📚 5C Hypothesis testing for a mean with unknown variance | 5C 均值未知方差的假设检验
In many real-world situations, we wish to test a claim about a population mean, but the population variance is unknown and must be estimated from a sample. This leads to the use of the t-distribution and the one-sample t-test. This chapter covers the theory, assumptions, and step-by-step procedures for conducting hypothesis tests for a mean when the variance is unknown, as required for IB Mathematics.
在许多实际场景中，我们需要检验关于总体均值的某个声明，但总体方差未知，必须通过样本进行估计。这就引出了 t 分布和单样本 t 检验。本章将涵盖方差未知时对均值进行假设检验的理论、前提条件以及完整步骤，完全对应 IB 数学课程的要求。
1. Introduction to the t-test | t 检验简介
When the population standard deviation σ is unknown, the standard normal z-test is no longer valid because using the sample standard deviation s introduces extra sampling error. William Sealy Gosset developed the t-distribution to account for this uncertainty, and the resulting procedure is known as the one-sample t-test.
当总体标准差 σ 未知时，标准正态 z 检验不再有效，因为使用样本标准差 s 会引入额外的抽样误差。威廉·西利·戈塞特提出了 t 分布来反映这种不确定性，由此产生的检验方法称为单样本 t 检验。
The t-distribution is similar in shape to the normal distribution but has heavier tails. These thicker tails reflect the greater variability that arises from estimating σ. As the sample size increases, the t-distribution approaches the standard normal distribution.
t 分布的形状与正态分布相似，但尾部更厚。较厚的尾部反映了由于估计 σ 而带来的更大变异性。随着样本容量增大，t 分布趋近于标准正态分布。
2. When to Use the t-distribution | 何时使用 t 分布
n is normal or if the sample size is sufficiently large (Central Limit Theorem, usually n ≥ 30).
对均值使用单样本 t 检验需要满足三个关键条件：(1) 总体标准差 σ 未知并用样本标准差 s 估计；(2) 数据来自简单随机样本；(3) 样本均值的抽样分布近似正态——如果总体服从正态分布，或者样本容量足够大（中心极限定理，通常 n ≥ 30），该条件成立。
In IB examinations, you will often be told whether to assume normality, or you may need to check a box plot or normal probability plot for symmetry and the absence of outliers.
在 IB 考试中，题目通常会明确是否可以假定正态性，或者你需要通过箱线图或正态概率图检验对称性及离群值情况。
3. Hypotheses Formulation | 假设的构建
tive hypothesis H₁ can be two-tailed (μ ≠ μ₀), left-tailed (μ < μ₀), or right-tailed (μ > μ₀), depending on the research question.
原假设总是声明总体均值 μ 等于某个指定值 μ₀：H₀: μ = μ₀。备择假设 H₁ 可以是双侧（μ ≠ μ₀）、左侧（μ < μ₀）或右侧（μ > μ₀），具体取决于研究问题。
Choosing a one-tailed or two-tailed test affects the p-value calculation and the position of the rejection region. Use a one-tailed test only when there is a clear directional suspicion before seeing the data.
选择单侧还是双侧检验会影响 p 值的计算及拒绝域的位置。只有在看到数据之前就有明确的方向性怀疑时，才应使用单侧检验。
4. The Test Statistic | 检验统计量
The test statistic measures how far the sample mean x̄ is from the hypothesised mean μ₀, scaled by the estimated standard error. It is given by:
检验统计量衡量样本均值 x̄ 与假设均值 μ₀ 之间的差距，并用估计标准误进行标准化。其公式为：
t = (x̄ − μ₀) / (s / √n)
where x̄ is the sample mean, μ₀ is the hypothesised population mean, s is the sample standard deviation, and n is the sample size. The denominator s/√n is the estimated standard error of the mean.
其中 x̄ 为样本均值，μ₀ 为假设的总体均值，s 为样本标准差，n 为样本容量。分母 s/√n 是均值的估计标准误。
Under the null hypothesis, this statistic follows a t-distribution with degrees of freedom that depend on the sample size.
在原假设条件下，该统计量服从一个自由度依赖于样本容量的 t 分布。
5. Degrees of Freedom | 自由度
For a one-sample t-test, the degrees of freedom (df) are given by n − 1. The value of n − 1 represents the number of independent pieces of information available for estimating the variance after fixing the mean.
单样本 t 检验的自由度 (df) 为 n − 1。n − 1 表示在确定均值后可用于估计方差的独立信息数量。
ractically indistinguishable, which is why some textbooks approximate with z for large samples.
随着自由度增加，t 分布越来越接近标准正态分布。当 df ≥ 30 时，这两个分布几乎无法区分，因此一些教材在大样本情况下用 z 近似。
6. Computing the p-value | 计算 p 值
The p-value is the probability, under the null hypothesis, of obtaining a test statistic at least as extreme as the one observed. For a two-tailed test, it is the sum of both tail probabilities; for one-tailed tests, it is the probability in the direction of H₁. Typically, you will use a GDC, statistical software, or t-tables.
p 值是在原假设下，得到至少与观测值一样极端的检验统计量的概率。对于双侧检验，它是两个尾部概率之和；对于单侧检验，则是 H₁ 所指方向的尾部概率。通常会使用图形计算器、统计软件或 t 分布表。
iled test, the p-value is P(T < −1.988). With the help of a GDC, you might find p ≈ 0.031.
例如，若在左侧检验中 t = −1.988，df = 19，则 p 值为 P(T < −1.988)。借助图形计算器，可得 p ≈ 0.031。
If the p-value is less than the significance level α (commonly 0.05), we reject the null hypothesis. Otherwise, we fail to reject it.
如果 p 值小于显著性水平 α（通常为 0.05），则拒绝原假设；否则不拒绝原假设。
7. Critical Region and Decision Rule | 拒绝域与决策规则
Rather than comparing the p-value to α, we can compare the observed t-statistic to a critical value t* from the t-distribution that corresponds to α and df. The critical region is the set of t-values that lead to rejection of H₀.
除了比较 p 值与 α，我们也可以将观测 t 统计量与 t 分布中对应于 α 和 df 的临界值 t* 进行比较。拒绝域是那些导致拒绝 H₀ 的 t 值集合。
For a two-tailed test at α = 0.05, the critical values are ±t* such that P(|T| > t*) = 0.05. If the test statistic falls in the critical region (|t| > t*), we reject H₀.
对于 α = 0.05 的双侧检验，临界值为 ±t*，满足 P(|T| > t*) = 0.05。如果检验统计量落入拒绝域（|t| > t*），则拒绝 H₀。
In the IB context, the p-value approach with a GDC is more commonly used, but both methods are acceptable and lead to the same conclusion.
在 IB 课程中，更多使用 GDC 进行 p 值比较，但两种方法均可接受且得到一致结论。
8. Confidence Intervals for the Mean | 均值的置信区间
l for the population mean μ, when σ is unknown, is constructed using the t-distribution: x̄ ± t* × (s / √n), where t* is the critical value for a confidence level (1 − α)100% with df = n − 1.
当 σ 未知时，总体均值 μ 的置信区间利用 t 分布构造：x̄ ± t* × (s / √n)，其中 t* 对应置信水平 (1 − α)100% 且自由度为 n − 1 的临界值。
Hypothesis testing and confidence intervals are closely linked. If a two-tailed test at significance level α is conducted and μ₀ lies outside the (1 − α)100% confidence interval, we reject H₀. Conversely, if μ₀ lies inside the interval, we fail to reject H₀.
假设检验与置信区间紧密关联。如果在显著性水平 α 下进行双侧检验，且 μ₀ 落在 (1 − α)100% 置信区间之外，则拒绝 H₀；反之，若 μ₀ 在区间内，则不拒绝 H₀。
9. Assumptions for the t-test | t 检验的前提假设
The validity of the one-sample t-test rests on three main assumptions. First, the sample must be randomly selected from the population. Second, the observations must be independent (e.g., no pairing or clustering). Third, the population distribution should be approximately normal, or the sample size should be large enough for the Central Limit Theorem to hold well.
单样本 t 检验的有效性依赖于三个主要前提。第一，样本必须从总体中随机抽取。第二，观测值必须相互独立（例如，无配对或聚类结构）。第三，总体分布应近似正态，或者样本容量足够大以便中心极限定理良好地起作用。
For small samples (n < 15), serious departure from normality can invalidate the test. When n is between 15 and 30, moderate skewness is acceptable if there are no outliers. For n > 30, the t-test is robust except for extreme outliers.
