📚 Hypothesis Testing with the Normal Distribution | 正态分布假设检验
In Edexcel A-Level Mathematics, hypothesis testing with the normal distribution extends the earlier ideas of binomial hypothesis testing to continuous data. It is used when we wish to test a population mean μ under the assumption that the sample data are normally distributed, or that the sample size is large enough for the Central Limit Theorem to apply.
在 Edexcel A-Level 数学中，正态分布假设检验将二项分布假设检验的思想推广到连续数据。当样本数据服从正态分布，或样本量足够大使得中心极限定理适用时，我们用它来检验总体均值 μ。
1. What Is a Hypothesis Test? | 什么是假设检验？
A hypothesis test is a formal procedure for deciding whether a statement about a population parameter is supported by sample data. It begins with two competing hypotheses: the null hypothesis H₀ and the alternative hypothesis H₁.
假设检验是一种正式程序，用于判断关于总体参数的陈述是否得到样本数据的支持。它从两个对立的假设开始：原假设 H₀ 和备择假设 H₁。
The null hypothesis represents the default or established claim, usually that the parameter equals a specific value. The alternative hypothesis represents the new claim or the change we are testing for.
原假设代表默认的或已有的声明，通常表示参数等于某个特定值。备择假设代表我们要检验的新声明或变化。
We use sample data to decide whether there is enough evidence to reject H₀. A test never proves H₀ or H₁; it only measures whether the data are consistent with H₀.
我们用样本数据判断是否有足够证据拒绝 H₀。检验永远不能证明 H₀ 或 H₁，它只衡量数据与 H₀ 的一致程度。
Every test ends with a decision and a contextual conclusion. The decision is either 'reject H₀' or 'do not reject H₀', never 'accept H₀' as a proof.
每次检验最终都要给出决策和结合情境的结论。决策只能是 '拒绝 H₀' 或 '不拒绝 H₀'，而不能把 '接受 H₀' 当作证明。
2. Why Use the Normal Distribution? | 为什么使用正态分布？
Normal hypothesis testing is used for continuous data when the population mean μ is unknown but the population standard deviation σ is known, or when the sample size is large.
当总体均值 μ 未知但总体标准差 σ 已知，或样本量较大时，连续数据可以使用正态分布假设检验。
The Central Limit Theorem says that for a large sample size n, usually n > 30, the sample mean X̄ is approximately normally distributed regardless of the shape of the population distribution.
中心极限定理指出，当样本量 n 较大，通常 n > 30 时，无论总体分布的形状如何，样本均值 X̄ 都近似服从正态分布。
This is why the method applies so widely: we do not always need the original data to be normal, only the sampling distribution of the mean.
这正是该方法应用广泛的原因：我们并不总要求原始数据服从正态分布，只要求均值的抽样分布近似正态。
If the original population is already normal, the sampling distribution of X̄ is exactly normal for any sample size n.
如果原始总体本身已经是正态分布，那么无论样本量 n 是多少，X̄ 的抽样分布都精确服从正态分布。
3. Setting Up Hypotheses | 建立假设
an μ, the null hypothesis is usually written as H₀: μ = μ₀, where μ₀ is the claimed value.
对于单个总体均值 μ 的检验，原假设通常写作 H₀: μ = μ₀，其中 μ₀ 是声称的数值。
The alternative hypothesis can be one-tailed or two-tailed. A two-tailed test uses H₁: μ ≠ μ₀, while one-tailed tests use H₁: μ > μ₀ or H₁: μ < μ₀.
备择假设可以是单尾或双尾的。双尾检验用 H₁: μ ≠ μ
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