📚 Hypothesis Testing for the Variance of a Normal Distribution | 正态分布方差的假设检验
In A-Level Further Mathematics, hypothesis testing is not limited to the mean of a normal distribution. When the population variance is unknown, we can use the chi-squared (χ²) distribution to test claims about the population variance σ². This is a key topic in Edexcel Further Statistics, and it requires a solid understanding of the chi-squared distribution, degrees of freedom, and the assumptions behind the test.
在 A-Level 进阶数学中，假设检验不仅限于正态分布的均值。当总体方差未知时，我们可以使用卡方（χ²）分布来检验关于总体方差 σ² 的假设。这是 Edexcel 进阶统计的核心考点，要求学生对卡方分布、自由度以及检验背后的假设条件有扎实的理解。
1. When to Use the Chi-Squared Test for Variance | 何时使用卡方检验方差
The chi-squared test for variance is used when we have a random sample taken from a normally distributed population, and we want to test a claim about the population variance σ². This is different from the z-test or t-test, which focus on the population mean. The test statistic follows a chi-squared distribution with n − 1 degrees of freedom, where n is the sample size.
方差卡方检验适用于从正态分布总体中抽取随机样本的情况，用于检验关于总体方差 σ² 的假设。这与检验总体均值的 z 检验或 t 检验不同。该检验统计量服从自由度为 n − 1 的卡方分布，其中 n 为样本容量。
The sample variance must be calculated using the formula with n − 1 in the denominator, often denoted as s². This is the unbiased estimator of the population variance. If your sample data is given as raw values, you will need to compute s² first; if the sample variance is given directly, you can proceed with the test.
样本方差必须使用分母为 n − 1 的公式计算，通常记为 s²。这是总体方差的无偏估计量。如果题目给出的是原始数据，你需要先计算 s²；如果直接给出样本方差，则可以直接进行检验。
2. The Chi-Squared Distribution | 卡方分布
The chi-squared distribution is a continuous probability distribution that is defined for positive values only. It is positively skewed, but as the degrees of freedom increase, the distribution becomes more symmetric and approaches a normal shape. The shape depends entirely on the degrees of freedom, denoted ν (Greek letter nu).
卡方分布是一种连续概率分布，仅定义在正数范围内。它是正偏态的，但随着自由度的增加，分布逐渐变得对称，并趋近于正态分布形态。分布的形状完全取决于自由度，用 ν（希腊字母 nu）表示。
For a random sample of size n from a normal population, the quantity (n − 1)s² / σ² follows a chi-squared distribution with n − 1 degrees of freedom. This is the fundamental result that underpins the variance hypothesis test. You will use chi-squared tables to find critical values, or your calculator may provide inverse chi-squared functions.
对于来自正态总体的容量为 n 的随机样本，量 (n − 1)s² / σ² 服从自由度为 n − 1 的卡方分布。这是方差假设检验的基本原理。你需要查卡方分布表来找到临界值，或者使用计算器上的卡方分布逆函数。
3. Hypotheses and Test Statistic | 原假设与检验统计量
The null hypothesis is always written as H₀: σ² = σ₀², where σ₀² is the claimed population variance. The alternative hypothesis can be one-tailed or two-tailed, depending on the wording of the question: H₁: σ² < σ₀² (one-tailed lower), H₁: σ² > σ₀² (one-tailed upper), or H₁: σ² ≠ σ₀² (two-tailed).
原假设始终记为 H₀: σ² = σ₀²，其中 σ₀² 是声称的总体方差。备择假设可以是单尾或双尾，具体取决于题目的表述：H₁: σ² < σ₀²（单尾下限）、H₁: σ² > σ₀²（单尾上限）或 H₁: σ² ≠ σ₀²（双尾）。
The test statistic is calculated using the formula:
检验统计量使用以下公式计算：
χ² = (n − 1)s² / σ₀²
where n is the sample size, s² is the sample variance, and σ₀² is the value claimed in the null hypothesis. This test statistic is then compared to the critical value from the chi-squared distribution with n − 1 degrees of freedom.
其中 n 是样本容量，s² 是样本方差，σ₀² 是原假设中的声称值。然后将该检验统计量与自由度为 n − 1 的卡方分布临界值进行比较。
4. Degrees of Freedom | 自由度
The degrees of freedom for this test are always n − 1, where n is the sample size. This is because we lose one degree of freedom when we estimate the population mean by the sample mean, which is required to calculate the sample variance. It is crucial to use the correct degrees of freedom when looking up critical values.
该检验的自由度始终为 n − 1，其中 n 是样本容量。这是因为在计算样本方差时，我们用样本均值估计总体均值，因此损失了一个自由度。在查临界值时，使用正确的自由度至关重要。
For example, if a sample of 20 observations yields a sample variance of 6.5, the test statistic will follow a chi-squared distribution with 19 degrees of freedom. If you accidentally use 20 degrees of freedom, your critical values will be incorrect and you may reach the wrong conclusion.
