Confidence Interval for a Mean with Unknown Variance | 方差未知时均值的置信区间

📚 Confidence Interval for a Mean with Unknown Variance | 方差未知时均值的置信区间

When we construct a confidence interval for a population mean, we often do not know the population standard deviation σ. In such cases, we must estimate σ using the sample standard deviation s. This additional uncertainty requires a different distribution called the t-distribution, which leads to wider intervals compared to the z-interval. This chapter explores the theory and application of the confidence interval for a mean with unknown variance, an essential topic in IB Mathematics.

当我们构建总体均值的置信区间时，常常不知道总体标准差 σ。此时必须用样本标准差 s 来估计 σ。这种额外的不确定性需要使用另一种分布——t 分布，其区间比基于 z 分布的区间更宽。本章探讨方差未知时均值置信区间的理论与应用，这是 IB 数学的重要内容。

1. Introduction: Why Unknown Variance Matters | 为何方差未知如此重要

In real-world data analysis, the population variance is rarely known. If we incorrectly assume σ is known and use the z-interval, the confidence interval will be too narrow, leading to overconfidence in the estimate. Using the t-distribution adjusts for the extra variability introduced by estimating σ with s.

在真实数据分析中，总体方差很少已知。如果我们错误地假设 σ 已知并采用 z 区间，置信区间会偏窄，导致对估计值过于自信。使用 t 分布可以修正用 s 估计 σ 带来的额外变异性。

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2. Estimating the Population Standard Deviation | 估计总体标准差

The sample standard deviation s is calculated from the data using the formula with (n−1) in the denominator. This ensures s is an unbiased estimator of the population standard deviation σ.

样本标准差 s 根据数据计算，分母使用 (n−1)。这可以保证 s 是总体标准差 σ 的无偏估计量。

s = √( Σ(x_i − x̄)² / (n − 1) )

s = √( Σ(x_i − x̄)² / (n − 1) )

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3. Student's t-Distribution | 学生 t 分布

The t-distribution is a family of symmetric, bell-shaped curves that depend on the degrees of freedom. Compared with the standard normal distribution, it has heavier tails, reflecting greater uncertainty when σ must be estimated from the sample.

t 分布是一族对称的钟形曲线，其形状取决于自由度。与标准正态分布相比，它的尾部更厚，反映出由样本估计 σ 时更大的不确定性。

As the degrees of freedom increase, the t-distribution approaches the standard normal distribution N(0,1). This means that for very large samples, the t-interval and the z-interval give nearly identical results.

随着自由度增大，t 分布趋近于标准正态分布 N(0,1)。这意味着对于非常大的样本，t 区间和 z 区间几乎完全一致。

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4. Degrees of Freedom (ν = n − 1) | 自由度

For a single sample used to estimate a mean, the degrees of freedom ν equals n − 1. One degree of freedom is lost because we used the sample mean x̄ to estimate μ, which imposes one constraint on the deviations (x_i − x̄).

对于估计均值所用的单个样本，自由度 ν 等于 n − 1。消耗的一个自由度是因为我们使用了样本均值 x̄ 来估计 μ，这给离差 (x_i − x̄) 施加了一个约束。

ν = n − 1

ν = n − 1

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5. Confidence Interval Formula Using t | 用 t 构建置信区间公式

When the population standard deviation σ is unknown, the confidence interval for the population mean μ is based on the t-statistic. The general formula for a (1−α)·100% confidence interval is:

当总体标准差 σ 未知时，总体均值 μ 的置信区间基于 t 统计量。置信水平为 (1−α)·100% 的区间公式为：

x̄ ± t_{α/2, ν} · (s / √n)

x̄ ± t_{α/2, ν} · (s / √n)

Here, t_{α/2, ν} is the critical value from the t-distribution with ν degrees of freedom that captures the central (1−α) area, s is the sample standard deviation, and n is the sample size.

式中，t_{α/2, ν} 是自由度为 ν 的 t 分布中对应于中部面积 (1−α) 的临界值，s 为样本标准差，n 为样本量。

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6. Finding Critical t-Values | 查找临界 t 值

y attention to whether the table provides one-tailed or two-tailed probabilities. For a 95% confidence interval, the two-tailed α is 0.05, so each tail has α/2 = 0.025.

临界 t 值可通过 t 分布表或图形计算器查找。务必注意表格提供的概率是单侧还是双侧。对于 95% 置信区间，双侧 α 为 0.05，故每侧尾部为 α/2 = 0.025。

ν	t0.025 (two‑tailed)
8	2.306
11	2.201
20	2.086
30	2.042

For example, with n = 12 (ν = 11), the critical value t_0.025,11 .201. Notice that this is greater than the z-value 1.96, reflecting the extra uncertainty.

