📚 H Confidence Interval for the Mean with Known Variance | 已知方差时均值的置信区间
In statistical inference, a confidence interval gives a range of plausible values for an unknown population parameter. When we are interested in estimating a population mean μ and we know the population variance σ², we can construct a confidence interval based on the sample mean and the standard normal distribution. This method, often called the z-interval, is one of the most fundamental tools in IB Mathematics and Cambridge Statistics, laying the groundwork for hypothesis testing and decision making under uncertainty.
在统计推断中，置信区间为未知的总体参数提供了一个合理的取值范围。当我们希望估计总体均值 μ，并且已知总体方差 σ² 时，可以基于样本均值和标准正态分布来构造置信区间。这种方法通常被称为 z 区间，是 IB 数学和剑桥统计学中最基础的工具之一，也为不确定条件下的假设检验和决策奠定了基础。
1. Confidence Intervals: Purpose and Logic | 置信区间的目的与逻辑
A single sample mean is a point estimate of the population mean, but it is almost never exactly equal to the true value. A confidence interval quantifies the uncertainty by providing a range that, with a specified probability, captures the parameter. The logic rests on the sampling distribution of the mean: if we repeatedly took samples and computed intervals, a certain proportion of them would contain μ.
单个样本均值是对总体均值的点估计，但它几乎不可能恰好等于真实值。置信区间通过提供一个以指定概率包含参数的范围来量化这种不确定性。其逻辑建立在均值的抽样分布之上：如果我们重复抽取样本并计算区间，其中一定比例的区间会包含 μ。
For the mean with known variance, the sampling distribution of X̄ is exactly normal if the population is normal, and approximately normal by the Central Limit Theorem for large samples. This allows us to use critical values from the standard normal curve to build the interval with a chosen confidence level.
对于已知方差的均值，若总体呈正态分布，则 X̄ 的抽样分布是精确正态的；对于大样本，由中心极限定理保证其近似正态。这使得我们能够利用标准正态曲线的临界值，按照选定的置信水平来构造区间。
2. Sampling Distribution of the Sample Mean | 样本均值的抽样分布
Let X₁, X₂, …, Xₙ be a random sample from a population with mean μ and known variance σ². The sample mean is defined as X̄ = (1/n) Σ Xᵢ. The expected value of X̄ is E(X̄) = μ, and its variance is Var(X̄) = σ²/n. Therefore, the standard error of the mean is σ/√n.
设 X₁, X₂, …, Xₙ 为取自均值为 μ、已知方差为 σ² 的总体的一个随机样本。样本均值定义为 X̄ = (1/n) Σ Xᵢ。X̄ 的期望值为 E(X̄) = μ，方差为 Var(X̄) = σ²/n。因此，均值的标准误为 σ/√n。
Because the variance is known, we can standardise the sample mean to obtain a z-statistic:
Z = (X̄ − μ) / (σ/√n) ~ N(0, 1)
由于方差已知，我们可以将样本均值标准化，得到 z 统计量：
Z = (X̄ − μ) / (σ/√n) ~ N(0, 1)
This pivotal quantity does not depend on the unknown μ, which makes it ideal for constructing confidence intervals.
这个枢轴量不依赖于未知的 μ，因此非常适合用于构造置信区间。
3. The Z-Interval When Variance Is Known | 已知方差时的 Z 区间
If we want a confidence interval with confidence level (1 − α) × 100%, we find the critical value zα/2 such that P(−zα/2 < Z < zα/2) = 1 − α. Rearranging the inequality gives the interval for μ:
如果我们想要一个置信水平为 (1 − α) × 100% 的置信区间，就需要找到临界值 zα/2，使得 P(−zα/2 < Z < zα/2) = 1 − α。重新整理不等式便可得到 μ 的置信区间：
X̄ − zα/2 × (σ/√n) < μ < X̄ + zα/2 × (σ/√n)
This is the z-interval for the population mean with known variance. The interval is symmetric around the sample mean, and its width depends on the critical value, the population standard deviation, and the sample size.