例如，如果 20 个观测值的样本产生样本方差为 6.5，则检验统计量将服从自由度为 19 的卡方分布。如果你错误地使用了 20 个自由度，临界值将是错误的，并可能导致错误的结论。
5. Critical Values and Rejection Regions | 临界值与拒绝域
The critical values for a chi-squared test are found from the chi-squared distribution table. For a one-tailed test at significance level α, the critical value is χ²_α(n − 1) for an upper-tailed test, or χ²_(1−α)(n − 1) for a lower-tailed test. For a two-tailed test at significance level α, the critical values are χ²_(α/2)(n − 1) and χ²_(1−α/2)(n − 1).
卡方检验的临界值从卡方分布表中查找。对于显著性水平为 α 的单尾检验，上尾检验的临界值为 χ²_α(n − 1)，下尾检验的临界值为 χ²_(1−α)(n − 1)。对于显著性水平为 α 的双尾检验，临界值为 χ²_(α/2)(n − 1) 和 χ²_(1−α/2)(n − 1)。
The rejection region is the set of values of the test statistic that lead to rejecting the null hypothesis. For an upper-tailed test, reject H₀ if χ²_calculated > χ²_α; for a lower-tailed test, reject H₀ if χ²_calculated < χ²_(1−α); for a two-tailed test, reject H₀ if χ²_calculated < χ²_(1−α/2) or χ²_calculated > χ²_(α/2).
拒绝域是导致拒绝原假设的检验统计量取值集合。对于上尾检验，若χ²计算值 > χ²_α，则拒绝 H₀；对于下尾检验，若χ²计算值 < χ²_(1−α)，则拒绝 H₀；对于双尾检验，若χ²计算值 < χ²_(1−α/2) 或 χ²计算值 > χ²_(α/2)，则拒绝 H₀。
6. Worked Example 1: One-Tailed Upper Test | 实例 1：单尾上尾检验
A machine is designed to produce bolts with a variance in diameter of 0.04 mm². A random sample of 15 bolts is taken, and the sample variance is found to be 0.072 mm². Test at the 5% significance level whether the variance has increased.
某机器生产的螺栓直径方差设计值为 0.04 mm²。随机抽取 15 个螺栓，样本方差为 0.072 mm²。在 5% 显著性水平下检验方差是否增大了。
Step 1: State the hypotheses. H₀: σ² = 0.04, H₁: σ² > 0.04. This is a one-tailed upper test.
第一步：陈述假设。H₀: σ² = 0.04，H₁: σ² > 0.04。这是单尾上尾检验。
Step 2: Calculate the test statistic. Using the formula χ² = (15 − 1)(0.072) / 0.04 = 14 × 0.072 / 0.04 = 1.008 / 0.04 = 25.2.
第二步：计算检验统计量。使用公式 χ² = (15 − 1)(0.072) / 0.04 = 14 × 0.072 / 0.04 = 1.008 / 0.04 = 25.2。
Step 3: Find the critical value. From the chi-squared table with 14 degrees of freedom at the 5% significance level (upper tail), the critical value is χ²_0.05(14) = 23.685.
第三步：查找临界值。自由度为 14、显著性水平为 5%（上尾）的卡方分布表中，临界值为χ²_0.05(14) = 23.685。
Step 4: Compare. Since 25.2 > 23.685, the test statistic falls in the rejection region. Therefore, we reject H₀. There is sufficient evidence at the 5% significance level to conclude that the variance has increased.
第四步：比较。由于 25.2 > 23.685，检验统计量落在拒绝域内。因此，我们拒绝 H₀。在 5% 显著性水平下有充分证据表明方差增大了。
7. Worked Example 2: Two-Tailed Test | 实例 2：双尾检验
A packaging company claims that the variance of the weights of their cereal boxes is 0.25 kg². A consumer group takes a random sample of 20 boxes and finds a sample variance of 0.36 kg². Test at the 10% significance level whether the variance differs from the claimed value.
一家包装公司声称其麦片盒重量的方差为 0.25 kg²。消费者组织随机抽取 20 盒，发现样本方差为 0.36 kg²。在 10% 显著性水平下检验方差是否与声称值不同。
Step 1: State the hypotheses. H₀: σ² = 0.25, H₁: σ² ≠ 0.25. This is a two-tailed test.
第一步：陈述假设。H₀: σ² = 0.25，H₁: σ² ≠ 0.25。这是双尾检验。
Step 2: Calculate the test statistic. χ² = (20 − 1)(0.36) / 0.25 = 19 × 0.36 / 0.25 = 6.84 / 0.25 = 27.36.
第二步：计算检验统计量。χ² = (20 − 1)(0.36) / 0.25 = 19 × 0.36 / 0.25 = 6.84 / 0.25 = 27.36。
Step 3: Find the critical values. For a two-tailed test at the 10% significance level, we split α into two tails of 0.05 each. With 19 degrees of freedom, the critical values are χ²_0.05(19) = 30.144 and χ²_0.95(19) = 10.117.
第三步：查找临界值。对于 10% 显著性水平的双尾检验，我们将 α 分成两个尾各 0.05。自由度为 19 时，临界值为 χ²_0.05(19) = 30.144 和 χ²_0.95(19) = 10.117。
Step 4: Compare. Since 10.117 < 27.36 < 30.144, the test statistic falls in the acceptance region. Therefore, we do not reject H₀. There is insufficient evidence at the 10% significance level to conclude that the variance differs from 0.25 kg².
