📚 Mean of a Normal Distribution with Unknown Variance | 方差未知时正态分布均值的推断
When we estimate or test the mean of a normal distribution, the population variance σ² is rarely known in real data problems. In A-Level Edexcel Statistics, replacing the unknown σ with the sample standard deviation s changes the sampling distribution from the normal z-distribution to Student's t-distribution, provided the underlying population is normal. This article covers confidence intervals, hypothesis tests, critical values, and exam-style worked examples for the mean with unknown variance.
在实际数据问题中，正态总体的方差 σ² 很少是已知的。在 A-Level Edexcel 统计学中，用样本标准差 s 代替未知的 σ 后，抽样分布会从正态 z 分布变为学生 t 分布，前提是总体服从正态分布。本文涵盖方差未知时均值的置信区间、假设检验、临界值以及考试型例题。
1. The Core Problem: Unknown σ | 核心问题：σ 未知
If X ~ N(μ, σ²) and σ is known, the sample mean x̄ from a random sample of size n satisfies x̄ ~ N(μ, σ²/n), so the standardised statistic z = (x̄ − μ) / (σ / √n) follows N(0, 1).
如果 X ~ N(μ, σ²) 且 σ 已知，来自容量为 n 的随机样本的样本均值 x̄ 满足 x̄ ~ N(μ, σ²/n)，因此标准化统计量 z = (x̄ − μ) / (σ / √n) 服从 N(0, 1)。
In most real examinations and experiments, σ is not known. We must estimate it from the same sample using the sample standard deviation s. This extra uncertainty from estimating σ makes the normal approximation too narrow, especially for small samples.
在大多数考试题和实验中，σ 是未知的。我们必须用同一样本的样本标准差 s 来估计它。这种因估计 σ 带来的额外不确定性使正态近似的置信区间或检验过于窄，尤其在小样本时。
2. From Z to T: Replacing σ by s | 从 Z 到 T：用 s 替换 σ
The natural idea is to replace σ by s in the z statistic. The resulting statistic is not normally distributed; it has a t-distribution with n − 1 degrees of freedom, assuming the sample comes from a normal population.
一个自然的想法是在 z 统计量中用 s 替换 σ。所得统计量并不服从正态分布；在样本来自正态总体的假设下，它服从自由度为 n − 1 的 t 分布。
t = ( x̄ − μ ) / ( s / √n ) ~ t( n − 1 )
Here s is the sample standard deviation calculated using the divisor n − 1. Using n − 1 rather than n gives an unbiased estimate of σ² and matches the degrees of freedom of the t-distribution.
这里 s 是用除数 n − 1 计算的样本标准差。使用 n − 1 而不是 n 可以得到 σ² 的无偏估计，并与 t 分布的自由度保持一致。
3. The t-Distribution | t 分布
The t-distribution is symmetric about zero and bell-shaped like the standard normal distribution, but it has heavier tails. This means that for the same tail probability, the t critical value is larger than the z critical value, reflecting the extra uncertainty from estimating σ.
t 分布关于 0 对称，形状与标准正态分布相似，但尾部更厚。这意味着在相同的尾部概率下，t 临界值比 z 临界值更大，反映了估计 σ 带来的额外不确定性。
Key properties of the t-distribution include:
t 分布的主要性质包括：
- It is symmetric, so P(T ≤ −a) = P(T ≥ a).
它是对称的，因此 P(T ≤ −a) = P(T ≥ a)。 - It has one parameter, ν = n − 1, called the degrees of freedom.
它有一个参数 ν = n − 1，称为自由度。 - As ν increases, the t-distribution approaches N(0, 1); for large n, z and t critical values become almost equal.
随着 ν 增大，t 分布趋近于 N(0, 1)；当 n 较大时，z 和 t 临界值几乎相等。
4. Degrees of Freedom
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