📚 4C Confidence interval for the population mean | 总体均值的置信区间
In statistical inference we often need to estimate an unknown population parameter, such as the mean μ. A confidence interval provides a range of plausible values for μ, based on sample data, together with a specified level of confidence. This topic, frequently labelled as 4C in the Cambridge syllabus, covers the construction and interpretation of confidence intervals for a population mean, both when the population standard deviation is known (using the z‑distribution) and when it is unknown (using the t‑distribution).
在统计推断中，我们常常需要估计未知的总体参数，例如均值μ。置信区间基于样本数据，给出了μ的一个合理取值范围，并附有一个指定的置信水平。本主题在剑桥课程中通常编号为4C，涵盖总体均值置信区间的构造与解释，包括已知总体标准差（使用z分布）和未知总体标准差（使用t分布）两种情况。
1. Point Estimation and Its Limitations | 点估计及其局限性
A natural starting point for estimating μ is the sample mean x̄. The value x̄ is called a point estimate because it gives a single number as the best guess for the population mean.
估计μ的自然起点是样本均值x̄。数值x̄被称为点估计，因为它只用一个单一的数值作为对总体均值的最佳猜测。
However, a point estimate does not convey any information about the precision or reliability of the estimate. For example, stating that the average height of students is 170 cm based on a sample of 10 does not tell us how close this estimate is to the true population mean.
然而，点估计并不能传递关于估计精确性或可靠性的任何信息。例如，基于10个样本声称学生平均身高为170厘米，并不能告诉我们这个估计与真实的总体均值有多接近。
To address this shortcoming, we use an interval estimate — a range of values that is likely to contain the true parameter with a specified probability.
为了弥补这一缺陷，我们使用区间估计——一个很可能以指定概率包含真实参数的值域。
2. The Concept of an Interval Estimate | 区间估计的概念
An interval estimate is constructed around the point estimate, extending equally on both sides to form a range. The width of the interval reflects the uncertainty associated with sampling variability.
区间估计围绕点估计构建，向两侧对称延伸形成一个范围。区间的宽度反映了与抽样变异性相关的不确定性。
If we were to repeat the sampling procedure many times and construct a 95 % confidence interval each time, we would expect about 95 % of those intervals to contain the true population mean μ.
如果我们多次重复抽样过程，并且每次都构建一个95 %置信区间，那么大约有95 %的区间会包含真实的总体均值μ。
This frequency‑based interpretation is central to the concept of confidence intervals. It is essential to note that the confidence level refers to the long‑run success rate of the method, not to the probability that a specific computed interval contains μ.
这种基于频率的解释是置信区间概念的核心。需要特别指出的是，置信水平指的是该方法在长期运行中的成功率，而不是某个具体计算出的区间包含μ的概率。
3. Confidence Level (1 − α) and Significance α | 置信水平与显著性水平α
The confidence level, denoted 1 − α, is chosen by the researcher. Common choices are 90 %, 95 % and 99 %. The complementary probability α is the significance level, which represents the risk that the interval does not contain the parameter.
置信水平记为1 − α，由研究者选定。常见的选择有90 %、95 %和99 %。互补概率α是显著性水平，它表示区间不包含该参数的风险。
For a 95 % confidence level, α = 0.05. Since we construct a two‑ terval, the critical value corresponds to the upper α/2 tail of the sampling distribution.
对于95 %的置信水平，α = 0.05。因为我们构建的是双侧对称区间，临界值对应抽样分布的上侧α/2分位数。
4. The Z-Interval when σ is Known | σ已知时的Z区间
When the population standard deviation σ is known, the sampling distribution of the sample mean x̄ is approximately normal (by the Central Limit Theorem) with mean μ and standard error σ/√n, provided the sample size n is sufficiently large or the population is itself normal.
当总体标准差σ已知时，只要样本量n足够大或总体本身服从正态分布，样本均值x̄的抽样分布就近似服从正态分布（根据中心极限定理），均值为μ，标准误差为σ/√n。
The confidence interval is given by:
置信区间由下式给出：
x̄ ± z* × (σ ÷ √n)
tical value from the standard normal distribution such that P(−z* < Z < z*) = 1 − α.
其中z*是来自标准正态分布的临界值，满足P(−z* < Z < z*) = 1 − α。
We often write the interval as ( x̄ − z* × σ/√n , x̄ + z* × σ/√n ).
我们通常将区间写作 ( x̄ − z* × σ/√n , x̄ + z* × σ/√n )。
5. Margin of Error and Width | 误差边际与宽度
The quantity E = z* × (σ ÷ √n) is called the margin of error. It determines the half‑width of the confidence interval, the full width being 2E.
量E = z* × (σ ÷ √n)称为误差边际。它决定了置信区间的半宽，全宽为2E。
A larger sample size n reduces the standard error, thereby decreasing the margin of error and yielding a narrower, more precise interval.
较大的样本量n会减小标准误差，从而减小误差边际，得到更窄、更精确的区间。
Conversely, a higher confidence level increases the critical value z*, which widens the interval. There is therefore a trade‑off between confidence and precision.
反之，较高的置信水平会增大临界值z*，从而使区间变宽。因此，在置信程度与精确性之间存在着权衡。
6. Critical Values from the Standard Normal | 标准正态分布的临界值
For commonly used confidence levels, the critical values z* (also written as zα/2) are obtained from standard normal tables or memorised:
对于常用的置信水平，临界值z*（也写作zα/2）可从标准正态表查得或直接记忆：
| Confidence Level | 置信水平 | α | α/2 | 广告 / Sponsored 现货 Cambridge International AS & ALevel IT Coursebook剑桥国际IGC… 计算机 备考教辅 · 正版现货 · 与本文内容相关 查看详情 / View on Taobao参考价 ¥467.0
更多咨询请联系16621398022（同微信） 备考资讯每日更新，关注不错过 考纲变化 · 科目分析 · 学习方法  在微信里长按二维码识别关注 或截图后用「扫一扫 → 相册」选择本图 公众号：tutorhao |
|---|