📚 Confidence Intervals for a Population Mean with Known Variance | 方差已知时总体均值的置信区间
In IB Mathematics Analysis and Approaches (AI and AA), one of the most important applications of the normal distribution is constructing a confidence interval for a population mean. This article focuses on the case where the population variance σ² is known, which allows us to use the standard normal distribution to quantify uncertainty in our estimate. We will derive the formula step by step, interpret its meaning, and work through exam-style examples.
在IB数学分析与方法（AI和AA）课程中，正态分布最重要的应用之一就是构造总体均值的置信区间。本文聚焦于总体方差σ²已知的情形，此时我们可以借助标准正态分布来量化估计的不确定性。我们将逐步推导公式、解释其含义，并通过考试风格的例题进行演练。
1. What Is a Confidence Interval? | 什么是置信区间？
A confidence interval (CI) is a range of values, calculated from sample data, that is likely to contain the true population mean μ with a given level of confidence. For example, a 95% confidence interval means that if we repeated the sampling process many times, about 95% of the intervals we construct would capture the true mean.
置信区间（CI）是根据样本数据计算出的一个数值范围，它以给定的置信水平包含总体均值μ。例如，95%置信区间意味着：如果我们重复多次抽样，那么约95%的区间会包含真实的总体均值。
The confidence interval provides more information than a single point estimate because it accounts for sampling variability. A point estimate such as x̄ gives one plausible value, while an interval acknowledges that the true mean could reasonably lie anywhere within a range.
置信区间比单纯的点估计提供更多信息，因为它考虑了抽样变异性。点估计如x̄只给出一个可能值，而区间则承认真实均值可能合理地落在某一范围内。
When the population variance is known, the confidence interval is based on the standard normal (z) distribution. This occurs in questions where σ is given in the problem, or where a previous investigation has established the population standard deviation with high reliability.
当总体方差已知时，置信区间基于标准正态（z）分布。这种情形常出现在题目直接给出σ，或者先前的研究已高可靠性地确定了总体标准差时。
2. Population, Sample and the Concept of a Statistic | 总体、样本与统计量的概念
A statistic is a numerical quantity computed from a sample, such as the sample mean x̄ or the sample variance s². In contrast, a parameter is a numerical characteristic of the population, such as the population mean μ or population variance σ². The essential idea of inferential statistics is to use a statistic to estimate an unknown parameter.
统计量是从样本中计算得出的数值量，例如样本均值x̄或样本方差s²。相比之下，参数是总体的数值特征，如总体均值μ或总体方差σ²。推断统计的核心思想就是利用统计量去估计未知的参数。
Because samples are randomly selected, a statistic is itself a random variable. If we take many different samples, we obtain many different values of x̄. This randomness is governed by the sampling distribution of the statistic, which is central to calculating confidence intervals.
由于样本是随机选取的，统计量本身也是随机变量。若抽取多个不同的样本，就会得到多个不同的x̄值。这种随机性由统计量的抽样分布所支配，而抽样分布正是计算置信区间的核心。
In this topic, we treat x̄ as an unbiased estimator of μ, meaning E(x̄) = μ. The variability of x̄ around μ is measured by the standard error, which will be used explicitly in the interval construction.
在本专题中，我们将x̄视为μ的无偏估计量，即E(x̄) = μ。x̄围绕μ的变异性由标准误衡量，构造区间时会直接使用这一指标。
3. The Sampling Distribution of the Sample Mean | 样本均值的抽样分布
The sampling distribution of x̄ describes how the sample mean varies across all possible samples of a fixed size n taken from a population. Regardless of the population’s original distribution, the Central Limit Theorem states that for sufficiently large n, x̄ is approximately normally distributed. Even for small n, if the population itself is normal, then x̄ is exactly normal.
x̄的抽样分布描述了从总体中抽取固定大小n的所有可能样本时，样本均值如何变化。无论总体原本的分布如何，中心极限定理表明：当n足够大时，x̄近似服从正态分布。即使n较小，若总体本身是正态的，那么x̄也精确地服从正态分布。
For a population with mean μ and known variance σ², the sample mean has the following distribution:
对于均值为μ、方差σ²已知的总体，样本均值具有如下分布：
x̄ ~ N(μ, σ²/n)
The mean of the sampling distribution equals the population mean μ, and the variance is σ²/n. The standard deviation of x̄ is therefore σ/√n, which is called the standard error of the mean.
抽样分布的均值等于总体均值μ，方差为σ²/n。因此x̄的标准差为σ/√n，称为均值的标准误。
Since σ is known, the standard error is a known quantity. This is the crucial advantage that allows us to use the z-distribution rather than the t-distribution, which would be required if σ were unknown.
