Hypothesis Test for the Mean with Known Variance | 已知方差下的均值假设检验

📚 Hypothesis Test for the Mean with Known Variance | 已知方差下的均值假设检验

In many real-world situations, we know or can safely assume the population variance. For example, machine specifications or long-term historical data often provide a known standard deviation. When the goal is to test a claim about the population mean, we can use a z‑test. This article walks through every aspect of the hypothesis test for a mean with known variance, from setting up the hypotheses to interpreting the final decision.

在许多实际情境中，总体方差是已知的或可以合理假定为已知的。比如机器规格参数或长期历史数据常常会提供已知的标准差。当我们需要检验关于总体均值的某个声明时，就可以使用 z 检验。本文将从建立假设开始，一直走到如何解读最终结论，全面解析方差已知时均值的假设检验。

1. Introduction to Hypothesis Testing | 假设检验介绍

Hypothesis testing is a formal statistical method for making decisions or inferences about a population parameter. It starts with a claim, then uses sample data to assess whether there is enough evidence to reject that claim. In this method, we always assume a certain status quo, and we only reject it if the sample evidence is strong enough.

假设检验是一种正式的统计方法，用于对总体参数做出决策或推断。它从一个声明出发，然后利用样本数据来评估是否有足够的证据拒绝这个声明。在检验过程中，我们始终默认一个现状，只有当样本证据足够强时才会拒绝它。

2. The Null and Alternative Hypotheses | 零假设与备择假设

The null hypothesis, denoted H₀, is a statement of no effect or no difference. For a test of the population mean μ, we often write H₀: μ = μ₀. The alternative hypothesis, H₁, is what we suspect might be true instead. It can take one of three forms: H₁: μ ≠ μ₀ (two‑tailed), H₁: μ > μ₀ (right‑tailed), or H₁: μ < μ₀ (left‑tailed).

零假设，记作 H₀，是一个“无效应”或“无差异”的陈述。对于总体均值 μ 的检验，通常写作 H₀: μ = μ₀。备择假设 H₁ 则是我们怀疑可能成立的陈述。它可以有三种形式：H₁: μ ≠ μ₀（双尾检验），H₁: μ > μ₀（右尾检验），或 H₁: μ < μ₀（左尾检验）。

The choice between a one‑tailed and a two‑tailed test depends entirely on the research question or the wording of the claim. If the claim uses words like ‘higher than’, ‘increased’, or ‘exceeds’, a one‑tailed test is appropriate. If it uses ‘different from’ or ‘changed’, a two‑tailed test should be used.

选择单尾还是双尾检验完全取决于研究问题或声明的措辞。如果声明使用“高于”“增加”“超过”等词语，适合用单尾检验；如果使用“不同于”“发生了变化”等词语，则应使用双尾检验。

3. Significance Level and Critical Region | 显著性水平与拒绝域

The significance level α is the probability of rejecting a true null hypothesis (a Type I error). Common choices are 0.05, 0.01 or 0.10. Before collecting data, we must fix α. The critical region (or rejection region) is the set of values of the test statistic that leads to rejection of H₀. Its boundary is determined by α.

显著性水平 α 是拒绝一个真的零假设的概率（即第一类错误）。常用的 α 取值有 0.05、0.01 或 0.10。在收集数据之前就必须确定 α。拒绝域（或临界区域）是检验统计量的一个取值集合，当检验统计量的值落在这个区域时就拒绝 H₀。拒绝域的边界由 α 决定。

4. Test Statistic for the Mean with Known Variance | 已知方差下的均值检验统计量

When the population variance σ² is known and either the population follows a normal distribution or the sample size n is large (n ≥ 30 by the central limit theorem), the test statistic follows a standard normal distribution under H₀. The formula is:

当总体方差 σ² 已知，并且总体服从正态分布或样本量 n 足够大（根据中心极限定理，通常 n ≥ 30），在 H₀ 成立时检验统计量服从标准正态分布。其公式为：

z = (x̄ − μ₀) / (σ / √n)

Here x̄ is the sample mean, μ₀ is the hypothesised population mean, σ is the known population standard deviation, and n is the sample size.

这里 x̄ 是样本均值，μ₀ 是假设的总体均值，σ 是已知的总体标准差，n 是样本量。

5. Calculating the Test Statistic | 计算检验统计量

Once the sample has been collected, compute the observed value of z by plugging the numbers into the formula. Always check that the data meet the assumptions: the sample should be random, the population variance must be truly known, and the sample size should be sufficiently large if the population is not normal.

收集样本后，代入公式计算 z 的观测值。必须始终检查数据是否满足前提条件：样本应为随机样本，总体方差必须是真正已知的，并且如果总体不服从正态分布，样本量应足够大。

For instance, if a sample of 50 light bulbs gives x̄ = 1020 hours, with a known σ = 50 hours, and H₀: μ = 1000, then z = (1020 − 1000) / (50 / √50) = 20 / (50/7.071) = 20 / 7.071 ≈ 2.828.

例如，一个包含 50 只灯泡的样本给出 x̄ = 1020 小时，已知 σ = 50 小时，且 H₀: μ = 1000，则 z = (1020 − 1000) / (50 / √50) = 20 / (50/7.071) = 20 / 7.071 ≈ 2.828。

6. Critical Values and Decision Rule | 临界值与决策规则

For a two‑tailed test at α = 0.05, the critical values are ±1.96. If |z| > 1.96, we reject H₀. For a right‑tailed test at α = 0.05, the critical value is 1.645; we reject H₀ if z > 1.645. For a left‑tailed test, the critical value is −1.645, and we reject H₀ if z < −1.645.

