Hypothesis Testing Based on Normal Distribution | 基于正态分布的假设检验步骤

📚 Hypothesis Testing Based on Normal Distribution | 基于正态分布的假设检验步骤

Hypothesis testing is a core topic in A-Level statistics, and the normal distribution often serves as the theoretical foundation for these tests. This article provides a clear, step-by-step guide to conducting hypothesis tests when the underlying distribution is normal or can be approximated by one.

假设检验是 A-Level 统计学的核心内容，而正态分布通常构成这类检验的理论基础。本文将提供一套清晰的逐步指南，帮助你在总体服从正态分布（或可近似为正态分布）时完成假设检验。

1. Setting Up Hypotheses | 建立假设

Every hypothesis test begins with two competing statements about a population parameter. The null hypothesis, denoted H₀, is a statement of no effect or no change, typically expressed as an equality. The alternative hypothesis, denoted H₁ or Hₐ, is what you are trying to provide evidence for.

每一次假设检验都始于关于总体参数的两个相互竞争的陈述。零假设（记为 H₀）是无效应或无变化的陈述，通常以等式的形式表达。备择假设（记为 H₁ 或 Hₐ）是你要寻找证据支持的主张。

For a test concerning the population mean μ based on a normal distribution, we write:

对于基于正态分布的总体均值 μ 检验，我们这样书写：

H₀: μ = μ₀ vs H₁: μ ≠ μ₀ (two-tailed)
H₁: μ > μ₀ or H₁: μ < μ₀ (one-tailed)

Here, μ₀ is a known constant, often the population mean under the null hypothesis. The choice between a one-tailed and two-tailed alternative must be made before collecting data, based on the research question.

其中 μ₀ 是一个已知常数，通常是零假设下的总体均值。单尾与双尾备择假设的选择必须基于研究问题，在收集数据之前作出决定。

2. Significance Level and Critical Values | 显著性水平与临界值

The significance level, denoted α (alpha), is the probability of rejecting H₀ when it is in fact true. Common significance levels in A-Level exams are 10% (0.10), 5% (0.05), and 1% (0.01).

显著性水平（记为 α）是在零假设实际为真时拒绝它的概率。A-Level 考试中常见的显著性水平为 10%（0.10）、5%（0.05）和 1%（0.01）。

Critical values are the boundary values of the test statistic that separate the rejection region from the acceptance region. For a standard normal distribution Z ~ N(0, 1), the critical z-values are determined from the significance level.

临界值是检验统计量中划分拒绝域与接受域的边界值。对于标准正态分布 Z ~ N(0, 1)，临界 z 值由显著性水平决定。

Significance Level	One-tailed z-critical	Two-tailed z-critical
10%	1.645	1.645 (both tails)
5%	1.645	1.960 (both tails)
1%	2.326	2.576 (both tails)

3. The Test Statistic for Normal Distribution | 正态分布的检验统计量

When the population variance is known, or the sample size is sufficiently large (n ≥ 30), we use the z-test statistic. For a sample mean x̄ taken from a normal population, the test statistic is:

当总体方差已知，或样本量足够大（n ≥ 30）时，我们使用 z 检验统计量。对于取自正态总体的样本均值 x̄，检验统计量为：

z = (x̄ − μ₀) / (σ / √n)

Here, σ is the population standard deviation, n is the sample size, and μ₀ is the hypothesised population mean. If σ is unknown, we substitute the sample standard deviation s as an estimate, provided n is large.

其中 σ 是总体标准差，n 是样本量，μ₀ 是假设的总体均值。如果 σ 未知，只要 n 足够大，可以用样本标准差 s 作为估计值替代。

A closely related approach uses the sample mean distribution. Since x̄ ~ N(μ₀, σ²/n) under H₀, we can compare the observed x̄ directly against critical values in the original units.

一种紧密相关的方法是使用样本均值的分布。由于在 H₀ 下 x̄ ~ N(μ₀, σ²/n)，我们可以直接将观察到的 x̄ 与原始单位下的临界值进行比较。

4. One-tailed vs Two-tailed Tests | 单尾检验与双尾检验

Two-tailed test (H₁: μ ≠ μ₀): The rejection region is split equally between both tails of the normal distribution. At the 5% significance level, we reject H₀ if z > 1.960 or z < −1.960.

双尾检验（H₁: μ ≠ μ₀）： 拒绝域平均分布在正态分布的两个尾端。在 5% 显著性水平下，当 z > 1.960 或 z < −1.960 时拒绝 H₀。

One-tailed test (H₁: μ > μ₀): The entire rejection region is placed in the upper tail. At the 5% significance level, we reject H₀ if z > 1.645.

单尾检验（H₁: μ > μ₀）： 整个拒绝域位于上尾。在 5% 显著性水平下，当 z > 1.645 时拒绝 H₀。

One-tailed test (H₁: μ < μ₀): The rejection region is placed entirely in the lower tail. At the 5% significance level, we reject H₀ if z < −1.645.