由于σ已知，标准误是一个已知量。这正是我们可以使用z分布而非t分布的关键优势；若σ未知，则需改用t分布。
4. The Standard Normal Distribution and Critical Values | 标准正态分布与临界值
To construct a confidence interval, we standardise the sample mean. Recall that if x̄ follows a normal distribution with mean μ and standard deviation σ/√n, then the z-score is:
为构造置信区间，我们对样本均值进行标准化。回忆：若x̄服从均值为μ、标准差为σ/√n的正态分布，则z分数为：
Z = (x̄ − μ) / (σ/√n) ~ N(0, 1)
We then choose a confidence level, such as 95%. The area under the standard normal curve between −z(α/2) and +z(α/2) equals the confidence level, where α = 1 − confidence level. The remaining area α is split equally into the two tails, each of area α/2.
我们随后选择一个置信水平，例如95%。标准正态曲线在−z(α/2)与+z(α/2)之间的面积等于置信水平，其中α = 1 − 置信水平。剩余面积α被均分到两个尾部，每个尾部的面积为α/2。
The critical value z(α/2) is the z-score such that P(Z > z(α/2)) = α/2. For a 95% confidence interval, α = 0.05, α/2 = 0.025, and z(0.025) = 1.96. For 90% confidence, z(0.05) = 1.645; for 99% confidence, z(0.005) = 2.576.
临界值z(α/2)是满足P(Z > z(α/2)) = α/2的z分数。对于95%置信区间，α = 0.05，α/2 = 0.025，z(0.025) = 1.96。90%置信水平对应z(0.05) = 1.645；99%置信水平对应z(0.005) = 2.576。
These critical values are typically obtained from a standard normal distribution table or from a calculator such as the TI-Nspire CX or Casio fx-CG50. In IB examinations, the relevant z-values are often provided, but memorising the common three is strongly recommended for speed.
这些临界值通常查标准正态分布表或使用TI-Nspire CX、Casio fx-CG50等计算器获得。IB考试通常会提供相应的z值，但强烈建议熟记上面三个常用值以提高解题速度。
5. Deriving the Confidence Interval Formula | 置信区间公式的推导
Starting from the standardised statistic, we set the probability that Z lies within the interval of interest:
从标准化统计量出发，我们设定Z落在目标区间内的概率：
P(−z(α/2) ≤ (x̄ − μ)/(σ/√n) ≤ z(α/2)) = 1 − α
Rearranging the inequality to isolate μ, we multiply all three parts by σ/√n and then substitute x̄ from the centre:
为将μ分离出来，我们对不等式的三部分同时乘以σ/√n，再从中间减去x̄：
P(x̄ − z(α/2)·σ/√n ≤ μ ≤ x̄ + z(α/2)·σ/√n) = 1 − α
The resulting confidence interval is:
由此得到的置信区间为：
x̄ ± z(α/2) · σ/√n
The quantity z(α/2)·σ/√n is called the margin of error (or error bound). It measures how much the sample mean may deviate from the population mean at the chosen confidence level. The interval is wider when the confidence level increases, as a higher confidence requires a larger range to be sure of capturing μ.
量z(α/2)·σ/√n称为误差幅度（或误差界）。它衡量在所选置信水平下样本均值与总体均值可能偏离的程度。置信水平越高，区间越宽，因为更高置信度需要更大的范围才能确保覆盖μ。
Note that the formula assumes: (1) the sample is random, (2) the population is normal or n is large enough for the Central Limit Theorem to apply, and (3) the population standard deviation σ is known. Violating any of these conditions invalidates the result.
注意该公式假设：（1）样本是随机的；（2）总体为正态或n足够大从而中心极限定理适用；（3）总体标准差σ已知。违反任何一条都会使结果失效。
6. Choosing the Confidence Level | 选择置信水平
Common confidence levels in IB questions are 90%, 95% and 99%. The choice of confidence level affects the width of the interval but not the point estimate x̄. A 90% interval is narrower, but we are less certain that it contains μ; a 99% interval is wider, giving greater certainty.
IB问题中常见的置信水平为90%、95%和99%。置信水平的选择影响区间的宽度，但不影响点估计x̄。90%区间较窄，但我们较不确定它包含μ；99%区间较宽，确定性更高。
The table below summarises the critical z-values used most frequently:
下表总结了最常用的临界z值：
| Confidence Level | α | α/2 | z(α/2) |
| 90% | 0.10 | 0.05 | 1.645 |
| 95% | 0.05 | 0.025 | 1.960 |
| 99% | 0.01 | 0.005 | 2.576 |
In exam questions, always check whether the confidence level is stated as a percentage, a decimal, or described in words such as ‘a 95% level of confidence’. Convert to α correctly before selecting the critical value.