对于 α = 0.05 的双尾检验，临界值为 ±1.96。如果 |z| > 1.96，则拒绝 H₀。对于 α = 0.05 的右尾检验，临界值为 1.645；如果 z > 1.645 就拒绝 H₀。对于左尾检验，临界值为 −1.645，如果 z < −1.645 就拒绝 H₀。

Test type	α = 0.05 Critical value(s)	Reject H₀ if
Two‑tailed	±1.96	|z| > 1.96
Right‑tailed	1.645	z > 1.645
Left‑tailed	−1.645	z < −1.645

7. One‑Tailed vs Two‑Tailed Tests | 单尾检验与双尾检验

A two‑tailed test splits the significance level equally between both tails of the distribution. It is used when the direction of the deviation is not specified. A one‑tailed test places the entire α in one tail and is more powerful for detecting an effect in the specified direction, but it should only be used when there is a strong prior reason to expect a particular direction.

双尾检验将显著性水平均分到分布的两个尾部，在偏离方向没有指明时使用。单尾检验将整个 α 放在一侧尾部，对于检测特定方向上的效应更为有力，但只有在有充分先验理由预期特定方向时才能使用。

8. p‑Value Approach | p值法

The p‑value is the probability of obtaining a test statistic at least as extreme as the observed one, assuming H₀ is true. For a two‑tailed test, p‑value = 2 × P(Z > |z|). For a right‑tailed test, p‑value = P(Z > z), and for a left‑tailed test, p‑value = P(Z < z). We reject H₀ if the p‑value is less than α.

p 值是在 H₀ 成立的条件下，观察到至少与当前观测值同样极端的检验统计量的概率。对于双尾检验，p 值 = 2 × P(Z > |z|)；对于右尾检验，p 值 = P(Z > z)；对于左尾检验，p 值 = P(Z < z)。如果 p 值小于 α，就拒绝 H₀。

9. Interpreting the Conclusion | 结论的解释

Rejecting H₀ means the data provide sufficient evidence to support the alternative hypothesis. Failing to reject H₀ does not prove that H₀ is true; it simply means the sample evidence is not strong enough to conclude otherwise. Always phrase the conclusion in the context of the problem: e.g. ‘There is significant evidence that the mean lifetime has increased.’

拒绝 H₀ 意味着数据提供了足够的证据支持备择假设。未能拒绝 H₀ 并不证明 H₀ 为真，而只是表示样本证据尚不足以得出相反的结论。一定要结合问题背景陈述结论，例如“有显著证据表明平均寿命已经提高”。

10. Common Mistakes | 常见错误

Avoid using a one‑tailed test simply to make a result significant. Do not change α after seeing the data. Remember that the test statistic z is valid only when σ is truly known – if σ is estimated from the sample, a t‑test must be used. Also, a significant result does not imply a large practical effect; always consider the effect size.

切忌为了得到显著结果而临时改用单尾检验。不要在查看数据后更改 α。请记住检验统计量 z 只有在 σ 真正已知时才适用——如果 σ 是从样本估计的，必须改用 t 检验。此外，统计显著并不意味着实际效应很大；始终要考虑效应量的大小。

11. Worked Example | 例题详解

A manufacturer claims that the average breaking strength of their cables is 800 kg. The known standard deviation from production data is 40 kg. A sample of 36 cables gives a mean breaking strength of 785 kg. Test the claim at the 5% significance level, considering the possibility that the strength has decreased.

某制造商声称其缆索的平均断裂强度为 800 kg。根据生产数据已知标准差为 40 kg。一个包含 36 根缆索的样本给出的平均断裂强度为 785 kg。在 5% 显著性水平下检验该声明，考虑强度已下降的可能性。

Hypotheses: H₀: μ = 800; H₁: μ < 800 (left‑tailed). Compute z = (785 − 800) / (40 / √36) = −15 / (40/6) = −15 / 6.667 = −2.25. The critical value is −1.645. Since −2.25 < −1.645, we reject H₀. There is sufficient evidence at the 5% level to conclude that the mean breaking strength is less than 800 kg.

假设：H₀: μ = 800；H₁: μ < 800（左尾检验）。计算 z = (785 − 800) / (40 / √36) = −15 / (40/6) = −15 / 6.667 = −2.25。临界值为 −1.645。由于 −2.25 < −1.645，我们拒绝 H₀。在 5% 显著性水平下有充分证据认为平均断裂强度低于 800 kg。

12. Connection to Confidence Intervals | 与置信区间的联系

A two‑tailed hypothesis test at significance level α is closely related to a (1−α) × 100% confidence interval for the population mean. If the hypothesised value μ₀ lies outside the confidence interval, we reject H₀ at level α. For the known‑variance case, the confidence interval is x̄ ± z_α/2 × (σ / √n).

显著水平 α 下的双尾假设检验与总体均值的 (1−α) × 100% 置信区间密切相关。如果假设值 μ₀ 落在置信区间之外，我们就在水平 α 下拒绝 H₀。对于方差已知的情形，置信区间为 x̄ ± z_α/2 × (σ / √n)。

This duality provides an intuitive check: the test rejects H₀ precisely when the confidence interval does not capture μ₀. It also reinforces that hypothesis testing and interval estimation are two sides of the same inference coin.

这种对偶关系提供了一种直观的检验：当且仅当置信区间没有覆盖 μ₀ 时，检验才会拒绝 H₀。这也有力地说明，假设检验和区间估计是统计推断这枚硬币的两面。

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