📚 Hypothesis Testing with the Normal Distribution | 正态分布假设检验
Hypothesis testing is a fundamental technique in inferential statistics which allows us to use sample data to decide whether a claim about a population parameter is supported by evidence. When the underlying data follow a normal distribution, or when sample sizes are sufficiently large for the Central Limit Theorem to apply, we can use the normal distribution to calculate probabilities, critical values and p‑values. This method is widely applied in Edexcel A‑Level Mathematics to test population means, differences between means and proportions where the normal approximation is justified. Understanding the logic of significance levels, test statistics and errors is essential for success in the statistics components of the course.
假设检验是推断统计学中的一项基本技术，它可以让我们利用样本数据判断关于总体参数的说法是否得到证据支持。当数据服从正态分布，或者样本量足够大以至于中心极限定理适用时，我们就可以利用正态分布来计算概率、临界值和 p 值。在 Edexcel A‑Level 数学中，这种方法被广泛应用于检验总体均值、均值之差以及可以使用正态近似的比例。理解显著性水平、检验统计量和错误类型的逻辑，对于学好课程中的统计部分至关重要。
1. Introduction to Hypothesis Testing | 假设检验简介
A statistical hypothesis is a statement about a population parameter, such as the population mean μ. Hypothesis testing provides a formal framework to challenge an initial assumption using observed data. For example, a manufacturer claims that the average lifetime of a light bulb is 1000 hours. We collect a random sample, calculate its mean and determine whether the difference from 1000 hours is statistically significant or just due to random chance. If the sample result is highly unlikely under the claimed value, we reject the claim in favour of an alternative.
统计假设是关于总体参数（例如总体均值 μ）的一种陈述。假设检验提供了一个正式的框架，使我们能够利用观测数据来质疑初始假设。例如，某制造商声称灯泡的平均使用寿命为 1000 小时。我们收集一个随机样本，计算其均值，然后判断与 1000 小时的差异在统计上是否显著，还是仅仅由随机因素造成。如果在声称的数值下样本结果出现的概率极低，我们就拒绝该声称，转而支持备择假设。
2. Null and Alternative Hypotheses | 原假设与备择假设
Every hypothesis test begins by stating two complementary hypotheses. The null hypothesis or no difference. It usually takes the form H₀: μ = μ₀, where μ₀ is a specific value. The alternative hypothesis, denoted H₁ or Hₐ, challenges the null by stating that the parameter is different, greater than, or less than the claimed value. A two‑tailed test uses H₁: μ ≠ μ₀, while one‑tailed tests use H₁: μ > μ₀ or H₁: μ < μ₀ depending on the direction of interest.
每个假设检验都从陈述两个互补的假设开始。原假设记为 H₀，是默认状况，即没有效应或没有差异。它通常采用 H₀: μ = μ₀ 的形式，其中 μ₀ 是一个特定值。备择假设记为 H₁ 或 Hₐ，通过声明参数不同于、大于或小于声称值来质疑原假设。双尾检验使用 H₁: μ ≠ μ₀，而单尾检验则根据关心的方向使用 H₁: μ > μ₀ 或 H₁: μ < μ₀。
3. Significance Level and Critical Region | 显著性水平与临界区域
The significance level of a test, denoted α, is the probability of rejecting the null hypothesis when it is actually true (a Type I error). Common choices for α are 0.05 (5%) or 0.01 (1%). The critical region (or rejection region) is the set of values of the test statistic that leads to the rejection of H₀. Its boundary is defined by critical values obtained from the standard normal distribution. For a two‑ are ±1.9600, meaning we reject H₀ if the test statistic falls below −1.96 or above +1.96.
检验的显著性水平记为 α，是当原假设实际上为真时拒绝它的概率（即 I 型错误）。α 的常用取值是 0.05（5%）或 0.01（1%）。临界区域（或称拒绝域）是检验统计量所有会导致拒绝 H₀ 的取值集合。其边界由从标准正态分布中获得的临界值确定。对于 α = 0.05 的双尾检验，临界值为 ±1.9600，这意味着如果检验统计量落于 −1.96 以下或 +1.96 以上，我们就拒绝 H₀。
4. Test Statistic for the Normal Distribution | 正态分布的检验统计量
When the population variance σ² is known and the sample is drawn from a normal population (or when n is large), the sampling distribution of the sample mean x̄ is normal with mean μ and standard error σ/√n. The test statistic is the standardised Z‑score:
Z = (x̄ − μ₀) / (σ / √n)
This Z‑ eviation s can be used as an approximation of σ, and the normal distribution remains appropriate due to the Central Limit Theorem.