在考试题目中，务必检查置信水平是以百分数、小数还是文字形式给出，例如“95%置信水平”。在选取临界值之前，要正确转换出α。
A larger sample size n reduces the standard error σ/√n, thus narrowing the interval for the same confidence level. This reflects the intuitive fact that more data gives more precise estimates.
更大的样本容量n会减小标准误σ/√n，从而在同一置信水平下使区间变窄。这反映出一个直观事实：数据越多，估计越精确。
7. Worked Example 1 | 例题1
A random sample of 25 IB students is selected from a large international school. Their mean score on a standardised mathematics test is 72. The population standard deviation is known to be 10. Construct a 95% confidence interval for the population mean score μ.
从一所大型国际学校中随机抽取25名IB学生。他们在标准化数学测试中的平均分为72。已知总体标准差为10。构造总体均分μ的95%置信区间。
Step 1: Identify the known quantities. We have x̄ = 72, σ = 10, n = 25, and for 95% confidence, z(0.025) = 1.96.
第一步：确定已知量。我们有x̄ = 72，σ = 10，n = 25，对于95%置信水平，z(0.025) = 1.96。
Step 2: Calculate the standard error:
第二步：计算标准误：
σ/√n = 10/√25 = 10/5 = 2
Step 3: Compute the margin of error:
第三步：计算误差幅度：
1.96 × 2 = 3.92
Step 4: Construct the interval:
第四步：构造区间：
72 ± 3.92 = (68.08, 75.92)
We are 95% confident that the true population mean score lies between 68.08 and 75.92. Round to 3 significant figures where appropriate: the interval may be written as (68.1, 75.9).
我们有95%的把握认为总体均分位于68.08与75.92之间。按需要保留三位有效数字：区间可写为(68.1, 75.9)。
8. Worked Example 2 | 例题2
A factory produces metal rods with a known population standard deviation of 0.5 mm. A random sample of 100 rods has a mean diameter of 12.3 mm. (a) Construct a 99% confidence interval for the true mean diameter. (b) Determine the sample size required to reduce the margin of error to 0.1 mm at the 99% confidence level.
某工厂生产金属棒，已知总体标准差为0.5 mm。随机抽取100根棒，平均直径为12.3 mm。（a）构造真实平均直径的99%置信区间。（b）在99%置信水平下，要将误差幅度减小到0.1 mm，需要多大的样本容量？
Part (a): For 99% confidence, z(0.005) = 2.576. The standard error is 0.5/√100 = 0.5/10 = 0.05.
第（a）部分：对于99%置信水平，z(0.005) = 2.576。标准误为0.5/√100 = 0.5/10 = 0.05。
The margin of error is 2.576 × 0.05 = 0.1288. Therefore the 99% confidence interval is:
误差幅度为2.576 × 0.05 = 0.1288。因此99%置信区间为：
12.3 ± 0.1288 = (12.1712, 12.4288)
Rounding to 3 decimal places, the interval is (12.171, 12.429) mm.
保留三位小数，区间为(12.171, 12.429) mm。
Part (b): We require the margin of error E = 0.1. Setting the margin of error formula equal to 0.1:
第（b）部分：要求误差幅度E = 0.1。令误差幅度公式等于0.1：
z(0.005) · σ/√n = 0.1
2.576 × 0.5/√n = 0.1
Squaring both sides, we find n = (2.576 × 0.5 / 0.1)² = (12.88)² = 165.89. Since n must be an integer and we need the margin of error to be no greater than 0.1, we round up to n = 166.
两边平方，得n = (2.576 × 0.5 / 0.1)² = (12.88)² = 165.89。由于n必须为整数且要求误差幅度不超过0.1，故向上取整为n = 166。
This example illustrates a common IB exam pattern: calculating the interval, then finding the sample size needed to achieve a specified precision. Remember to always round up when determining the minimum sample size.
此例展示了IB考试常见的题型：先计算区间，再求达到指定精度所需的样本量。切记在确定最小样本量时始终向上取整。
9. Interpreting the Interval Correctly | 正确解释置信区间
The correct interpretation of a 95% confidence interval must refer to the long-run frequency of the method, not to the probability that μ lies in this specific interval. Since μ is a fixed constant, it either lies in the interval or it does not. The correct statement is: ‘In repeated sampling, approximately 95% of confidence intervals constructed this way will contain μ.’