单尾检验（H₁: μ < μ₀）： 整个拒绝域位于下尾。在 5% 显著性水平下，当 z < −1.645 时拒绝 H₀。

5. The Step-by-Step Procedure | 逐步检验流程

Follow this systematic procedure for any normal-distribution hypothesis test:

对于任何正态分布假设检验，请遵循以下系统化流程：

Step 1: Define the parameter and state H₀ and H₁ clearly. Identify μ₀ and decide on the alternative direction.
步骤 1： 明确参数并写出 H₀ 和 H₁。确定 μ₀ 并判断备择假设的方向。

Step 2: Choose the significance level α. This is usually given in the question (e.g., 5%, 1%).
步骤 2： 选择显著性水平 α。题目中通常会给出（例如 5%、1%）。

Step 3: Determine the distribution of the sample mean under H₀. If X ~ N(μ, σ²), then x̄ ~ N(μ₀, σ²/n). Calculate the test statistic z.
步骤 3： 确定 H₀ 下样本均值的分布。如果 X ~ N(μ, σ²)，则 x̄ ~ N(μ₀, σ²/n)。计算检验统计量 z。

Step 4: Find the critical value(s) from the normal distribution table corresponding to α and the test tail type.
步骤 4： 从正态分布表中查找与 α 及检验尾型对应的临界值。

Step 5: Compare the test statistic with the critical value. Reject H₀ if z falls in the rejection region; otherwise, do not reject H₀.
步骤 5： 将检验统计量与临界值进行比较。如果 z 落在拒绝域内则拒绝 H₀；否则不拒绝 H₀。

Step 6: Write a conclusion in the context of the original problem, stating whether there is sufficient evidence to support H₁.
步骤 6： 结合原问题语境写出结论，说明是否有充分证据支持 H₁。

6. Worked Example: Two-tailed Test | 实例分析：双尾检验

Problem: A factory produces steel rods with a mean length of 20.0 cm and a standard deviation of 0.5 cm. A sample of 100 rods is taken, yielding a mean length of 19.8 cm. Test at the 5% significance level whether the production process has changed.

题目： 某工厂生产的钢棒平均长度为 20.0 cm，标准差为 0.5 cm。现抽取 100 根钢棒，样本均值为 19.8 cm。在 5% 显著性水平下检验生产过程是否发生变化。

Solution: Let μ be the population mean length of all rods. We test:

解答： 设 μ 为所有钢棒的总体平均长度。我们检验：

H₀: μ = 20.0 cm H₁: μ ≠ 20.0 cm

The significance level is α = 0.05, two-tailed, so the critical values are z = ±1.960. The test statistic is:

显著性水平为 α = 0.05，双尾检验，因此临界值为 z = ±1.960。检验统计量为：

z = (19.8 − 20.0) / (0.5 / √100) = (−0.2) / 0.05 = −4.00

Since |z| = 4.00 > 1.960, the test statistic lies well inside the rejection region. We therefore reject H₀ at the 5% level. There is strong evidence that the mean rod length has changed from 20.0 cm.

因为 |z| = 4.00 > 1.960，检验统计量明显落入拒绝域。因此我们在 5% 水平上拒绝 H₀。有充分证据表明钢棒平均长度已偏离 20.0 cm。

7. Worked Example: One-tailed Test | 实例分析：单尾检验

Problem: A drinks machine is designed to dispense on average 250 ml per cup, with a standard deviation of 8 ml. An inspector suspects the machine is dispensing too much. A sample of 40 cups gives a mean of 253 ml. Test at the 1% significance level whether the machine is over-filling.

题目： 某饮料机设计为每杯平均出液 250 ml，标准差为 8 ml。检查员怀疑机器出液过多。抽取 40 杯测得均值为 253 ml。在 1% 显著性水平下检验机器是否过量出液。

Solution: Here, the suspicion is that μ > 250, so we use a one-tailed test:

解答： 这里怀疑 μ > 250，因此使用单尾检验：

H₀: μ = 250 ml H₁: μ > 250 ml

At α = 0.01, the critical value for a one-tailed test is z = 2.326. The test statistic is:

在 α = 0.01 时，单尾检验的临界值为 z = 2.326。检验统计量为：

z = (253 − 250) / (8 / √40) = 3 / 1.2649 = 2.372

Since 2.372 > 2.326, we reject H₀ at the 1% level. There is sufficient evidence to conclude that the machine is dispensing more than 250 ml on average.

由于 2.372 > 2.326，我们在 1% 水平上拒绝 H₀。有充分证据得出结论：机器平均出液量超过 250 ml。

8. Type I and Type II Errors | 第一类错误与第二类错误

Type I error: Rejecting H₀ when it is actually true. The probability of a Type I error is equal to the significance level α. For example, at the 5% level, there is a 5% chance of concluding a difference exists when none truly does.

第一类错误： 当 H₀ 实际为真时拒绝了它。犯第一类错误的概率等于显著性水平 α。例如在 5% 水平下，有 5% 的概率在事实上没有差异时得出了存在差异的结论。

Type II error: Failing to reject H₀ when it is actually false. The probability of a Type II error is denoted β. Its calculation is more involved, as it depends on the true value of the parameter under H₁.

