📚 Confidence Intervals for a Population Mean with Known Variance (Data) | 方差已知时总体均值的置信区间（数据）
In statistics, we often need to estimate an unknown population mean μ using sample data. When the population variance is known, we can construct a confidence interval based on the standard normal distribution (z-distribution). This article explains the concept, formula, calculation, and interpretation of such intervals in the context of IB Mathematics.
在统计学中，我们经常需要利用样本数据估计未知的总体均值 μ。当总体方差已知时，我们可以基于标准正态分布（z分布）构造置信区间。本文将解释这一概念、公式、计算及其在 IB 数学背景下的应用与解释。
1. The Idea of a Confidence Interval | 置信区间的思想
A point estimate, such as the sample mean x̄, gives a single best guess for μ. However, it is unlikely to equal μ exactly. A confidence interval provides a range of plausible values for μ, together with a level of confidence, usually 90%, 95% or 99%.
点估计（如样本均值 x̄）给出了对 μ 的一个最佳猜测。然而，它不太可能恰好等于 μ。置信区间为 μ 提供了一个合理的取值范围，并附带一个置信水平，通常为 90%、95% 或 99%。
For example, a 95% confidence interval for μ of (2.4, 3.6) means that we are 95% confident that the true mean lies between 2.4 and 3.6. The interval is constructed from sample data and the known variance.
例如，μ 的 95% 置信区间为 (2.4, 3.6)，意味着我们有 95% 的把握认为真实均值位于 2.4 和 3.6 之间。该区间由样本数据和已知方差构造而成。
2. Sampling Distribution of the Sample Mean | 样本均值的抽样分布
When we take a random sample of size n from a population with mean μ and known variance σ², the sample mean x̄ has a distribution with mean μ and standard deviation σ/√n. This standard deviation is called the standard error of the mean.
当从均值为 μ、方差为 σ² 的总体中抽取大小为 n 的随机样本时，样本均值 x̄ 的分布均值为 μ，标准差为 σ/√n。这个标准差称为均值的标准误差。
If the population is normally distributed, x̄ is exactly normal. Even if the population is not normal, for large n (by the Central Limit Theorem) x̄ is approximately normal. Therefore we can use the z-distribution when σ is known.
若总体服从正态分布，则 x̄ 精确服从正态分布。即使总体不服从正态分布，当 n 较大时（根据中心极限定理），x̄ 近似服从正态分布。因此，当 σ 已知时，我们可以使用 z 分布。
3. The Z-Statistic | Z 统计量
To construct a confidence interval, we standardize the sample mean using the known standard error. The z-statistic is defined as:
为了构造置信区间，我们使用已知的标准误差对样本均值进行标准化。Z 统计量定义为：
Z = (x̄ − μ) / (σ/√n)
Under the null distribution, Z follows a standard normal distribution N(0,1). For a given confidence level, we find the critical value z(α/2) such that the area between −z(α/2) and +z(α/2) equals the desired confidence level.
在零分布下，Z 服从标准正态分布 N(0,1)。对于给定的置信水平，我们找到临界值 z(α/2)，使得 −z(α/2) 与 +z(α/2) 之间的面积等于所需的置信水平。
4. Formula for the Confidence Interval | 置信区间公式
The general formula for a confidence interval for the population mean μ when σ is known is:
当 σ 已知时，总体均值 μ 的置信区间的一般公式为：
x̄ ± z(α/2) × σ/√n
Here, x̄ is the sample mean, z(α/2) is the critical value from the standard normal distribution, σ is the population standard deviation, and n is the sample size. The term σ/√n is the standard error.
其中，x̄ 是样本均值，z(α/2) 是标准正态分布的临界值，σ 是总体标准差，n 是样本容量。σ/√n 是标准误差。
The interval is sometimes written as (x̄ − z(α/2) × σ/√n, x̄ + z(α/2) × σ/√n). The width of the interval depends on the confidence level, the standard deviation, and the sample size.
该区间也可写为 (x̄ − z(α/2) × σ/√n, x̄ + z(α/2) × σ/√n)。区间的宽度取决于置信水平、标准差和样本容量。
5. Choosing the Confidence Level | 选择置信水平
Common confidence levels and their critical values z(α/2) are:
常见的置信水平及其临界值 z(α/2) 为：
| Confidence Level | 置信水平 | α | z(α/2) |
| 90% | 0.10 | 1.645 |
| 95% | 0.05 | 1.960 |
| 99% | 0.01 | 2.576 |
Higher confidence levels produce wider intervals. If you want more certainty that the interval contains μ, you must accept a larger margin of error. In contrast, a lower confidence level gives a narrower but less certain interval.
置信水平越高，区间就越宽。如果你想更确信区间包含 μ，就必须接受更大的误差范围。相反，较低的置信水平给出的区间更窄，但确定性较低。
6. Worked Example | 计算示例
A factory produces metal rods whose lengths are normally distributed with a known standard deviation σ = 0.5 cm. A random sample of 25 rods has a mean length of x̄ = 12.3 cm. Construct a 95% confidence interval for the true mean length μ.