95%置信区间的正确解释必须涉及方法的长期频率，而不是μ落在这个特定区间内的概率。因为μ是固定常数，它要么在区间内，要么不在。正确的表述是：“在重复抽样中，按这种方式构造的置信区间约有95%会包含μ。”
A common misconception is to say ‘there is a 95% probability that μ lies within this interval.’ This is incorrect in the strict classical (frequentist) framework. Once the interval is computed, μ is either inside or outside; the uncertainty refers to the procedure that generated the interval.
一个常见误解是“μ落在此区间内的概率为95%”。在严格的经典（频率学派）框架下这是不正确的。区间一旦算出，μ要么在内部要么在外部；不确定性指的是生成区间的过程。
In IB answers, examiners accept phrases such as ‘we are 95% confident that the population mean is between A and B,’ provided the meaning of ‘confident’ is clear. Avoid the phrase ‘probability that μ is in the interval’ to prevent losing marks.
在IB答卷中，考官接受“我们有95%的置信度认为总体均值介于A与B之间”这样的表述，前提是“置信”的含义清晰。避免使用“μ在此区间内的概率”这一短语，以防丢分。
10. Common Errors and Exam Tips | 常见错误与考试技巧
One frequent error is using the t-distribution instead of the normal distribution. When σ is known, the z-distribution is always appropriate, regardless of the sample size. The t-distribution is reserved for σ unknown.
一个常见错误是错用t分布代替正态分布。当σ已知时，任何样本容量下都应使用z分布。t分布仅在σ未知时使用。
A second error is incorrectly computing the standard error. Some students use σ instead of σ/√n, or mistakenly multiply by n. Always check that the standard error is smaller than σ, since averaging reduces variability.
第二个错误是计算标准误有误。有些学生使用σ而不是σ/√n，或错误乘以n。务必检查：标准误应小于σ，因为取平均会减小变异性。
A third common issue is misreading the critical value. For a 95% interval, z = 1.96; for a 90% interval, z = 1.645. Ensure you identify the two-tailed critical value correctly and do not use the one-tailed value from a hypothesis test by mistake.
第三个常见问题是读错临界值。95%区间对应z = 1.96；90%区间对应z = 1.645。务必正确识别双尾临界值，不要误用假设检验中的单尾值。
Finally, remember to state the interval in the correct order and with appropriate rounding. In IB examinations, intervals are usually required to 3 significant figures or 3 decimal places. Always include the units in your final answer when context is given.
最后，记得以正确顺序写出区间并采用合适的舍入。在IB考试中，区间通常要求保留三位有效数字或三位小数。当题目有实际背景时，最终答案要写上单位。
11. Summary | 总结
When the population variance is known, the confidence interval for the population mean is computed using the formula x̄ ± z(α/2)·σ/√n. This relies on the sampling distribution of the sample mean being normal, which is guaranteed either by an underlying normal population or by the Central Limit Theorem for large samples.
当总体方差已知时，总体均值置信区间由公式x̄ ± z(α/2)·σ/√n计算得出。这依赖于样本均值抽样分布的正态性，而正态性可通过总体本身为正态或大样本下中心极限定理来保证。
The key steps in any exam question are: identify x̄, σ, n and the confidence level; find the correct critical value z(α/2); compute the standard error σ/√n; multiply by the critical value to obtain the margin of error; and finally write the interval as (x̄ − E, x̄ + E).
任何考试题目的关键步骤是：确定x̄、σ、n和置信水平；找出正确的临界值z(α/2)；计算标准误σ/√n；乘以临界值得出误差幅度E；最后写出区间(x̄ − E, x̄ + E)。
Understanding the derivation and interpretation of confidence intervals is essential for both Paper 2 questions and the Internal Assessment in IB Mathematics, where statistical analysis is frequently applied to real data. Mastery of this topic also builds a solid foundation for hypothesis testing, which follows naturally in the statistics unit.
理解置信区间的推导和解释，对IB数学Paper 2题目和内部评估（IA）都至关重要——IA中经常需要对真实数据进行统计分析。掌握本专题也为后续假设检验打下坚实基础，假设检验是统计单元中自然衔接的内容。
Practice constructing intervals for different confidence levels and sample sizes until the procedure becomes automatic. This is one of the most reliably tested skills in the IB statistics syllabus, and careful procedural accuracy will earn full marks.
练习在不同置信水平和样本容量下构造区间，直到这一流程完全熟练。这是IB统计考纲中最常考的技能之一，严谨的过程准确性可以让你拿到满分。
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