例如，当 n = 12 (ν = 11) 时，临界值 t_0.025,11 ≈ 2.201。注意它比 z 值 1.96 大，体现了额外的不确定性。

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7. Assumptions for the t-Interval | t 区间的假设条件

The t-confidence interval for a mean is valid when the following assumptions are reasonably met:

t 置信区间在以下假设条件基本满足时有效：

If normality is in doubt, a larger sample size helps because of the Central Limit Theorem.

样本来自总体的简单随机抽样；观测值相互独立；总体分布近似正态，或样本量足够大（通常 n ≥ 30）。如果正态性存疑，较大的样本量可借助中心极限定理弥补。

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8. Worked Example: 95% Confidence Interval | 计算示例：95% 置信区间

Twelve students took a test. Their scores give a sample mean x̄ = 68 and sample standard deviation s = 8. Construct a 95% confidence interval for the true mean test score.

随机抽取 12 名学生进行测试，样本均值 x̄ = 68，样本标准差 s = 8。求总体平均成绩的 95% 置信区间。

Step 1: Compute the standard error SE = s / √n = 8 / √12 ≈ 2.309.

步骤1：计算标准误 SE = s / √n = 8 / √12 ≈ 2.309。

Step 2: Degrees of freedom ν = 12 − 1 = 11. For 95% confidence, the two-tailed critical value is t_0.025,11 = 2.201.

步骤2：自由度 ν = 12 − 1 = 11。对于 95% 置信水平，双侧临界值 t_0.025,11 = 2.201。

Step 3: Margin of error = t_0.025,11 × SE = 2.201 × 2.309 ≈ 5.08.

步骤3：误差边际 = 2.201 × 2.309 ≈ 5.08。

Step 4: Confidence interval = x̄ ± margin of error = 68 ± 5.08, i.e. (62.92, 73.08).

步骤4：置信区间 = 68 ± 5.08，即 (62.92, 73.08)。

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9. Interpreting the Confidence Interval | 解释置信区间

We say “We are 95% confident that the true population mean test score lies between 62.92 and 73.08.” This interpretation is based on long-run behavior: if we took many random samples of size 12 and constructed such intervals, about 95% of them would contain the true mean μ.

我们说“我们有 95% 的把握认为真正的总体平均测验成绩介于 62.92 到 73.08 之间”。这一解释基于长期行为：假如我们抽取许多大小为 12 的随机样本并构建这样的区间，大约 95% 的区间会包含真值 μ。

It does NOT mean that there is a 95% probability that μ lies in that specific interval. The interval is either correct or not; the confidence level describes the success rate of the method.

这并不意味着 μ 落入该特定区间的概率是 95%。该区间要么包含真值要么不包含；置信水平描述的是方法在重复抽样中的成功率。

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10. Effect of Sample Size on the t-Distribution | 样本量对 t 分布的影响

As n increases, two effects narrow the confidence interval: the t-critical value decreases toward the standard normal z-value, and the standard error s/√n becomes smaller. With ν = 29 (n = 30), the t-critical value for 95% confidence is about 2.045, already very close to 1.96.

随着 n 增大，两种效应使置信区间变窄：临界 t 值减小并趋近标准正态 z 值，同时标准误 s/√n 也变得更小。当 ν = 29 (n = 30) 时，95% 置信水平的临界 t 值约为 2.045，已非常接近 1.96。

For small samples, the heavier tails of the t-distribution produce noticeably wider intervals, correctly reflecting the lack of precise knowledge about σ.

对于小样本，t 分布的厚尾部会导致区间明显更宽，恰当地反映了对 σ 缺乏精确了解的现实。

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11. Comparison: z-Interval vs t-Interval | 比较：z 区间与 t 区间

When σ is known, the z-interval is used: x̄ ± z_α/2 · (σ / √n). This gives a narrower and more precise interval because there is no uncertainty from estimating σ.

当 σ 已知时，使用 z 区间：x̄ ± z_α/2 · (σ / √n)。由于无需估计 σ，该区间更窄、更精确。

When σ is unknown, the t-interval is always wider for the same confidence level and sample size. The difference is most pronounced for n < 30. Always prefer the t-interval unless the true population standard deviation is known.

当 σ 未知时，对于相同的置信水平和样本量，t 区间始终更宽。这一差异在 n < 30 时最为显著。除非已知真正的总体标准差，否则应总是使用 t 区间。

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12. Common Pitfalls and Key Takeaways | 常见错误与关键要点

− 1) and use the correct t-critical value. Ensure the assumptions of randomness, independence, and approximate normality or large n are satisfied.

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