第二类错误： 当 H₀ 实际为假时没有拒绝它。犯第二类错误的概率记为 β。其计算较为复杂，因为它取决于 H₁ 下参数的真实值。

The power of a test is 1 − β, the probability of correctly rejecting a false H₀. Increasing the sample size reduces both types of error simultaneously, while lowering α decreases Type I error but increases Type II error.

检验的功效为 1 − β，即正确拒绝错误 H₀ 的概率。增大样本量可以同时减少两类错误；而降低 α 会减少第一类错误但增加第二类错误。

9. Normal Approximation to the Binomial Distribution | 二项分布的正态近似

When testing a proportion p using a binomial model, X ~ B(n, p), a normal approximation can be used if np > 5 and n(1 − p) > 5. The test statistic then becomes:

当使用二项模型 X ~ B(n, p) 检验比例 p 时，若 np > 5 且 n(1 − p) > 5，可以使用正态近似。此时检验统计量变为：

z = (x − np₀) / √(np₀(1 − p₀))

Here, x is the observed number of successes, n is the number of trials, and p₀ is the hypothesised probability under H₀. A continuity correction can be applied for improved accuracy, though many A-Level syllabi omit it.

其中 x 是观察到的成功次数，n 是试验次数，p₀ 是 H₀ 下假设的概率。为提高精度可以应用连续性修正，不过许多 A-Level 考纲不要求这一点。

10. p-value Method | p 值方法

An alternative to the critical-value approach is the p-value method. The p-value is the probability of obtaining a test statistic at least as extreme as the observed value, assuming H₀ is true. For a two-tailed test, this probability is doubled.

除临界值方法外，还可以使用 p 值方法。p 值是在假设 H₀ 为真的前提下，得到与观测值至少同样极端的检验统计量的概率。对于双尾检验，该概率要乘以 2。

Decision rule: Reject H₀ if p-value < α. The p-value method produces the same conclusion as the critical-value method but directly quantifies the strength of evidence against H₀.

判定规则：若 p 值 < α 则拒绝 H₀。p 值方法与临界值方法得出的结论相同，但能直接量化反对 H₀ 的证据强度。

For the two-tailed example in Section 6, z = −4.00 gives a p-value of 2 × P(Z < −4.00) ≈ 2 × 0.00003 = 0.00006, which is far less than 0.05, confirming the rejection of H₀.

对于第 6 节中的双尾示例，z = −4.00 对应的 p 值为 2 × P(Z < −4.00) ≈ 2 × 0.00003 = 0.00006，远小于 0.05，从而确认拒绝 H₀。

11. Common Pitfalls and Exam Tips | 常见误区与考试技巧

Pitfall 1: Using the wrong tail. Always check whether the alternative hypothesis implies > , < , or ≠ before selecting critical values.

误区 1：用错尾型。 在选取临界值之前，务必确认备择假设是 >、< 还是 ≠。

Pitfall 2: Forgetting to divide by √n. The standard deviation of the sample mean is σ/√n, not σ. This is the most frequently lost mark in hypothesis testing questions.

误区 2：忘记除以 √n。 样本均值的标准差是 σ/√n，而不是 σ。这是假设检验试题中最常见的失分点。

Pitfall 3: Stating the conclusion in statistical language only. Marks are awarded for a conclusion in the context of the problem. Write a full sentence about what the result means for the situation described.

误区 3：仅用统计语言下结论。 结合问题情境给出结论才能获得相应分数。请用完整的句子说明结果对所述情境的意义。

Pitfall 4: Confusing sample standard deviation with population standard deviation. When σ is unknown and n is large, use s, but clearly state this substitution in your working.

误区 4：混淆样本标准差与总体标准差。 当 σ 未知且 n 较大时，可用 s 替代，但要在答题过程中明确说明这一替换。

Exam tip: Always write H₀ and H₁ first — this shows the examiner your framework is correct. Then show the calculation of z in full, with at least two decimal places in intermediate steps.

考试技巧： 务必先写出 H₀ 和 H₁——这向考官表明你的框架是正确的。然后完整展示 z 的计算过程，中间步骤至少保留两位小数。

12. Summary | 总结

Hypothesis testing based on the normal distribution follows a systematic series of steps: state hypotheses, choose the significance level, compute the test statistic, determine critical values, and draw a contextual conclusion. Mastery of this procedure is essential for A-Level success.

基于正态分布的假设检验遵循一套系统化步骤：陈述假设、选择显著性水平、计算检验统计量、确定临界值，并给出结合情境的结论。掌握这一流程是 A-Level 考试取得高分的关键。

Remember that the choice between one-tailed and two-tailed tests depends entirely on the research question, and the z-test formula requires the standard deviation of the sample mean, not that of the population. Consistent practice with past-paper questions is the most effective way to avoid common errors.

请记住：单尾与双尾检验的选择完全取决于研究问题；z 检验公式使用的是样本均值的标准差而非总体标准差。通过历年真题进行持续练习，是避免常见错误的最有效方法。

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Hypothesis Testing Based on Normal Distribution | 基于正态分布的假设检验步骤