Confidence Intervals and Tests Using the t-Distribution | 基于 t 分布的置信区间与假设检验

📚 Confidence Intervals and Tests Using the t-Distribution | 基于 t 分布的置信区间与假设检验

In A-Level Statistics, many inference methods begin with the normal distribution. If the population standard deviation σ is known, the sample mean can be standardised using z = (x̄ − μ)/(σ/√n). However, in practice σ is rarely known and must be estimated from the sample. When σ is replaced by the sample standard deviation s, the sampling distribution is no longer exactly normal for small samples; it is modelled by Student’s t-distribution.

在 A-Level 统计学中，许多推断方法都从正态分布开始。若总体标准差 σ 已知，可以用 z = (x̄ − μ)/(σ/√n) 对样本均值进行标准化。但实际中 σ 很少已知，必须用样本去估计。当用样本标准差 s 替代 σ 时，小样本下抽样分布不再恰好是正态分布，而是用学生 t 分布来建模。

1. When to Use the t-Distribution | 何时使用 t 分布

Use the t-distribution when the population distribution is assumed to be normal, the population standard deviation σ is unknown, and the sample size n is small, typically n < 30. If n is large, the t-distribution is close to the normal distribution, but exam questions often require the t-distribution whenever σ is unknown and the sample size is not large enough to justify the normal approximation.

当总体分布假定为正态、总体标准差 σ 未知且样本量 n 较小（通常 n < 30）时，应使用 t 分布。如果 n 很大，t 分布接近正态分布，但只要 σ 未知且样本量不足以支持正态近似，考试题通常要求使用 t 分布。

If σ is known and the population is normal, use the z-procedure. If σ is unknown, the t-procedure accounts for the extra uncertainty introduced by estimating σ with s.

如果 σ 已知且总体正态，则使用 z 方法。如果 σ 未知，t 方法可以反映用 s 估计 σ 时引入的额外不确定性。

2. Degrees of Freedom and Shape of the t-Distribution | 自由度与 t 分布的形状

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