这就是已知方差时总体均值的 z 区间。区间以样本均值为中心对称，其宽度取决于临界值、总体标准差和样本量。
4. Confidence Level and Critical Values | 置信水平与临界值
The most commonly used confidence levels are 90%, 95% and 99%. The corresponding significance level α is 0.10, 0.05 and 0.01 respectively. The critical value zα/2 is the positive number that cuts off the upper tail area α/2 of the standard normal distribution.
最常用的置信水平是 90%、95% 和 99%，相应的显著性水平 α 分别为 0.10、0.05 和 0.01。临界值 zα/2 是标准正态分布上侧尾部面积为 α/2 时所对应的正数。
| Confidence Level | α | α/2 | zα/2 |
|---|---|---|---|
| 90% | 0.10 | 0.05 | 1.645 |
| 95% | 0.05 | 0.025 | 1.960 |
| 99% | 0.01 | 0.005 | 2.576 |
It is useful to remember that 1.96 is the critical value for a 95% confidence interval. In most IB and Cambridge problems, you will be expected to use the inverse normal function or provided tables to obtain these values.
记住 1.96 是 95% 置信区间的临界值会很有用。在大多数 IB 和剑桥的题目中，你都需要使用逆正态函数或提供的表格来得到这些值。
5. Margin of Error and Interval Width | 误差边际与区间宽度
The term zα/2 × (σ/√n) is called the margin of error (ME). It represents the maximum likely distance between the sample mean and the true population mean at the given confidence level. The confidence interval can then be written compactly as
X̄ ± ME, where ME = zα/2 × (σ/√n).
zα/2 × (σ/√n) 这一项被称为误差边际 (ME)。它表示在给定的置信水平下，样本均值与真实总体均值之间可能的最大差距。置信区间可以简洁地写为
X̄ ± ME，其中 ME = zα/2 × (σ/√n)。
Everything else being equal, a larger sample size n decreases the standard error σ/√n, which narrows the interval. Likewise, a higher confidence level increases zα/2, widening the interval. These relationships are essential when planning a study or interpreting published results.
在其他条件不变时，更大的样本量 n 会减小标准误 σ/√n，从而使区间变窄。同理，更高的置信水平会增大 zα/2，使区间变宽。这些关系在规划研究或解读已发表的结果时至关重要。
6. Correct Interpretation of a Confidence Interval | 如何正确解释置信区间
A 95% confidence interval does not mean that there is a 95% probability that μ lies inside the particular interval you have computed. Once the interval is calculated, μ is either inside it or not; the probability statement refers to the method. We say: “If we were to take many random samples and compute a 95% confidence interval from each, about 95% of those intervals would contain the true population mean.”
95% 置信区间并不意味着 μ 有 95% 的概率落在你所计算的这个具体区间内。一旦区间计算出来，μ 要么在区间内，要么不在；概率陈述针对的是方法本身。我们应该说：“如果我们抽取许多随机样本，并从每个样本计算一个 95% 置信区间，那么大约 95% 的区间会包含真实的总体均值。”
This frequentist understanding is a key assessment objective in the IB syllabus. Candidates often lose marks by phrasing the conclusion as “the probability that μ lies in the interval is 0.95”, which is technically incorrect for a computed interval.
这种频率学派的解释是 IB 教学大纲中的一个重要评估目标。考生常常因为将结论表述为“μ 落在这个区间内的概率为 0.95”而失分，对于已经计算出的区间，这种说法在技术上是错误的。
7. Assumptions Behind the Z-Interval | Z 区间背后的假设
For the z-interval to be valid, several conditions must be met. First, the population variance σ² must be known, which is rarely the case in practice but is a useful starting point for learning. Second, the data should come from a simple random sample to ensure independence. Third, the population distribution should be normal, or the sample size should be large enough (typically n ≥ 30) for the Central Limit Theorem to apply.
要使 z 区间有效，必须满足几个条件。首先，总体方差 σ² 必须是已知的——虽然这在实践中很少见，但它是学习的有用起点。其次，数据应来自简单随机样本，以确保独立性。第三，总体分布应为正态，或者样本量足够大（通常 n ≥ 30），以便中心极限定理适用。
When the sample size is small and the population is not normal, the z-interval should not be used; instead, a t-interval would be more appropriate, even if the variance were somehow known. The IB syllabus often contrasts the known-variance case with the unknown-variance case to highlight the role of the t-distribution.
