μ, and s² with divisor n−1 has expectation σ².
s² = Σ(xᵢ − x̄)²/(n−1); the divisor n−1 guarantees unbiasedness.
An interval estimate of a parameter centred on a statistic with a stated confidence level, e.g. x̄ ± 1.96σ/√n.
The long-run success rate of the interval procedure; 95% corresponds to the critical value 1.96 and 99% to 2.575.
3Step by step Steps
- Step 1: Population and sampleThe core problem of inference: the population is too large to measure exhaustively, so we sample and infer parameters from statistics. The population mean μ and variance σ² are fixed unknown constants; the sample mean x̄ and sample variance s² vary from sample to sample and are random variables. For example, with about 80,000 S5 students in Hong Kong, the mean height μ is unknown; a random sample of 100 gives x̄ = 168 centimetres, a figure that differs between samples yet always fluctuates around μ. Distinguishing parameter from statistic is the starting point of the whole lesson.
- Step 2: The sampling distribution of the sample meanIf the population mean is μ and variance σ², then the sample mean satisfies E(x̄) = μ and Var(x̄) = σ²/n; the standard deviation of x̄, called the standard error, is σ/√n. The larger the sample, the more tightly x̄ clusters around μ. For example, with σ = 12 and n = 36 the standard error is 12÷6 = 2; enlarge the sample to n = 144 and the standard error halves to 1, doubling the precision. The standard error is inversely proportional to the square root of n — the mathematical statement of 'bigger samples are better' and the basis for planning sample sizes.
- Step 3: The shape of the distribution and the central limit theoremIf the population is normal, X ~ N(μ, σ²), then regardless of n, x̄ ~ N(μ, σ²/n). If the population is not normal, the central limit theorem says: provided n is large enough, the distribution of x̄ is still approximately normal. For example, one fair die gives a uniform distribution with expectation 3.5; but throw 60 dice at once and average, and the distribution of the average is already bell-shaped about 3.5. This theorem makes normal methods valid for almost all large samples — the engine of statistical inference.
- Step 4: Point estimation and unbiasednessPoint estimation uses a single value to estimate a parameter: x̄ estimates μ and s² estimates σ². One criterion of a good estimator is unbiasedness: its expectation equals the parameter it estimates. E(x̄) = μ, so x̄ is unbiased for μ; the sample variance is defined as s² = Σ(xᵢ − x̄)²/(n−1), the denominator deliberately n−1 so that E(s²) = σ². For example, five cans measured at 249, 251, 250, 248, 252 millilitres give x̄ = 250; dividing by n instead of n−1 would make the variance systematically too small, underestimating the spread in the long run.
- Step 5: The idea of a confidence intervalA point estimate says nothing about error; interval estimation adds it: build an interval centred at x̄ and declare a confidence level. The correct reading of a 95% confidence interval: repeat the sampling and construction many times, and in the long run about 95% of the intervals cover μ; μ is a fixed constant, and it is the interval that is random. For example, a poll reporting '52% support, margin of error plus or minus three percentage points (95% confidence)' means the interval 49% to 55% covers the true support with long-run success rate 95%.
- Step 6: Constructing the interval (known σ)Sampling from a normal population with known variance σ², a 95% confidence interval for μ is x̄ ± 1.96 × σ/√n; in general a 100(1−α)% interval uses the corresponding critical value (99% uses 2.575). Three steps: compute x̄, compute the standard error σ/√n, multiply by the critical value for the margin of error. For example, n = 36, x̄ = 52, σ = 6: standard error = 1, and the 95% interval is 52 ± 1.96, i.e. (50.04, 53.96); a 99% interval would be 52 ± 2.575 — wider, in exchange for higher confidence.
- Step 7: Unknown σ and the role of nThe population variance is usually unknown; the curriculum states that for n large enough the sample standard deviation s may replace σ, giving x̄ ± 1.96 × s/√n. The width is inversely proportional to the square root of n: halving the error requires quadrupling the sample. For example, a factory wants the 95% margin of error for mean volume within 1 millilitre; with s = 4, we need n ≥ (1.96×4)² = 61.5, so take n = 64. This is the standard calculation for planning surveys — and explains why large studies interview thousands.
5Worked example Example
Example: a soft-drink plant randomly samples 36 cans, with sample mean volume 250.4 millilitres. Given the population standard deviation is 6 millilitres, find the 95% confidence interval for the population mean volume.
- Given n = 36, x̄ = 250.4 and σ = 6; the 95% level corresponds to the critical value 1.96
- Standard error = σ/√n = 6/√36 = 1
- Margin of error = 1.96 × 1 = 1.96
- Confidence interval = 250.4 ± 1.96, i.e. (248.44, 252.36)
- Answer: The 95% confidence interval for the population mean volume is (248.44, 252.36) millilitres. Solution format: state the formula x̄ ± 1.96σ/√n, compute the standard error, substitute for the limits, and give units; forgetting to divide by √n is the commonest error.
6Common mistakes Mistakes
7Quick check Quick check
Q1Of μ and x̄, which is a parameter and which is a statistic?
Q2With σ = 12 and n = 36, find the standard deviation of x̄.
Q3If the sample size quadruples, what happens to the standard error?
Q4Why does the sample variance divide by n−1 rather than n?
Q5With x̄ = 100, σ = 15 and n = 25, find the 95% confidence interval for μ.
8Recap Recap
- Inference moves from sample to population: x̄ estimates μ and s² estimates σ²; statistics are random, parameters fixed.
- Sampling distribution of x̄: E(x̄) = μ, Var(x̄) = σ²/n, standard error σ/√n; approximately normal for large n by the central limit theorem.
- Sample variance s² = Σ(xᵢ − x̄)²/(n−1); the divisor n−1 guarantees unbiasedness.
- With known σ, the 95% interval is x̄ ± 1.96σ/√n; for large n with unknown σ, replace σ by s.
- Interval width is inversely proportional to √n: halving the error quadruples the sample; confidence and precision trade off.