📚 Confidence Intervals for a Population Mean with Unknown Variance (Data) | 方差未知时总体均值的置信区间（数据）
When the population variance is unknown, constructing a confidence interval for the population mean requires the t-distribution. This article explains the theory, outlines the step-by-step calculation, and works through a data example.
当总体方差未知时，为总体均值构造置信区间需要使用 t 分布。本文将解释相关理论、逐步计算方法，并通过一个数据示例进行演示。
1. Population Mean and Variance: Known vs Unknown | 总体均值与总体方差：已知与未知
Statistical inference often focuses on the population mean μ. If the population variance σ² is known, a z-interval based on the standard normal distribution is appropriate. If σ² is unknown, we must estimate it from the sample using the sample variance s², which introduces extra uncertainty.
统计推断通常关注总体均值 μ。如果总体方差 σ² 已知，可以使用基于标准正态分布的 z 区间。如果 σ² 未知，我们只能利用样本方差 s² 来估计它，这会引入额外的不确定性。
The sample mean x̄ is an unbiased estimator of μ, and the sample standard deviation s is an estimator of σ. Because s varies from sample to sample, the width of the confidence interval must account for this variation. Using the t-distribution achieves this automatically.
样本均值 x̄ 是 μ 的无偏估计量，样本标准差 s 是 σ 的估计量。由于 s 会随样本不同而变化，置信区间的宽度必须考虑这一波动。使用 t 分布可以自动实现这一要求。
2. Why Use the t-Distribution | 为什么使用 t 分布
The t-distribution is used when the population variance is unknown because it accounts for the extra variability introduced by estimating σ with s. As a result, t-intervals are generally wider than z-intervals with the same confidence level, especially for small samples.
当总体方差未知时使用 t 分布，是因为它考虑了用 s 估计 σ 所带来的额外变异。因此，在相同置信水平下，t 区间通常比 z 区间更宽，在小样本时尤为明显。
As the sample size n increases, the t-distribution approaches the standard normal distribution. This means the difference between the t and z critical values becomes negligible for large n, but for small n the correct t critical value is essential for accurate intervals.
随着样本量 n 增大，t 分布趋近于标准正态分布。这意味着当 n 很大时，t 与 z 临界值之间的差异可以忽略，但在小样本时，使用正确的 t 临界值对于构造准确的区间至关重要。
3. The t-Distribution and Degrees of Freedom | t 分布与自由度
The shape of the t-distribution is controlled by the degrees of freedom, abbreviated df. For a single-sample inference about the mean, the degrees of freedom are df = n − 1, where n is the sample size.
t 分布的形状由自由度 df 控制。对于关于均值的单样本推断，自由度 df = n − 1，其中 n 是样本量。
The degrees of freedom represent the number of independent pieces of information available in the sample. The t-distribution has thicker tails when df is small, reflecting more uncertainty; as df increases, it looks increasingly like the normal distribution.
自由度代表样本中可用的独立信息量。当 df 较小时，t 分布尾部更厚，体现更大的不确定性；随着 df 增大，它越来越接近正态分布。
4. The Confidence Interval Formula | 置信区间公式
The general form of a confidence interval for a population mean when σ² is unknown is given by:
当 σ² 未知时，总体均值的置信区间一般形式为：
x̄ ± t* · (s / √n)
In this formula, x̄ is the sample mean, s is the sample standard deviation, n is the sample size, and t* is the critical t-value with n − 1 degrees of freedom that puts an area of α/2 in the upper tail. For a 95% confidence level, α = 0.05, and the area in each tail is 0.025.
在该公式中，x̄ 是样本均值，s 是样本标准差，n 是样本量，t* 是具有 n − 1 个自由度、使得上侧尾部面积为 α/2 的临界 t 值。对于 95% 置信水平，α = 0.05，每侧尾部面积为 0.025。
5. Step-by-Step Procedure | 构建置信区间的步骤
To construct a confidence interval for the population mean with unknown variance, follow these steps.
当方差未知时，为总体均值构造置信区间的步骤如下。
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Check the conditions: the sample must be a simple random sample, and either the population distribution is approximately normal or the sample size is large enough for the Central Limit Theorem to apply.
检查条件：样本必须是简单随机样本，且总体分布近似正态，或者样本量足够大从而使中心极限定理成立。
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Calculate the sample mean x̄ and the sample standard deviation s from the data.
根据数据计算样本均值 x̄ 和样本标准差 s。
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Choose the confidence level, for example 95%. Determine α = 1 − confidence level, and find the critical t* with df = n − 1 and tail area α/2.
