📚 Confidence Intervals: From Sample to Population Mean | 置信区间：从样本到总体均值
In A-Level Statistics, a confidence interval gives a range of plausible values for an unknown population parameter, usually the population mean μ. This article explains how to construct, interpret and apply confidence intervals for the mean when the population standard deviation is known, or when the sample is large enough for the Central Limit Theorem to be used.
在 A-Level 统计学中，置信区间给出了未知总体参数（通常是总体均值 μ）的合理取值范围。本文将讲解当总体标准差已知，或样本量足够大、可以使用中心极限定理时，如何构造、解释并应用均值的置信区间。
1. What Is a Confidence Interval? | 什么是置信区间？
A confidence interval is an interval estimate calculated from sample data. It is centred on a point estimate, typically the sample mean x̄, and its width reflects sampling variability. Instead of giving one single guess for μ, we give a range of values that are consistent with the observed data.
置信区间是根据样本数据计算的区间估计。它以点估计（通常是样本均值 x̄）为中心，其宽度反映了抽样变异性。我们不是给出 μ 的单一猜测值，而是给出与观测数据一致的一个取值范围。
For example, a 95% confidence interval for a population mean μ has the form x̄ ± margin of error. If we repeated the sampling process many times under the same conditions, about 95% of the intervals constructed in this way would contain the true value of μ.
例如，总体均值 μ 的 95% 置信区间形式为 x̄ ± 误差界限。如果我们在相同条件下多次重复抽样过程，按此方法构造的区间中约有 95% 会包含 μ 的真实值。
This range is not a guarantee for one particular interval, but it is a statement about the reliability of the method over many repetitions.
这个范围并不是对某一个具体区间的保证，而是关于该方法在多次重复中可靠性的陈述。
2. Point Estimate Versus Interval Estimate | 点估计与区间估计
A point estimate, such as the sample mean x̄, is a single value used to estimate μ. Although x̄ is an unbiased estimator of μ, it almost never equals μ exactly because of random sampling error.
点估计（例如样本均值 x̄）是用于估计 μ 的单一数值。虽然 x̄ 是 μ 的无偏估计量，但由于随机抽样误差，它几乎不可能恰好等于 μ。
An interval estimate adds a margin of error around the point estimate. This gives a range that accounts for the uncertainty caused by using a sample rather than the whole population. The wider the interval, the more confident we can be that it captures μ, but the less precise the estimate becomes.
区间估计在点估计的基础上增加误差界限。这样给出的范围考虑了因使用样本而非整个总体而产生的不确定性。区间越宽，我们对其包含 μ 的信心就越大，但估计的精度会降低。
- Point estimate: x̄
- Interval estimate: x̄ ± z* × (σ / √n)
中文对应：点估计为 x̄；区间估计为 x̄ ± z* × (σ / √n)。
3. The Role of the Normal Distribution | 正态分布的作用
When the population is normally distributed, or when the sample size n is large, the sample mean x̄ is approximately normally distributed with mean μ and standard deviation σ / √n. This result is fundamental to confidence interval construction.
当总体服从正态分布，或样本量 n 很大时，样本均值 x̄ 近似服从正态分布，其均值为 μ，标准差为 σ / √n。这一结果是构造置信区间的基础。
This standard deviation of the sampling distribution is called the standard error of the mean. It measures how much x̄ is expected to vary from one sample to another. A smaller standard error means the sample mean is more precise as an estimate of μ.
抽样分布的这个标准差称为均值的标准误差。它衡量每次抽样时 x̄ 预计会波动多大。标准误差越小，样本均值作为 μ 的估计就越精确。
Standard error = σ / √n
标准误差 = σ / √n
4. Critical Values and Confidence Levels | 临界值与置信水平
The confidence level is the long-run success rate of the method. Common levels are 90%, 95% and 99%, with corresponding two-tailed z-critical values. These values come from the standard normal distribution.