当样本量较小且总体非正态时，不应使用 z 区间；此时，即使方差已知，也最好使用 t 区间。IB 教学大纲常常将已知方差的情况与未知方差的情况进行对比，以凸显 t 分布的作用。
8. Effect of Changing Confidence Level and Sample Size | 改变置信水平与样本量的影响
Increasing the confidence level from 95% to 99%, while keeping everything else fixed, makes the critical value larger and therefore widens the confidence interval. A wider interval uses a bigger margin of error, which makes the estimate less precise but increases our confidence that it captures the parameter.
将置信水平从 95% 提高到 99%，其他条件不变，会使临界值变大，从而加宽置信区间。更宽的区间使用了更大的误差边际，这会降低估计的精确度，但提高了我们捕捉到参数的信心。
On the other hand, increasing the sample size n reduces the standard error σ/√n and narrows the interval without sacrificing confidence. Doubling the precision (halving the margin of error) requires quadrupling the sample size, because ME is inversely proportional to √n.
另一方面，增大样本量 n 会减小标准误 σ/√n，在不牺牲置信度的情况下缩窄区间。将精确度提高一倍（使误差边际减半）需要将样本量增大到四倍，因为 ME 与 √n 成反比。
9. A Worked Example | 计算示例
A machine fills cereal boxes, and the filling weight is normally distributed with standard deviation σ = 15 g. A random sample of 25 boxes yields a mean weight X̄ = 502 g. Construct a 95% confidence interval for the population mean filling weight μ.
一台机器填充谷物盒，填充重量服从正态分布，标准差 σ = 15 g。随机抽取 25 盒，得到平均重量 X̄ = 502 g。请构造总体平均填充重量 μ 的 95% 置信区间。
For a 95% confidence level, z0.025 = 1.960. The standard error is σ/√n = 15/√25 = 3 g. The margin of error is ME = 1.960 × 3 = 5.88 g.
对于 95% 置信水平，z0.025 = 1.960。标准误为 σ/√n = 15/√25 = 3 g。误差边际为 ME = 1.960 × 3 = 5.88 g。
CI: 502 ± 5.88 → (496.12 g, 507.88 g)
We interpret this interval by saying: “We are 95% confident that the true mean filling weight lies between 496.12 g and 507.88 g.” Remember that this confidence is based on the long-run performance of the method.
我们这样解释这个区间：“我们有 95% 的把握认为，真实的平均填充重量介于 496.12 g 至 507.88 g 之间。”请注意，这种信心是基于该方法在长期中的表现。
10. Common Pitfalls and Exam Tips | 常见错误与考试提示
Students often confuse the standard deviation σ with the standard error σ/√n. The standard error is always smaller than the population standard deviation and decreases with larger n. Another frequent mistake is using the wrong critical value – for example, using 1.645 for a 95% interval or incorrectly reading two-tailed values from the standard normal table.
学生们常常将标准差 σ 与标准误 σ/√n 混淆。标准误总是小于总体标准差，并随 n 的增大而减小。另一个常见错误是使用了错误的临界值——例如，对 95% 区间使用 1.645，或从标准正态表中错误地读取了双侧值。
In exam questions, you may be asked to find the minimum sample size required to achieve a desired margin of error. The formula can be rearranged as
n ≥ (zα/2 × σ / ME)²
and you should always round up to the next integer to ensure the confidence interval is at least as narrow as required.
在考试问题中，你可能会被要求计算达到期望误差边际所需的最小样本量。可以将公式变形为
n ≥ (zα/2 × σ / ME)²
并且必须始终向上取整，以确保置信区间至少达到所需的窄度。
Finally, practice writing interpretation statements carefully: always mention the confidence level, the parameter in context, and the long-run success rate of the method. Avoid saying “the interval contains 95% of the data” or “there is a 95% chance that the mean is in this interval”.
最后，请认真练习撰写解释性语句：务必提到置信水平、情境中的参数以及该方法的长期成功率。避免说“该区间包含了 95% 的数据”或“均值在这个区间内的概率为 95%”。
Published by TutorHao | Statistics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply