📚 Setting the Hypotheses in Hypothesis Testing | 假设检验中假设的设定
In A-Level Mathematics, hypothesis testing is a powerful tool for making inferences about a population based on sample data. The very first – and arguably most important – step is to set up the null and alternative hypotheses correctly. A mistake here can invalidate all subsequent calculations, regardless of how accurate they are. This article focuses purely on how to formulate these hypotheses for Edexcel A-Level, covering binomial proportions, normal means, and correlation coefficients. You will learn the logic behind H₀ and H₁, how to decide between one-tailed and two-tailed tests, and how to express the hypotheses using proper statistical language.
在 A-Level 数学中，假设检验是根据样本数据对总体进行推断的有力工具。第一步——也可以说是最重要的一步——就是正确地设定原假设和备择假设。这一步一旦出错，无论后面的计算多么精确，整个检验都会失效。本文专门讲授如何为 Edexcel A-Level 课程构造这些假设，涵盖二项分布比例、正态分布均值和相关系数的情形。你将理解 H₀ 与 H₁ 背后的逻辑，学会如何选择单尾检验或双尾检验，以及如何用正确的统计语言表述它们。
1. The Role of Hypotheses in Statistical Testing | 假设在统计检验中的角色
Every hypothesis test is a formal decision-making process. We begin with two competing claims about a population parameter: the null hypothesis and the alternative hypothesis. The test uses sample evidence to judge whether the null hypothesis is tenable or should be rejected in favour of its alternative. The hypotheses must be stated before any data are examined – this protects the objectivity of the test.
每一次假设检验都是一个正式的决策过程。我们首先对总体参数提出两个对立的陈述：原假设和备择假设。检验利用样本证据来判断原假设是否成立，或者是否应被拒绝而支持备择假设。假设必须在查看任何数据之前就陈述清楚——这能保护检验的客观性。
In the Edexcel specification, the parameters you will most often encounter are the population proportion p (for a binomial distribution), the population mean μ (for a normal distribution), and the population correlation coefficient ρ. The ability to set up H₀ and H₁ for each scenario is a core skill tested in both pure statistics questions and word problems.
在 Edexcel 考纲中，你最常遇到的参数是总体比例 p（用于二项分布）、总体均值 μ（用于正态分布）以及总体相关系数 ρ。能够为每种情景设定 H₀ 和 H₁ 是一项核心技能，无论是纯统计题还是文字应用题都会考查。
2. The Null Hypothesis H₀ | 原假设 H₀
The null hypothesis, denoted H₀, is the statement that there is no effect, no difference, or no change. It represents the status quo or a baseline value that we assume to be true until evidence suggests otherwise. In A-Level hypothesis tests, H₀ always contains an equality: it states that the population parameter equals a specific numerical value.
原假设，记作 H₀，是假设没有效应、没有差异或没有变化的陈述。它代表了现状或我们假定为真的基线值，除非证据表明并非如此。在 A-Level 的假设检验中，H₀ 总是包含等式：它声称总体参数等于某个具体的数值。
H₀: p = 0.4 (for a binomial test on proportion)
H₀: p = 0.4（二项分布比例检验）
H₀: μ = 50 kg (for a normal mean test)
H₀: μ = 50 kg（正态均值检验）
H₀: ρ = 0 (for a correlation test)
H₀: ρ = 0（相关系数检验）
The null hypothesis is the one we attempt to disprove. If the sample data are highly unlikely under H₀, we reject H₀. Otherwise, we do not reject it. Note that we never “accept” H₀ in the sense of proving it true; we simply lack sufficient evidence to reject it.
原假设是我们试图去否定的假设。如果样本数据在 H₀ 成立的条件下出现的概率极低，我们就拒绝 H₀。否则，我们不拒绝它。请注意，我们从来不会“接受” H₀ 并视其为已被证真；我们只是缺乏足够的证据去拒绝它罢了。
3. The Alternative Hypothesis H₁ | 备择假设 H₁
The alternative hypothesis, denoted H₁ (or Hₐ), is the statement that challenges the null. It describes what we suspect might be true – the research claim or the suspected change. H₁ must contradict the equality in H₀. It can take one of three forms: a less-than alternative, a greater-than alternative, or a not-equal-to alternative.
备择假设，记作 H₁（或 Hₐ），是对原假设提出挑战的陈述。它描述了我们怀疑可能为真的情形——即研究主张或疑似的变动。H₁ 必须与原假设中的等式相矛盾。它可以采取以下三种形式之一：小于、大于或不等于。
H₁: p < 0.4 (the proportion has decreased)
H₁: p < 0.4（比例下降）
H₁: μ > 50 (the mean has increased)
H₁: μ > 50（均值上升）
H₁: ρ ≠ 0 (there is some linear correlation)
H₁: ρ ≠ 0（存在某种线性相关）
The choice of H₁ is driven by the context of the problem. If the question asks “has the proportion decreased?”, the alternative will be a one-tailed less-than hypothesis. If it asks “has the mean changed?”, a two-tailed not-equal alternative is required. The direction must come from the wording, not from the sample data.
H₁ 的选取由题目背景决定。如果问题问的是“比例是否下降？”，那么备择假设就是单尾的小于假设。如果问的是“均值是否发生了变化？”，则需要采用双尾的不等于假设。方向必须来自题干的措辞，而非来自样本数据。
4. One-Tailed vs Two-Tailed Tests | 单尾检验与双尾检验
A one-tailed (or one-sided) test is used when the alternative hypothesis specifies a direction: either a decrease (less than) or an increase (greater than). The entire significance level α is placed in one tail of the sampling distribution. For example, “test whether the new drug reduces the recovery time” leads to H₁: μ < 30 days.
当备择假设指明了方向——要么减小（小于），要么增大（大于）——时，我们使用单尾（或单侧）检验。整个显著性水平 α 被放在抽样分布的单一尾部中。例如，“检验新药是否缩短了康复时间”会导出 H₁: μ < 30 天。
with ≠. The significance level is split equally between the two tails. A typical scenario: “has the company’s claim that p = 0.7 changed?” results in H₁: p ≠ 0.7.
当我们只是寻找任何变化而不预测方向时，则采用双尾（或双侧）检验。备择假设用 ≠ 表示。显著性水平被平分到两个尾部中。典型的情景是：“该公司关于 p = 0.7 的说法是否发生了变化？”由此得到 H₁: p ≠ 0.7。
It is vital to decide the number of tails before looking at the data. Edexcel questions often contain keywords: “more than”, “less than”, “reduced”, “increased” indicate one-tailed tests; “changed”, “different”, “not equal” point to two-tailed tests.
在查看数据之前就决定使用单尾还是双尾至关重要。Edexcel 考题中常包含关键词：“more than”、“less than”、“reduced”、“increased” 表示单尾检验；“changed”、“different”、“not equal” 则指向双尾检验。
5. Setting Hypotheses for a Binomial Proportion | 二项分布比例的假设设定
When the test concerns a population proportion p based on a binomial distribution, the null hypothesis fixes p at a specific value. The alternative will be <, >, or ≠. The hypotheses are always about the population parameter p, never about the sample proportion p̂.
当检验涉及基于二项分布的总体比例 p 时，原假设把 p 固定为一个特定的数值。备择假设则为 <、> 或 ≠。这些假设总是针对总体参数 p，绝不针对样本比例 p̂。
H₀: p = 0.35
H₁: p < 0.35 (one-tailed, suspecting a decrease)
H₁: p < 0.35（单尾，怀疑下降）
Suppose a manufacturer claims that at most 5% us quo) and H₁: p > 0.05. The equality always stays with H₀. Even if the manufacturer says “at most”, the null retains p = 0.05, and the alternative covers the suspected direction.
假设某制造商声称缺陷率最多为 5%。为了质疑这一声称，消费者组织怀疑缺陷率更高。此时的假设变为 H₀: p = 0.05（维持现状）与 H₁: p > 0.05。等式永远留在 H₀ 中。即便制造商说的是“至多”，原假设仍然保留 p = 0.05，而备择假设则涵盖所怀疑的方向。
For a two-tailed binomial test, the hypotheses could be:
H₀: p = 0.2
H₁: p ≠ 0.2
This is used when the question asks whether the proportion “has changed” from the historical value of 0.2.
当题目询问比例相对于历史值 0.2 “是否发生了变化”时，便使用这种双尾检验。
6. Setting Hypotheses for a Normal Mean | 正态分布均值的假设设定
For a test on a population mean μ, the data are assumed to come from a normal distribution (or the sample size is large enough to invoke the Central Limit Theorem). The null hypothesis states that μ equals a particular value. The alternative can take any of the three inequality forms.
对于总体均值 μ 的检验，数据被假定来自正态分布（或者样本量足够大，可以应用中心极限定理）。原假设声称 μ 等于某个特定值。备择假设可以取三种不等号形式的任何一种。
Typical Edexcel exam wording: “The manager believes the mean waiting time has increased from 4 minutes.” The hypotheses are:
H₀: μ = 4
H₁: μ > 4
If the statement is “test whether the mean content is different from 330 ml”, then:
H₀: μ = 330
H₁: μ ≠ 330
It is tempting to set H₁: μ < 330 if the sample mean happens to be smaller, but this would be data-snooping. The direction must come from the question’s intention, not from looking at the sample results first.
如果样本均值恰好偏小，人们容易设成 H₁: μ < 330，但这就成了数据偷窥。方向必须来自题目的意图，而不是先看了样本结果再决定。
When the population variance is known, the test statistic follows a standard normal distribution; when it is unknown, the t-distribution is used. In both cases, the structure of H₀ and H₁ remains identical – they are always about μ.
当总体方差已知时，检验统计量服从标准正态分布；当方差未知时，则使用 t 分布。无论哪种情况，H₀ 和 H₁ 的结构始终不变——它们总是关于 μ 的陈述。
7. Setting Hypotheses for a Correlation Coefficient | 相关系数的假设设定
t ρ. The default null hypothesis is always that there is no linear correlation: H₀: ρ = 0.
在 Edexcel A-Level 统计学考纲中，假设检验也可以应用于总体相关系数 ρ。默认的原假设始终是不存在线性相关：H₀: ρ = 0。
A two-tailed alternative H₁: ρ ≠ 0 simply tests for any evidence of correlation, either positive or negative. A one-tailed alternative is specified when we have a prior reason to test for positive correlation (H₁: ρ > 0) or negative correlation (H₁: ρ < 0). The context, such as a scientific theory predicting a positive relationship, dictates the tail.
双尾备择假设 H₁: ρ ≠ 0 仅仅是检验是否存在任何相关性的证据，无论正负。当我们有先验理由去检验正相关时 (H₁: ρ > 0) 或负相关时 (H₁: ρ < 0)，才会指定单尾备择假设。具体情境——例如科学理论预测的正向关系——会决定使用哪一侧尾部。
For example, “test the claim that there is negative correlation between age and reaction time” leads to H₀: ρ = 0 and H₁: ρ < 0. Never write H₁: r < 0 – r is the sample statistic, not the population parameter.
例如，“检验年龄与反应时间之间存在负相关的说法” 会导出 H₀: ρ = 0 与 H₁: ρ < 0。千万不要写成 H₁: r < 0 —— r 是样本统计量，不是总体参数。
8. Common Mistakes When Setting Hypotheses | 设定假设时的常见错误
is using the sample statistic in the hypotheses. For a test on a mean, do not write H₀: x̄ = 50. The hypotheses must refer to μ. Similarly, for a proportion test, use p, not p̂, and for correlation, use ρ, not r. The symbols x̄, p̂ d r describe the sample, whereas hypothesis testing makes inferences about the population.
最常见的错误之一是在假设中使用样本统计量。对于均值的检验，不要写成 H₀: x̄ = 50。假设必须涉及 μ。类似地，对于比例检验，使用 p 而不是 p̂；对于相关检验，使用 ρ 而不是 r。符号 x̄、p̂ 和 r 描述的是样本，而假设检验是对总体作出推断。
Another pitfall is placing the inequality in the null hypothesis. H₀ must always contain an equality. Expressions like H₀: p > 0.5 or H₀: μ < 100 are incorrect. The null is the hypothesis of no change or no effect, and it is the benchmark we test against.
另一个陷阱是把不等号放进原假设中。H₀ 必须始终包含等号。诸如 H₀: p > 0.5 或 H₀: μ < 100 的写法是错误的。原假设是“无变化”或“无效应”的假设，也是我们赖以检验的基准。
Choosing the wrong tail is also common. When the question asks “has the proportion changed?”, a two-tailed test is required; using a one-tailed test will halve the critical region on the wrong side and potentially lead to an incorrect conclusion. Always identify the keywords carefully.
选错尾部也很常见。当题目问的是“比例是否发生了变化？”时，就需要双尾检验；若使用单尾检验，会将临界区减半并放在错误的一侧，可能导致错误结论。务必仔细识别关键词。
Finally, remember that the alternative hypothesis is what you are trying to establish. In a court of law, the null is “the defendant is innocent”; the alternative is “the defendant is guilty”. You need strong evidence to reject innocence. Hypothesis testing works under the same logic.
最后，记住备择假设是你试图去确立的论断。在法庭上，原假设是“被告无罪”；备择假设是“被告有罪”。你需要强有力的证据才能推翻无罪假定。假设检验遵循完全相同的逻辑。
9. Worked Examples from Edexcel-Style Questions | Edexcel 风格考题实例分析
Example 1 – Binomial proportion (one-tailed): A teacher claims that more than 60% of her students achieve a grade C or above. A sample of 40 students is taken. Formulate the hypotheses.
实例 1 – 二项比例（单尾）： 一位老师声称，她超过 60% 的学生取得了 C 等或以上成绩。抽取了 40 名学生作为样本。请构造假设。
H₀: p = 0.6
H₁: p > 0.6
The claim “more than” points to a greater-than one-tailed test. The equal sign stays with H₀. The parameter p is the true proportion of all her students achieving the grade.
声称中的“more than”指向一个大于的单尾检验。等号留在 H₀ 中。参数 p 代表她所有学生中取得相应成绩的真实比例。
Example 2 – Normal mean (two-tailed): A bottling plant states that the mean volume of cola in a can is 330 ml. The quality control manager suspects the mean has changed. Formulate the hypotheses.
实例 2 – 正态均值（双尾）： 一家罐装厂声称每罐可乐的平均容量为 330 毫升。品控经理怀疑均值已经发生了变化。请构造假设。
H₀: μ = 330
H₁: μ ≠ 330
The word “changed” does not indicate direction, so a two-tailed test is appropriate. The hypotheses are about the population mean μ, not the sample mean volume.
“changed” 一词没有指示方向，因此用双尾检验是恰当的。假设是关于总体均值 μ 的，而非样本平均容量。
Example 3 – Correlation test: A researcher believes there is a negative linear correlation between hours spent on social media and exam marks. State the hypotheses for a correlation test.
实例 3 – 相关检验： 一位研究者认为，花在社交媒体上的小时数与考试成绩之间存在负线性相关。请陈述相关检验的假设。
H₀: ρ = 0
H₁: ρ < 0
The word “negative” gives the direction, hence a one-tailed alternative. The null assumes no linear correlation.
“negative” 一词给出了方向，因此采用单尾备择假设。原假设假定没有线性相关。
10. Summary and Exam Tips | 总结与考试技巧
Setting the hypotheses correctly is a relatively small but high-stakes part of an A-Level statistics question. Remember the following golden rules:
正确设定假设是 A-Level 统计题中看似轻巧却举足轻重的一步。请记住以下黄金法则：
Always use population parameters (p, μ, ρ), not sample statistics (p̂, x̄, r).
始终使用总体参数 (p, μ, ρ)，而非样本统计量 (p̂, x̄, r)。
H₀ must contain an equality (=).
H₀ 必须包含等号 (=)。
H₁ contradicts H₀ and contains <, >, or ≠.
H₁ 与 H₀ 相矛盾，并包含 <、> 或 ≠。
Choose one-tailed only when the wording gives a clear direction; otherwise use two-tailed.
只有在题目措辞给出清晰方向时才选用单尾检验；否则使用双尾。
Write the hypotheses in mathematical form as soon as you have identified the parameter.
一旦识别出参数，就立刻用数学形式写出假设。
eck the symbols: a common slip is writing H₀: μ = 0.5 when the data are actually binomial.
检查符号：一个常见的笔误是当数据实际上来自二项分布时，却写成 H₀: μ = 0.5。
In the exam, the hypotheses will usually be the first line of your answer. A clear, correct statement sets the tone for the entire solution and often earns the first mark. Practise writing them for a variety of contexts, and this skill will become automatic.
考场上，假设通常是你答卷的第一行。一个清晰、正确的陈述为整个解答定下基调，并且常常能轻松拿到第一分。多加练习在多种情景下写出假设，这项技能就会变得自动化。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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