选择置信水平，例如 95%。确定 α = 1 − 置信水平，并查找 df = n − 1、尾部面积为 α/2 的临界值 t*。
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Compute the margin of error E = t* × (s / √n).
计算误差幅度 E = t* × (s / √n)。
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Write the confidence interval as x̄ − E < μ < x̄ + E, or equivalently (x̄ − E, x̄ + E).
将置信区间写为 x̄ − E < μ < x̄ + E，或等价地 (x̄ − E, x̄ + E)。
6. Worked Example: 95% Confidence Interval from Sample Data | 数值示例：由样本数据计算 95% 置信区间
Suppose a teacher records the test scores of 10 randomly selected students from a large class: 68, 72, 75, 78, 80, 82, 83, 85, 88, 91. The population variance is unknown, so we will construct a 95% confidence interval for the mean test score.
假设一位老师记录了一个大班中随机抽取的 10 名学生的测验成绩：68, 72, 75, 78, 80, 82, 83, 85, 88, 91。总体方差未知，因此我们为平均成绩构造 95% 置信区间。
| 68 | 72 | 75 | 78 | 80 | 82 | 83 | 85 | 88 | 91 |
The sample mean is x̄ = 80.2. The sample standard deviation is calculated using the formula s = √[ Σ(xᵢ − x̄)² / (n − 1) ] = √(459.6 / 9) ≈ 7.15.
样本均值为 x̄ = 80.2。样本标准差按公式 s = √[ Σ(xᵢ − x̄)² / (n − 1) ] 计算，得到 s = √(459.6 / 9) ≈ 7.15。
For a 95% confidence level and df = 9, the critical t-value is t* = 2.262. The margin of error is E = 2.262 × (7.15 / √10) ≈ 2.262 × 2.26 ≈ 5.11.
对于 95% 置信水平和 df = 9，临界 t 值为 t* = 2.262。误差幅度为 E = 2.262 × (7.15 / √10) ≈ 2.262 × 2.26 ≈ 5.11。
Therefore, the 95% confidence interval for the population mean is 80.2 ± 5.11, giving (75.09, 85.31).
因此，总体均值的 95% 置信区间为 80.2 ± 5.11，即 (75.09, 85.31)。
7. Using a Calculator or GDC | 使用计算器或 GDC
Many graphing calculators have a built-in t-interval function. For example, on a TI-84 calculator, press STAT, choose TESTS, then select TInterval. You can enter the data list or the summary statistics x̄, s, and n.
许多图形计算器内置了 t 区间功能。例如，在 TI-84 计算器上，按 STAT，选择 TESTS，然后选 TInterval。你可以输入数据列表，或输入汇总统计量 x̄、s 和 n。
After choosing the confidence level, the calculator outputs the confidence interval directly. This is a quick validation tool, but understanding the underlying formula and assumptions is still essential for the IB exam.
选择置信水平后，计算器会直接输出置信区间。这是快捷的验证工具，但理解背后的公式和假设对于 IB 考试仍然至关重要。
8. Interpreting Confidence Intervals and Confidence Level | 解释置信区间与置信水平
A 95% confidence interval does not mean the probability that μ lies inside this particular interval is 95%. Instead, it means that if we repeated the sampling process many times and constructed the interval each time, approximately 95% of those intervals would contain the true population mean.
95% 置信区间并不表示 μ 落在这个特定区间内的概率为 95%。它的含义是：如果多次重复抽样过程并每次都构造区间，那么大约 95% 的区间会包含真实的总体均值。
Thus, we say we are “95% confident” that the true mean lies between the lower and upper bounds of the calculated interval. This interpretation is a common exam question, so it must be expressed carefully.
因此，我们说“有 95% 的置信度”认为真实均值位于所计算区间的上下限之间。这种解释是常见的考点，需要表述准确。
9. Assumptions and Common Mistakes | 假设与常见错误
The t-interval relies on three key assumptions: independent random sampling, an approximately normal population distribution (or a large enough sample), and no extreme outliers. Violating these assumptions can make the interval unreliable.
t 区间依赖三个关键假设：独立随机抽样、总体分布近似正态（或样本量足够大）、以及没有极端异常值。违反这些假设会使区间不可靠。
Common mistakes include using a z-value instead of a t-value, confusing σ with s, using the wrong degrees of freedom, or misinterpreting the confidence level. For example, using z* = 1.96 instead of t* = 2.262 in the example above would produce a narrower and incorrectly precise interval.
常见错误包括：使用 z 值而不是 t 值、混淆 σ 和 s、使用错误的自由度、或错误解释置信水平。例如，在上面的例子中用 z* = 1.96 代替 t* = 2.262 会得到更窄但不正确的区间。
10. Summary | 总结
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