第四步：比较。由于 10.117 < 27.36 < 30.144，检验统计量落在接受域内。因此，我们不拒绝 H₀。在 10% 显著性水平下没有充分证据表明方差与 0.25 kg² 不同。
8. Assumptions and Common Pitfalls | 假设条件与常见误区
The most important assumption for this test is that the population from which the sample is drawn must be normally distributed. If the population is not normal, the chi-squared test for variance becomes unreliable. This is a common point that examiners like to test.
该检验最重要的假设是样本所来自的总体必须服从正态分布。如果总体不服从正态分布，方差卡方检验将不可靠。这是考官喜欢考查的常见考点。
Another common pitfall is confusing the hypotheses. Remember that we are testing the population variance σ², not the sample variance s². The sample variance is just an estimate. Also, be careful with the degrees of freedom: it is always n − 1, not n. Finally, for a two-tailed test, remember to use α/2 in each tail when finding critical values.
另一个常见误区是混淆假设。记住我们检验的是总体方差 σ²，而不是样本方差 s²。样本方差只是一个估计值。此外，要注意自由度始终是 n − 1，而不是 n。最后，对于双尾检验，查找临界值时要在每个尾部使用 α/2。
Here is a quick comparison of the z-test for mean and the chi-squared test for variance:
以下是均值 z 检验与方差卡方检验的快速比较：
| Feature | 特征 | Z-Test for Mean | 均值 Z 检验 | Chi-Squared Test for Variance | 方差卡方检验 |
| Parameter of interest | 感兴趣的参数 | Population mean μ | 总体均值 μ | Population variance σ² | 总体方差 σ² |
| Test statistic | 检验统计量 | z = (x̄ − μ₀) / (σ / √n) | χ² = (n − 1)s² / σ₀² |
| Distribution | 分布 | Standard normal N(0, 1) | Chi-squared with n − 1 df | 自由度为 n − 1 的卡方分布 |
| Key assumption | 关键假设 | Population variance known or large n | 总体方差已知或 n 较大 | Population is normally distributed | 总体服从正态分布 |
9. Using Calculator Functions | 使用计算器功能
Modern graphing calculators can greatly simplify the process of finding chi-squared critical values. On most calculators, you can use the inverse chi-squared function. For example, on a Casio calculator, you would access the distribution menu, select inverse chi-squared, and enter the tail probability and the degrees of freedom.
现代图形计算器可以大大简化查找卡方临界值的过程。在大多数计算器上，你可以使用卡方分布逆函数。例如，在 Casio 计算器上，你可以进入分布菜单，选择逆卡方，然后输入尾部概率和自由度。
For an upper-tailed test with α = 0.05 and df = 14, you would enter the area to the right of the critical value as 0.05. The calculator will return the critical value χ² = 23.685. For a lower-tailed test, you would enter the area to the left as 0.95, which gives the same numerical result.
对于 α = 0.05、df = 14 的上尾检验，你应输入临界值右侧面积为 0.05。计算器将返回临界值χ² = 23.685。对于下尾检验，你应输入左侧面积为 0.95，得到相同的数值结果。
It is strongly recommended that you show the formula and the substitution in your working, even if you use a calculator to find the critical values. Examiners award method marks for the setup and the comparison, as well as the final conclusion.
强烈建议你在解题过程中写出公式和代入过程，即使你使用计算器查找临界值。考官会根据假设的设定、比较过程以及最终结论给分。
10. Exam Tips and Revision Summary | 考试技巧与复习总结
When tackling a hypothesis test for variance, always follow the structured approach: state the hypotheses clearly, calculate the test statistic accurately, find the correct critical value, make your comparison, and write a conclusion in the context of the question. Setting out your work in this way will maximise your marks.
在解答方差假设检验题目时，务必遵循结构化步骤：清晰地陈述假设、准确计算检验统计量、查找正确的临界值、进行比较，并结合题目背景写出结论。按这种方式排列你的解答过程可以最大化你的得分。
A conclusion must always be written in the context of the original problem. For example, instead of writing “we reject H₀”, you should write “we reject the manufacturer’s claim that the variance is 0.04 mm², and conclude that the variance has increased.” This demonstrates a full understanding of the test.
结论必须始终结合原始问题背景来写。例如，不要只写”我们拒绝 H₀”，而应写”我们拒绝制造商关于方差为 0.04 mm² 的声称，并得出结论方差已增大”。这展示了你对检验的全面理解。
Finally, practise with past exam questions. The chi-squared test for variance appears regularly in Edexcel Further Statistics papers. Work through examples with one-tailed and two-tailed tests, and make sure you are comfortable with finding critical values both from tables and using calculator functions.
最后，多做历年真题。方差卡方检验在 Edexcel 进阶统计试卷中经常出现。多练习单尾和双尾检验的题目，并确保你能够熟练地从表格和计算器功能中查找临界值。
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