📚 Estimation, Confidence Intervals and Tests Using a Normal Distribution | 正态分布下的估计、置信区间与假设检验
In Edexcel A Level Mathematics, the normal distribution is the foundation for making inferences about a population mean when the population variance is known or when the sample is large. This article covers point estimation, the sampling distribution of the mean, confidence intervals, and hypothesis tests using the standard normal distribution.
在 Edexcel A Level 数学中，正态分布是当总体方差已知或样本较大时对总体均值进行统计推断的基础。本文涵盖点估计、样本均值的抽样分布、置信区间以及使用标准正态分布的假设检验。
1. The Normal Distribution and the Sample Mean | 正态分布与样本均值
If X ~ N(μ, σ²), then for a random sample of size n, the sample mean X̄ has distribution X̄ ~ N(μ, σ²/n). This result is exact when the population is normal.
如果 X ~ N(μ, σ²)，那么对于容量为 n 的随机样本，样本均值 X̄ 服从分布 X̄ ~ N(μ, σ²/n)。当总体为正态分布时，该结论是精确的。
X̄ ~ N(μ, σ²/n)
For large n, usually n ≥ 30, the Central Limit Theorem says X̄ is not normal. This is why the normal distribution is so useful in estimation and testing.
对于大样本，通常 n ≥ 30，中心极限定理表明即使总体不服从正态分布，X̄ 也近似服从 N(μ, σ²/n)。这就是正态分布在估计和检验中非常实用的原因。
The standard deviation of the sample mean, σ/√n, measures how much sample means vary around the population mean and is called the standard error.
样本均值的标准差 σ/√n 衡量样本均值围绕总体均值的波动程度，称为标准误。
2. Point Estimation and Unbiased Estimators | 点估计与无偏估计量
The sample mean x̄ mean μ. It is an unbiased estimator because its expected value is equal to μ, that is E(X̄) = μ.
样本均值 x̄ 是总体均值 μ 的点估计。它是一个无偏估计量，因为其期望值等于 μ，即 E(X̄) = μ。
pulation variance σ² is unknown, the unbiased estimate of σ² from a sample is s² = Σ(x − x̄)² / (n − 1). Many exam questions use this when only sample data are given.
当总体方差 σ² 未知时，由样本得到的 σ² 的无偏估计为 s² = Σ(x − x̄)² / (n − 1)。当题目只给出样本数据时，常使用该公式。
s² = Σ(x − x̄)² / (n − 1)
A point estimate gives a single best guess for the parameter, but it does not show the precision of that guess. Confidence intervals are used to show the uncertainty around a point estimate.
点估计给出参数的一个最佳猜测值，但它不能体现猜测的精确度。置信区间用于表示点估计周围的不确定性。
3. Standard Error of the Mean | 样本均值的标准误
The standard error of the mean is σ/√n. If σ is unknown but the sample size n is large, use s/√n as an estimate of the standard error.
样本均值的标准误是 σ/√n。如果 σ 未知但样本容量 n 较大，可以用 s/√n 作为标准误的估计。
SE(X̄) = σ / √n
A smaller standard error means the sample mean is a more precise estimator of μ. Increasing the sample size n decreases the standard error, so larger samples give more reliable estimates.
标准误越小，样本均值对 μ 的估计越精确。增大样本容量 n 可以减小标准误，因此较大的样本能给出更可靠的估计。
For example, if the sample size is multiplied by 4, the standard error is divided by 2 because SE is proportional to 1/√n.
例如，如果样本容量变为原来的 4 倍，标准误会减半，因为标准误与 1/√n 成正比。
4. Confidence Intervals for the Mean | 均值的置信区间
l gives a range of plausible values for the population mean μ. For a known σ, a 100(1 − α)% confidence interval is x̄ ± z* × (σ/√n).
置信区间给出总体均值 μ 的合理取值范围。当 σ 已知时，100(1 − α)% 的置信区间为 x̄ ± z* × (σ/√n)。
CI = x̄ ± z* × (σ / √n)
Common z* values are shown below. They come from the standard normal distribution and depend on the confidence level.
常用 z* 值如下表所示。它们来自标准正态分布并取决于置信水平。
| Confidence level | z* |
|---|---|
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