置信水平是该方法的长期成功率。常见水平为 90%、95% 和 99%，对应双尾 z 临界值。这些值来自标准正态分布。
| Confidence level | 置信水平 | Critical value z* | 临界值 z* |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
These critical values are found from the inverse normal distribution. For example, for a 95% confidence interval, the central 95% leaves 2.5% in each tail, so z* = 1.960. For 99%, each tail has 0.5%, giving z* = 2.576.
这些临界值可由逆正态分布求得。例如，95% 置信区间的中央 95% 在两侧各留下 2.5%，因此 z* = 1.960。对于 99%，每侧尾部为 0.5%，因此 z* = 2.576。
5. Constructing a Confidence Interval for μ | 构造总体均值的置信区间
When the population standard deviation σ is known, a confidence interval for μ is given by:
当总体标准差 σ 已知时，μ 的置信区间为：
x̄ − z* × σ/√n ≤ μ ≤ x̄ + z* × σ/√n
或写作 x̄ ± z* × σ/√n
This formula requires either a normal population or a large sample. The term z* × σ/√n is called the margin of error. It determines the half-width of the interval.
该公式要求总体服从正态分布或样本量较大。z* × σ/√n 这一项称为误差界限，它决定了区间的半宽度。
If the sample size n is given and the data are summarised by x̄, you can substitute directly into the formula. Always keep at least three significant figures during intermediate calculations to avoid rounding errors in the final answer.
如果给出样本量 n，并用 x̄ 概括数据，就可以直接代入公式。计算过程中至少保留三位有效数字，以避免最终答案出现舍入误差。
6. Large Samples and the Central Limit Theorem | 大样本与中心极限定理
If the population is not known to be normal but the sample size n is large, the Central Limit Theorem says that x̄ is approximately normal. In most A-Level contexts, n ≥ 30 is considered large enough for this approximation to be reasonable.
如果总体不一定是正态分布，但样本量 n 很大，中心极限定理表明 x̄ 近似服从正态分布。在大多数 A-Level 情境中，n ≥ 30 被视为足够大，可以使用该近似。
When n is large and σ is unknown, the sample standard deviation s may be used to estimate σ, so the standard error becomes s / √n. This is the most common practical situation in exam questions: a large sample is summarised by x̄ and s, and you are asked to find a confidence interval.
当 n 很大且 σ 未知时，可以用样本标准差 s 估计 σ，因此标准误差变为 s / √n。这是考试题中最常见的实际情境：大样本由 x̄ 和 s 概括，要求你求出置信区间。
It is important to state clearly when you are using s in place of σ, even if the final formula looks the same. In A-Level Mathematics questions, this substitution is accepted for large samples.
当你用 s 代替 σ 时，即使最终公式看起来相同，也要明确说明。在 A-Level 数学题中，大样本下这种替代是可以接受的。
7. Interpreting a Confidence Interval Correctly | 正确解释置信区间
A 95% confidence interval does not mean there is a 95% chance that μ lies inside that specific interval. Once the interval is calculated, μ is either inside it or not; the probability is 0 or 1. This is a very common exam pitfall.
95% 置信区间并不意味着 μ 有 95% 的概率落在该特定区间内。一旦计算出区间，μ 要么在区间内，要么不在；概率为 0 或 1。这是考试中非常常见的误区。
The correct interpretation is long-run: if we took many random samples of the same size and constructed a 95% confidence interval from each, about 95% of those intervals would capture μ. The confidence level describes the method, not one particular result.
正确解释是长期频率：如果我们抽取许多相同大小的随机样本，并分别构造 95% 置信区间，那么这些区间中约有 95% 会包含 μ。置信水平描述的是方法，而非某一个特定结果。
For a single interval, we say we are 95% confident that the interval contains μ. This reflects the reliability of the procedure, not a probability for this particular interval.
对于单个区间，我们说有 95% 的把握认为该区间包含 μ。这反映程序的可靠性，而不是这个特定区间的概率。
8. Margin of Error and Determining Sample Size | 误差界限与样本量的确定
The margin of error is M = z* × σ/√n.
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