📚 Confidence Intervals for a Population Mean | 总体均值的置信区间
In statistics, a confidence interval (CI) gives a range of plausible values for an unknown population mean μ, based on sample data. It is one of the most important tools in inferential statistics and a core topic in IB Mathematics Applications & Interpretation (AI) and Analysis & Approaches (AA).
在统计学中，置信区间（CI）根据样本数据为未知的总体均值 μ 提供一个合理的取值区间。它是推断统计中最重要的工具之一，也是 IB 数学 Applications & Interpretation (AI) 和 Analysis & Approaches (AA) 的核心考点。
1. The Concept of a Confidence Interval | 置信区间的概念
When we take a random sample from a population, the sample mean x̄ is a point estimate of μ. A confidence interval extends this estimate to account for sampling variability, producing an interval of the form: point estimate ± margin of error.
当我们从总体中抽取随机样本时，样本均值 x̄ 是 μ 的点估计。置信区间在此基础上扩展，以反映抽样变异性，得到形如「点估计 ± 边际误差」的区间。
x̄ ± z* × (σ/√n) 或 x̄ ± t* × (s/√n)
The confidence level C (e.g. 95%) describes the long-run proportion of intervals that would contain the true mean if we repeated the sampling process many times.
置信水平 C（如 95%）描述的是：如果重复多次抽样，在所有构造出的区间中，包含真实均值的区间所占的长期比例。
2. The z-Interval: When σ Is Known | z 区间：当 σ 已知时
If the population standard deviation σ is known and the sample is large, or the population is normally distributed, we use the z-interval:
如果总体标准差 σ 已知且样本较大，或总体服从正态分布，我们使用 z 区间：
x̄ ± z* × (σ/√n)
The critical value z* is determined by the confidence level. Common values are:
临界值 z* 由置信水平决定，常用值如下：
| Confidence level 置信水平 | 90% | 95% | 99% |
| z* | 1.645 | 1.960 | 2.576 |
On the GDC, we can find these values using the inverse normal function, e.g. invNorm(0.975) = 1.960 for a 95% interval.
在图形计算器（GDC）上，可用正态分布的逆函数求得，例如 95% 区间对应 invNorm(0.975) = 1.960。
3. The t-Interval: When σ Is Unknown | t 区间：当 σ 未知时
In most real-life situations, σ is unknown, and we estimate it using the sample standard deviation s. The t-interval is:
在实际问题中，σ 通常是未知的，我们使用样本标准差 s 来估计。t 区间为：
x̄ ± t* × (s/√n)
Here t* is the critical value from the t-distribution with n − 1 degrees of freedom.
其中 t* 是自由度为 n − 1 的 t 分布的临界值。
On the GDC, the t-interval is often computed directly using the “TInterval” function, which returns both the interval and the sample statistics.
在 GDC 上，可直接使用 “TInterval” 功能计算 t 区间，它会同时返回区间和样本统计量。
4. Degrees of Freedom | 自由度
Degrees of freedom (df) measure the amount of independent information in the sample. For a single-sample t-interval, df = n − 1. As df increases, the t-distribution approaches the standard normal distribution.
自由度（df）衡量样本中独立信息的数量。对于单样本 t 区间，df = n − 1。随着自由度增大，t 分布会趋近标准正态分布。
Since the t-distribution has heavier tails than the normal, t* is larger than z* for the same confidence level, producing a slightly wider (more conservative) interval.
由于 t 分布的尾部比正态分布更厚，在相同置信水平下 t* 大于 z*，因此得到的区间也略宽（更为保守）。
5. The Central Limit Theorem | 中心极限定理
The Central Limit Theorem (CLT) states that for a sufficiently large sample size (commonly n ≥ 30), the sampling distribution of x̄ is approximately normal regardless of the population distribution.
中心极限定理（CLT）指出：当样本量足够大（通常 n ≥ 30）时，无论总体分布如何，x̄ 的抽样分布近似正态。
This justifies the use of the z-interval for large samples even when the population is not normally distributed. For small samples, we require that the population itself is approximately normal.
这为大样本下使用 z 区间提供了依据，即使总体并不服从正态分布。对于小样本，则要求总体本身近似正态。
6. Interpreting a Confidence Interval | 置信区间的解读
A correct interpretation of a 95% confidence interval (a, b) for μ is: “We are 95% confident that the true population mean μ lies between a and b.”
对 μ 的 95% 置信区间 (a, b) 的正确解读是：「我们有 95% 的把握认为总体均值 μ 位于 a 和 b 之间。」
It is incorrect to say “there is a 95% probability that μ is in this particular interval.” The parameter μ is fixed; the interval is random. The probability statement applies to the procedure, not to the particular interval.
说「μ 有 95% 的概率落在这个特定区间内」是错误的。参数 μ 是固定的，区间是随机的。概率描述针对的是构造方法本身，而非某一个特定区间。
7. Margin of Error and Interval Width | 边际误差与区间宽度
The margin of error E = critical value × standard error determines the width of the CI. Three factors affect E:
边际误差 E = 临界值 × 标准误，它决定了置信区间的宽度。影响 E 的因素有三个：
- Confidence level: higher C → larger z*/t* → wider interval 置信水平越高 → 临界值越大 → 区间越宽
- Sample size n: larger n → smaller standard error → narrower interval 样本量越大 → 标准误越小 → 区间越窄
- Variability σ or s: more variability → wider interval 变异性越大 → 区间越宽
To halve the margin of error, we must multiply the sample size by 4, because E ∝ 1/√n.
要使边际误差减半，样本量需扩大为原来的 4 倍，因为 E ∝ 1/√n。
8. Choosing the Sample Size | 确定样本量
Before collecting data, we often want the smallest n that achieves a desired margin of error E at a given confidence level:
在收集数据之前，我们往往需要确定在给定置信水平下达到目标边际误差 E 所需的最小样本量：
n = (z* × σ / E)²
This formula requires a known σ; if σ is unknown, we can use a pilot study’s s or a conservative estimate. Always round n up to the nearest whole number.
该公式需要已知 σ；若 σ 未知，可用预实验的标准差 s 或保守估计值。n 必须向上取整到最接近的整数。
9. Worked Example | 例题演示
A random sample of 25 students has a mean score of x̄ = 72 with a sample standard deviation s = 8. Find a 95% confidence interval for the population mean score.
随机抽取 25 名学生，其平均分数 x̄ = 72，样本标准差 s = 8。求总体平均分 μ 的 95% 置信区间。
Since σ is unknown, we use the t-interval with df = 24. From the GDC or t-table, t* = 2.064. The standard error is 8/√25 = 1.6, and the margin of error is 2.064 × 1.6 = 3.30.
由于 σ 未知，我们使用自由度为 24 的 t 区间。查表或使用 GDC 得 t* = 2.064。标准误为 8/√25 = 1.6，边际误差为 2.064 × 1.6 = 3.30。
72 ± 3.30 → (68.70, 75.30)
We are 95% confident that the true mean lies between 68.70 and 75.30.
我们有 95% 的把握认为总体均值介于 68.70 与 75.30 之间。
10. Common Errors in Exams | 考试常见错误
- Using z instead of t when σ is unknown — 当 σ 未知时误用 z 分布
- Misinterpreting the confidence level as a probability statement about μ — 将置信水平误读为关于 μ 的概率陈述
- Forgetting to check normality conditions for small samples — 忘记检验小样本的正态性条件
- Using the wrong degrees of freedom — 使用错误的自由度
- Confusing sample size n with degrees of freedom n − 1 — 混淆样本量 n 与自由度 n − 1
Always state the interval with the correct units and write a full interpretive sentence in the exam.
考试中务必写出带正确单位的区间，并完整地写出一句解读性文字。
11. GDC Skills | 计算器技能
In IB exams, using the GDC is often faster and more accurate than manual computation. Key menus include:
在 IB 考试中，使用 GDC 常比手算更快更准确。常用菜单包括：
- STAT → TESTS → ZInterval (for σ known) — ZInterval（σ 已知）
- STAT → TESTS → TInterval (for σ unknown) — TInterval（σ 未知）
- invNorm / inverse t for critical values — 使用 invNorm / 逆 t 求临界值
Remember to select “Stats” mode and enter summary statistics correctly; a single misplaced digit changes the entire answer.
记得选择 “Stats” 模式并正确输入汇总统计量；一个数字输错就会导致整个答案出错。
12. Summary | 要点总结
The confidence interval for a population mean is a fundamental inferential tool. Choose z when σ is known (or n ≥ 30), and t when σ is
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