某工厂生产的金属棒长度服从正态分布，已知标准差 σ = 0.5 cm。随机抽取 25 根棒，样本平均长度为 x̄ = 12.3 cm。试构造总体真实均值 μ 的 95% 置信区间。
For a 95% confidence level, z(α/2) = 1.96. The standard error is σ/√n = 0.5/√25 = 0.1 cm. The margin of error is 1.96 × 0.1 = 0.196 cm.
对于 95% 置信水平，z(α/2) = 1.96。标准误差为 σ/√n = 0.5/√25 = 0.1 cm。误差范围为 1.96 × 0.1 = 0.196 cm。
Interval = 12.3 ± 0.196 = (12.104, 12.496)
Thus, we are 95% confident that the true mean length lies between 12.104 cm and 12.496 cm.
因此，我们有 95% 的把握认为真实平均长度位于 12.104 cm 和 12.496 cm 之间。
7. Interpretation of the Confidence Interval | 置信区间的解释
It is crucial to interpret a confidence interval correctly. A 95% confidence interval means that if we repeatedly took samples of the same size and constructed intervals in the same way, approximately 95% of those intervals would contain the true population mean μ.
正确解释置信区间至关重要。95% 置信区间意味着如果我们反复抽取相同大小的样本并以同样方式构造区间，大约 95% 的区间会包含真正的总体均值 μ。
For a specific interval like (12.104, 12.496), we cannot say that the probability of μ being inside is 95%, because μ is fixed. Instead, we say that the interval was constructed using a method that has a 95% success rate.
对于像 (12.104, 12.496) 这样的特定区间，我们不能说 μ 落在其中的概率是 95%，因为 μ 是固定的。我们应说该区间是用具有 95% 成功率的方法构造的。
This subtle distinction is often tested in IB exams. A common correct statement is: “We are 95% confident that μ lies in this interval.”
这个细微差别在 IB 考试中经常被考查。一个常见的正确表述是：“我们有 95% 的把握认为 μ 位于该区间内。”
8. Factors Affecting the Width | 影响区间宽度的因素
The width of a confidence interval is determined by three factors:
置信区间的宽度由三个因素决定：
- The confidence level: increasing the confidence level widens the interval.
- The population standard deviation σ: a larger σ produces a wider interval.
- The sample size n: increasing n reduces the standard error, thus narrowing the interval.
- 置信水平：提高置信水平会使区间变宽。
- 总体标准差 σ：σ 越大，区间越宽。
- 样本容量 n：增大 n 会减小标准误差，从而使区间变窄。
If you want a narrower interval without changing confidence, you must increase the sample size. This is a key relationship in experimental design.
如果希望在保持置信水平不变的情况下获得更窄的区间，就必须增大样本容量。这是实验设计中的一个关键关系。
9. Common Mistakes | 常见误区
Students often make the following errors when constructing confidence intervals:
学生在构造置信区间时经常犯以下错误：
- Using the sample standard deviation s instead of the known σ. When σ is known, use z; when σ is unknown, use t-distribution.
- Confusing σ and σ/√n. Remember to divide by √n.
- Misinterpreting the confidence level as the probability that μ falls in a specific interval.
- Using the wrong critical value, e.g., z(α) instead of z(α/2).
- 使用样本标准差 s 而不是已知的 σ。当 σ 已知时，使用 z；当 σ 未知时，使用 t 分布。
- 混淆 σ 与 σ/√n。切记要除以 √n。
- 将置信水平误解为 μ 落在某个特定区间的概率。
- 使用错误的临界值，例如使用 z(α) 而不是 z(α/2)。
Always check the wording of the problem: “known variance” or “known standard deviation” signals the z-interval, while “estimated from the sample” signals the t-interval.
务必注意题目表述：“方差已知”或“标准差已知”提示使用 z 区间，而“由样本估计”则提示使用 t 区间。
10. Using Technology | 使用计算器与技术
In IB mathematics, you may use a GDC (graphical display calculator) to compute confidence intervals quickly. For a known variance, select the ‘ZInterval’ or ‘1-Sample Z’ function, enter the relevant values (σ, x̄, n, confidence level), and the calculator outputs the interval.
在 IB 数学中，你可能使用 GDC（图形计算器）快速计算置信区间。对于方差已知的情况，选择 “ZInterval” 或 “1-Sample Z” 功能，输入相关数值（σ、x̄、n、置信水平），计算器便会输出区间。
It is still essential to understand the underlying formula so that you can verify the result and answer questions that involve reasoning or interpretation.
理解背后的公式仍然至关重要，这样你才能验证结果，并回答涉及推理或解释的问题。
11. Summary | 总结
The confidence interval for a population mean with known variance is a fundamental tool in statistical inference. Its formula is:
方差已知时总体均值的置信区间是统计推断的基本工具。其公式为：
x̄ ± z(α/2) × σ/√n
Remember: the interval is centered at the sample mean, the margin of error depends on the critical value and standard error, and the interpretation must always refer to the method, not to the probability of μ being in a fixed interval.
请记住：区间以样本均值为中心，误差范围取决于临界值和标准误差，且解释时必须始终指向所用方法，而不是 μ 落在某个固定区间内的概率。
With practice, constructing and interpreting these intervals becomes a straightforward skill that will serve you well in exams and beyond.
通过练习，构造和解释这些区间将成为一项简单的技能，在考试及以后的学习中都会大有裨益。
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