📚 Estimation, Confidence Intervals and Tests Using a Normal Distribution | 估计、置信区间和使用正态分布的检验
In A-Level Mathematics, statistical inference plays a vital role in making decisions or drawing conclusions about a population based on sample data. This article focuses on three interconnected topics: point estimation, construction of confidence intervals, and hypothesis testing, all underpinned by the normal distribution. These techniques allow us to estimate unknown parameters, quantify uncertainty, and objectively test claims.
在A-Level数学中，统计推断在根据样本数据对总体做出决策或得出结论方面起着至关重要的作用。本文重点介绍三个相互关联的主题：点估计、置信区间的构建以及假设检验，它们都以正态分布为基础。这些技术能让我们估计未知参数、量化不确定性并客观地检验主张。
1. Point Estimation and Unbiased Estimators | 点估计与无偏估计量
A point estimate is a single value calculated from sample data to estimate an unknown population parameter. For example, the sample mean x̄ is a natural point estimator of the population mean μ.
点估计是由样本数据计算得出的单个值，用于估计未知的总体参数。例如，样本均值 x̄ 是总体均值 μ 的自然点估计量。
unbiasedness: the expected value of the estimator equals the true parameter. The sample mean x̄ is an unbiased estimator of μ because E(X̄) = μ.
估计量是用于估计参数的统计量；它在特定样本中的具体取值称为估计值。一个理想的性质是无偏性：估计量的期望值等于真实参数。样本均值 x̄ 是 μ 的无偏估计量，因为 E(X̄) = μ。
For population variance σ², the sample variance s² defined as s² = Σ(xᵢ − x̄)&# he version dividing by n gives a biased estimator. We predominantly use the unbiased form.
对于总体方差 σ²，定义为 s² = Σ(xᵢ − x̄)²/(n−1) 的样本方差是无偏估计量；除以 n 的版本则是有偏的。我们主要使用无偏形式。
2. Distribution of the Sample Mean | 样本均值的分布
Suppose we draw random samples of size n from a population with mean μ and variance σ². The sample mean X̄ has mean μ and variance σ²/n. If the population is normally distributed, X̄ ~ N(μ, σ²/n) exactly.
假设我们从均值为 μ、方差为 σ² 的总体中抽取大小为 n 的随机样本。样本均值 X̄ 的均值为 μ，方差为 σ²/n。如果总体服从正态分布，则 X̄ 精确服从 N(μ, σ²/n)。
If the population is not normal but n is sufficiently large, the Central Limit Theorem tells us that the distribution of X̄ is approximately normal. This is crucial for conducting inference about μ.
如果总体不服从正态分布但 n 足够大，根据中心极限定理，X̄ 的分布近似正态。这对于进行关于 μ 的推断至关重要。
3. Central Limit Theorem | 中心极限定理
The Central Limit Theorem (CLT) states that for independent and identically distributed random variables with finite mean μ and variance σ², the sample mean X̄ tends toward a normal distribution N(μ, σ²/n) as n → ∞, regardless of the population’s shape.
中心极限定理指出，对于独立同分布且具有有限均值 μ 和方差 σ² 的随机变量，当 n → ∞ 时，样本均值 X̄ 趋向于正态分布 N(μ, σ²/n)，无论总体的形状如何。
In practice, a sample size n ≥ 30 is usually considered large enough for the normal approximation to hold. This justifies the use of normal-based confidence intervals and tests even when σ is unknown, by using the sample standard deviation s as an estimate of σ.
在实践中，样本量 n ≥ 30 通常被认为足够大，可进行正态近似。这为即使 σ 未知时使用基于正态的置信区间和检验提供了依据，此时用样本标准差 s 作为 σ 的估计。
4. Confidence Intervals for μ (Known Population Variance) | μ 的置信区间（已知总体方差）
A confidence interval provides a range of plausible values for a population parameter. When the population variance σ² is known, a 100(1−α)% confidence interval for μ is given by:
置信区间为总体参数提供了一个合理的取值范围。当已知总体方差 σ² 时，μ 的 100(1−α)% 置信区间由下式给出：
x̄ − zα/2 × σ/√n ≤ μ ≤ x̄ + zα/2 × σ/√n
Here zα/2 is the critical value from the standard normal distribution such that the area in each tail is α/2. For a 95% confidence level, z0.025 = 1.96.
其中 zα/2 是标准正态分布中的临界值，使每个尾部的面积为 α/2。对于 95% 置信水平，z0.025 = 1.96。
The margin of error is zα/2 × σ/√n. Increasing sample size n reduces the margin of error, yielding a narrower interval.
误差范围是 zα/2 × σ/√n。增加样本量 n 会减小误差范围，使区间更窄。
| Confidence Level | α | zα/2 |
|---|---|---|
| 90% | 0.10 | 1.645 |
| 95% | 0.05 | 1.96 |
| 99% | 0.01 | 2.576 |
The table summarises the critical z-values for commonly used confidence levels.
该表格总结了常用置信水平对应的临界 z 值。
5. Confidence Intervals for μ (Unknown
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