当总体方差 σ² 已知且样本来自正态总体（或者当 n 很大时），样本均值 x̄ 的抽样分布服从均值为 μ、标准误为 σ/√n 的正态分布。检验统计量为标准化 Z 值：
Z = (x̄ − μ₀) / (σ / √n)
在原假设下，该 Z 统计量服从标准正态分布。它衡量了样本均值与假设的总体均值 μ₀ 相差了多少个标准误。如果总体方差未知但样本量很大（n ≥ 30），样本标准差 s 可以作为 σ 的近似值，并且由于中心极限定理，使用正态分布依然是合适的。
5. One‑tailed vs Two‑tailed Tests | 单尾与双尾检验
The form of the alternative hypothesis determines whether the test is one‑tailed or two‑tailed. A two‑tailed test is used when we are testing for any difference, so H₁: μ ≠ μ₀, and the critical region is split equally between both tails. A one‑tailed test hile H₁: μ < μ₀ examines the lower tail. The critical values differ: for α = 0.05 in a one‑tailed test, the critical Z‑value is ±1.6449 depending on direction, rather than ±1.9600 for two tails.
备择假设的形式决定了检验是单尾还是双尾。双尾检验用于检验任何差异，故 H₁: μ ≠ μ₀，临界区域等分在两侧尾部。单尾检验则指明具体的方向：H₁: μ > μ₀ 考察上尾，H₁: μ < μ₀ 考察下尾。临界值也随之不同：对于 α = 0.05 的单尾检验，临界 Z 值为 ±1.6449（视方向而定），而双尾为 ±1.9600。
6. Critical Value Approach | 临界值法
In the critical value approach, we follow a structured process:
State H₀ and H₁.
陈述 H₀ 与 H₁。
Choose the significance level α and identify the corresponding critical value(s) from the standard normal table.
选择显著性水平 α，并从标准正态表中找出相应的临界值。
Compute the test statistic Z.
计算检验统计量 Z。
Compare Z with the critical value(s). If Z falls in the critical region, reject H₀; otherwise, do not reject H₀.
将 Z 与临界值比较。如果 Z 落入临界区域，则拒绝 H₀；否则不拒绝 H₀。
下面给出一些常用的 Z 临界值：
| Significance level α | One‑tailed (Z) | Two‑tailed (Z) |
| 0.10 (10%) | ±1.2816 | ±1.6449 |
| 0.05 (5%) | ±1.6449 | ±1.9600 |
| 0.01 (1%) | ±2.3263 | ±2.5758 |
Critical values are often found using the inverse normal function on a calculator. In an exam, you must clearly show which values you use and whether you are applying a one‑ or two‑tailed test.
临界值通常使用计算器上的逆正态函数求得。在考试中，你必须清楚说明所使用的数值以及你采用的是单尾还是双尾检验。
7. p‑Value Approach | p 值法
The p‑value est statistic at least as extreme as the one observed, assuming that H₀ is true. For a two‑tailed test, the p‑value is the sum of the probabilities in both tails. A small p‑ cates strong evidence against H₀, leading to rejection. For instance, if Z = −2.5 in a one‑tailed lower test, the p‑value is P(Z ≤ −2.5) = 0.0062. Since 0.0062 < 0.05, we reject H₀ at the 5% level. Reporting the p‑value provides more information than a simple reject/not reject decision.
p 值是在 H₀ 为真的前提下，得到至少与观测值一样极端的检验统计量的概率。对于双尾检验，p 值是两侧尾部概率之和。很小的 p 值（≤ α）表明存在强有力的证据反对 H₀，从而拒绝原假设。例如，在一个下尾单尾检验中，若 Z = −2.5，则 p 值为 P(Z ≤ −2.5) = 0.0062。由于 0.0062 < 0.05，我们在 5% 的水平上拒绝 H₀。报告 p 值能比简单的拒绝/不拒绝决定提供更丰富的信息。
8. Steps for Conducting a Hypothesis Test | 进行假设检验的步骤
A standard hypothesis test for the population mean using the normal distribution can be broken down into these steps:
使用正态分布对总体均值进行假设检验的标准流程可分解为以下步骤：
Step 1: State the null hypothesis H₀ and the alternative hypothesis H₁. Define μ₀ explicitly. | 第 1 步：陈述原假设 H₀ 与备择假设 H₁。明确定义 μ₀。
Step 2: Choose the significance level α (e.g., 0.05). | 第 2 步：选取显著性水平 α（例如 0.05）。
Step 3: Compute the test statistic Z = (x̄ − μ₀) / (σ/√n). If σ is unknown use s. | 第 3 步：计算检验统计量 Z = (x̄ − μ₀) / (σ/√n)。若 σ 未知，使用 s。
Step 4: Determine the critical value(s) or compute the p‑value. | 第 4 步：确定临界值或计算 p 值。
Step 5: ecision: reject H₀ if Z is in the critical region or if p ≤ α. Otherwise do not reject H₀. | 第 5 步：做出决策：如果 Z 落入临界区域或 p ≤ α，则拒绝 H₀；否则不拒绝 H₀。
Step 6: State the conclusion in the context of the problem, using non‑technical language. | 第 6 步：用非技术性语言在问题的背景下陈述结论。
9. Type I and Type II Errors | I 型与 II 型错误
No test is perfect, and two types of error can occur. A Type I error is rejecting a true null hypothesis; its probability is precisely α, the significance level. A Type II error is failing to reject a false null hypothesis; its probability is denoted β. There is a trade‑off: decreasing α reduces the chance of a Type I error but increases β unless the sample size is enlarged. In A‑Level questions, you may be asked to interpret these errors in context, for example explaining what a Type I error would mean for a production process.
没有哪种检验是完美的，可能出现两类错误。I 型错误是拒绝了真实原假设；其概率正好是显著性水平 α。II 型错误则是未能拒绝错误原假设；其概率记为 β。两者之间存在权衡：减小 α 会降低 I 型错误的机会，但除非增大样本量，否则 β 会上升。在 A‑Level 题目中，你可能会被要求结合实际情境解释这些错误，例如说明 I 型错误对生产流程意味着什么。
10. Worked Example | 实例解析
r selects a random sample of 10 bags and records a mean weight of 1.488 kg. Test, at the 5% significance level, whether there is evidence that the machine is under‑filling.
一台机器为面粉装袋，标称平均重量为 1.500 kg。已知灌装过程服从正态分布，标准差 σ = 0.020 kg。质量检验员随机抽取 10 袋，记录到样本平均重量为 1.488 kg。在 5% 显著性水平下检验是否有证据表明机器灌装量不足。
Solution | 解法
Step 1: H₀: μ = 1.500; H₁: μ < 1.500 (one‑tailed lower test). | 第 1 步：H₀: μ = 1.500；H₁: μ < 1.500（下尾单尾检验）。
Step 2: α = 0.05. | 第 2 步：α = 0.05。
Step 3: Z = (1.488 − 1.500) / (0.020/√10) = −0.012 / (0.020/3.1623) ≈ −1.897. | 第 3 步：Z = (1.488 − 1.500) / (0.020/√10) = −0.012 / (0.020/3.1623) ≈ −1.897。
Step 4: The one‑tailed critical Z at α = 0.05 is −1.6449. The p‑value is P(Z ≤ −1.897) = 0.0289. | 第 4 步：α = 0.05 时的单尾临界 Z 为 −1.6449。p 值为 P(Z ≤ −1.897) = 0.0289。
Step 5: Z = −1.897 < −1.6449, falling in the critical region. Also p = 0.0289 < 0.05. Reject H₀. | 第 5 步：Z = −1.897 < −1.6449，落入临界区域。且 p = 0.0289 < 0.05。拒绝 H₀。
Step 6: There is sufficient evidence, at the 5% level, to conclude that the mean filling weight is less than 1.500 kg. The machine is under‑filling and requires adjustment. | 第 6 步：在 5% 水平下有充分证据表明平均灌装重量低于 1.500 kg。机器灌装不足，需要调整。
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