对于小样本（n < 15），严重偏离正态性会使检验失效。当 n 介于 15 与 30 之间时，若没有离群值，中等程度的偏态尚可接受。当 n > 30 时，除非存在极端离群值，t 检验较为稳健。
10. Worked Example | 实例分析
pects the true mean is lower and obtains a random sample of 20 bulbs. The sample mean is x̄ = 1462 hours and the sample standard deviation is s = 90 hours. Test at the 5% significance level whether the true mean is less than 1500 hours.
某制造商声称其灯泡的平均寿命为 1500 小时。检查员怀疑真实均值偏低，随机抽取了 20 个灯泡，样本均值为 x̄ = 1462 小时，样本标准差为 s = 90 小时。在 5% 显著性水平下检验真实均值是否小于 1500 小时。
Step 1: State the hypotheses. H₀: μ = 1500, H₁: μ < 1500 (left-tailed test).
第一步：陈述假设。 H₀: μ = 1500，H₁: μ < 1500（左侧检验）。
Step 2: Compute the test statistic. t = (1462 − 1500) / (90 / √20) = −38 / (90 / 4.4721) = −38 / 20.124 = −1.888.
第二步：计算检验统计量。 t = (1462 − 1500) / (90 / √20) = −38 / (90 / 4.4721) = −38 / 20.124 = −1.888。
Step 3: Determine the p-value. Using a GDC with df = 19, the left-tail probability P(T < −1.888) ≈ 0.037.
第三步：确定 p 值。 使用 GDC，df = 19，左侧尾部概率 P(T < −1.888) ≈ 0.037。
Step 4: Conclusion. Since p = 0.037 < 0.05, we reject H₀. There is sufficient evidence at the 5% level to support the inspector's belief that the mean lifetime is less than 1500 hours.
第四步：结论。 因为 p = 0.037 < 0.05，拒绝 H₀。在 5% 显著性水平下有足够证据支持检查员的看法，即平均寿命低于 1500 小时。
11. Interpreting Results | 结果解释
A small p-value indicates that the observed sample mean is far from what we would expect if H₀ were true. It does not measure the size of the effect; for that, a confidence interval or effect size such as Cohen's d is more informative.
较小的 p 值表明，如果 H₀ 为真，观测到的样本均值与期望值相差甚远。p 值并不度量效应大小；要了解效应大小，应参考置信区间或 Cohen's d 等效应量指标。
Be cautious about Type I and Type II errors. Rejecting H₀ when it is true is a Type I error (probability α). Failing to reject H₀ when H₁ is true is a Type II error (probability β). The power of the test (1 − β) increases with sample size.
注意 I 类错误和 II 类错误。拒绝真实的 H₀ 构成 I 类错误（概率为 α）；当 H₁ 为真时未能拒绝 H₀ 构成 II 类错误（概率为 β）。检验的效能 (1 − β) 随样本容量增加而提高。
12. Summary and Tips | 总结与提示
The t-test for a mean with unknown variance is a fundamental statistical tool. Remember: identify the test as one-sample t when σ is unknown and normality can be assumed; always define hypotheses clearly; calculate t with x̄, μ₀, s, and √n tical value using df = n − 1; and always state a conclusion in the context of the problem.
方差未知时对均值的 t 检验是一个基本统计工具。记忆要点：当 σ 未知且可假定正态性时选择单样本 t 检验；明确写出假设；用 x̄、μ₀、s 和 √n 计算 t；用自由度 df = n − 1 求 p 值或临界值；最后务必结合问题情境给出结论。
Many IB exam questions ask you to interpret a p-value or link a confidence interval to the test result. Practise using your GDC to generate both the test statistic and p-value efficiently. Familiarise yourself with the 2-SampTTest and other t‑based procedures as well.
许多 IB 考题会要求解释 p 值，或将置信区间与检验结果联系起来。多练习使用图形计算器高效生成检验统计量和 p 值。同时熟悉双样本 t 检验等其他基于 t 分布的方法。
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