Hypothesis Testing for the Variance of a Normal Distribution | 正态分布方差的假设检验

📚 Hypothesis Testing for the Variance of a Normal Distribution | 正态分布方差的假设检验

In A-Level Statistics, especially within Edexcel S2, we often test claims about population means. However, when the focus is on the consistency of a process, we need to examine the population variance σ². This article presents the chi-squared test for a single normal variance, covering the test statistic, critical regions, p-values, and worked examples designed for A-Level students.

在A-Level统计学（特别是Edexcel S2）中，我们经常检验关于总体均值的断言。但当关注点在于过程的稳定性时，我们需要考察总体方差σ²。本文介绍针对单个正态总体方差的卡方检验，涵盖检验统计量、拒绝域、p值以及适合A-Level学生的例题。

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1. When to Test for Variance? | 何时检验方差？

If we have a single sample from a normal distribution, we may need to verify whether the population variance equals a specified value. For instance, a manufacturer claims that a filling machine operates with a variance of at most 2 ml². Testing such a claim requires a hypothesis test for σ².

当我们有一个来自正态分布的单样本时，可能需要检验总体方差是否等于某个给定值。例如，一个制造商声称灌装机的运行方差不超过2毫升²。检验这类断言就需要对σ²进行假设检验。

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2. The Chi-Squared Distribution | 卡方分布

If X ~ N(µ ared (χ²) distribution with ν = n−1 degrees of freedom. The χ² distribution is defined only for positive values and is right-skewed, especially for small ν.

若 X ~ N(µ, σ²) 且我们抽取容量为 n 的随机样本，样本方差 s² 可用于构建统计量 (n−1)s²/σ²。该统计量服从自由度为 ν = n−1 的卡方（χ²）分布。χ² 分布仅定义在正值上且右偏，尤其当 ν 较小时。

The shape of the χ² distribution becomes more symmetric as ν increases; its mean is ν and its variance is 2ν. Because the distribution is not symmetric, two-tailed critical regions are not symmetric around the centre—you must find separate left and right critical values.

随着 ν 增大，χ² 分布的形状变得更对称；其均值为 ν，方差为 2ν。由于该分布不对称，双尾临界域不以中心对称——必须分别找出左尾和右尾的临界值。

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3. Degrees of Freedom | 自由度

rees of freedom. It represents the number of independent pieces of information used to estimate the population variance. Because we use the sample mean x̄ s², we lose one degree of freedom. Always use n−1, not n, when looking up critical values.

参数 ν = n−1 称为自由度。它表示用于估计总体方差的独立信息数目。由于计算 s² 时使用了样本均值 x̄，我们损失了一个自由度。在查表时务必使用 n−1，而不是 n。

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4. Test Statistic for Variance | 方差的检验统计量

For a hypothesis about σ², the test statistic is:

对于关于 σ² 的假设，检验统计量为：

T = (n − 1) s² / σ₀²

where σ₀² is the hypothesised variance under the null hypothesis H₀. If H₀ is true, T ~ χ²₍ₙ₋₁₎. We will compare the observed value t against critical values or use it to compute a p-value.

其中 σ₀² 是零假设 H₀ 下假定的方差值。若 H₀ 成立，则 T ~ χ²₍ₙ₋₁₎。我们将观测值 t 与临界值作比较，或利用它计算 p 值。

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5. Formulating Hypotheses | 提出假设

one-tailed test). The alternative hypothesis H₁ depends on the claim being tested: σ² < σ₀², σ² > σ₀², or σ² ≠ σ₀². Always define your hypotheses clearly before choosing a critical region.

零假设通常为 H₀: σ² = σ₀²（对于单尾检验有时为 H₀: σ² ≤ σ₀²）。备择假设 H₁ 取决于待检验的断言：σ² < σ₀², σ² > σ₀² 或 σ² ≠ σ₀²。在选择拒绝域之前，一定要先清晰定义假设。

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6. One-Tailed vs Two-Tailed Tests | 单尾与双尾检验

If we suspect the variance has decreased, we set H₁: σ² < σ₀² (lower-tail test). The entire rejection region lies in the left tail of the χ² distribution. If we suspect an increase, we use H₁: σ² > σ₀² (upper-tail test) and reject in the right tail.

若怀疑方差减小，设 H₁: σ² < σ₀²（下尾检验）。整个拒绝域位于 χ² 分布的左尾。若怀疑方差增大，使用 H₁: σ² > σ₀²（上尾检验）并在右尾拒绝。

A two-tailed test is used when we are only interested in whether the variance has changed in either direction: H₁: σ² ≠ σ₀². The rejection region is split equally between the two tails, each with area α/2.

当我们只关心方差是否发生任何方向的变化时，采用双尾检验：H₁: σ² ≠ σ₀²。拒绝域被平分到两尾，每尾面积 α/2。

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7. Critical Region Approach | 临界域方法

Select a significance level α. For a lower-tail test, reject H₀ if T < χ²₁₋α, ν , where χ²₁₋α, ν is the value leaving an area α in the left tail. For an upper-tail test, reject H₀ if T > χ²α, ν (the upper-tail critical value).

选择显著性水平 α。对于下尾检验，若 T < χ²₁₋α, ν 则拒绝 H₀，其中 χ²₁₋α, ν 是使得左尾面积为 α 的值。对于上尾检验，若 T > χ²α, ν（上尾临界值）则拒绝 H₀。

For a two-tailed test, reject H₀ if T < χ²₁₋α/₂, ν or T > χ²α/₂, ν . These critical values are found from χ² tables. Note that χ²₁₋α/₂, ν is much smaller than χ²α/₂, ν .

对于双尾检验，若 T < χ²₁₋α/₂, ν 或 T > χ²α/₂, ν 则拒绝 H₀。这些临界值可从 χ² 表中查到。注意 χ²₁₋α/₂, ν 远小于 χ²α/₂, ν 。

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8. P-Value Method | P值方法

st, find the smaller of the two tail probabilities and double it: p-value = 2 × min[P(χ²ν ≤ t), P(χ²ν ≥ t)]. Reject H₀ if p-value < α.

计算观测检验统计量 t。下尾检验的 p 值 = P(χ²ν ≤ t)；上尾检验的 p 值 = P(χ²ν ≥ t)。对于双尾检验，取两个尾部概率中较小的一个并翻倍：p 值 = 2 × min[P(χ²ν ≤ t), P(χ²ν ≥ t)]。若 p 值 < α，则拒绝 H₀。

In A-Level exams, you may be given probability tables or use a calculator to find tail probabilities. P-value reasoning often strengthens your conclusion.

在 A-Level 考试中，你可能会被提供概率表或用计算器求尾部概率。p 值推理常常能强化你的结论。

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9. Worked Example: One-Tailed Test | 例题：单尾检验

A quality inspector suspects that the variance of the filling machine exceeds the claimed 25 g². A random sample of 10 packets gives a sample variance of 32.4 g². Test, at the 5% significance level, whether the variance has increased, assuming a normal distribution.

一位质量检查员怀疑灌装机的方差超过了所声称的 25 g²。随机抽取 10 包，得到样本方差 32.4 g²。在 5% 显著性水平下检验方差是否增大，假设总体服从正态分布。

= 32.4, ν = 9. Compute t = (9 × 32.4)/25 = 291.6/25 = 11.664. From χ² tables, the upper 5% < 16.919, we do not reject H₀. There is insufficient evidence of an increase in variance.

假设：H₀: σ² = 25，H₁: σ² > 25（上尾）。n = 10,

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