📚 Review Set 16C: Probability Distributions and Statistical Hypothesis Testing | 复习集16C：概率分布与统计假设检验
In this Review Set we bring together fundamental concepts of probability distributions and statistical inference. The problems move from calculating probabilities for discrete and continuous variables to constructing confidence intervals and performing formal hypothesis tests. Mastery of these techniques is essential for the IB Mathematics analysis and approaches course and for any data-driven study.
在本复习集中，我们将整合概率分布与统计推断的基本概念。题目涵盖离散和连续变量的概率计算、构建置信区间以及执行正式的假设检验。掌握这些技巧对 IB 数学分析与方法的课程以及任何数据驱动的研究都至关重要。
1. Discrete Random Variables and Expectation | 离散随机变量与期望
A discrete random variable X takes a countable set of possible outcomes, each with a probability P(X = x). The sum of all probabilities must equal 1. The expected value E(X) represents the long‑run average, while variance Var(X) measures spread.
离散随机变量 X 取一组可数的可能结果，每个结果具有概率 P(X = x)。所有概率之和必须等于 1。期望值 E(X) 代表长期平均值，而方差 Var(X) 衡量离散程度。
- E(X) = Σ x · P(X = x)
- Var(X) = E(X²) – [E(X)]²
For example, a spinner gives scores 0, 2, 5 with probabilities 0.5, 0.3, 0.2. Then E(X) = 0×0.5 + 2×0.3 + 5×0.2 = 1.6. The variance is computed from E(X²) = 0²×0.5 + 2²×0.3 + 5²×0.2 = 6.2, so Var(X) = 6.2 – 1.6² = 3.64.
例如，一个转盘给出分数 0、2、5，概率分别为 0.5、0.3、0.2。则 E(X) = 0×0.5 + 2×0.3 + 5×0.2 = 1.6。方差通过 E(X²) = 0²×0.5 + 2²×0.3 + 5²×0.2 = 6.2 计算，因此 Var(X) = 6.2 – 1.6² = 3.64。
2. The Binomial Distribution | 二项分布
A binomial experiment consists of n independent trials, each with two outcomes (success/failure) and constant success probability p. The random variable X ~ B(n, p) counts the number of successes.
二项试验由 n 次独立试验组成，每次试验只有两种结果（成功/失败），且成功的概率 p 恒定。随机变量 X ~ B(n, p) 计数成功的次数。
P(X = k) = C(n, k) pᵏ (1 – p)ⁿ⁻ᵏ, where C(n, k) = n! / [k!(n – k)!]
For instance, if a fair die is rolled 8 times and we count the number of sixes, then n = 8, p = 1/6. The probability of exactly two sixes is C(8,2)×(1/6)²×(5/6)⁶ ≈ 0.260.
例如，若投掷一颗公平的骰子 8 次，计数出现 6 的次数，则 n = 8，p = 1/6。恰好出现两次 6 的概率为 C(8,2)×(1/6)²×(5/6)⁶ ≈ 0.260。
3. Mean and Variance of the Binomial | 二项分布的均值与方差
For X ~ B(n, p), the expected value and variance are given by simple formulas that avoid the need for summation. These are widely used in hypothesis testing and quality control.
对于 X ~ B(n, p)，期望值和方差由简单的公式给出，无需逐项求和。这些公式广泛用于假设检验与质量控制。
- E(X) = np
- Var(X) = np(1 – p)
If 12% of a large batch of components are defective and we test a random sample of 50, the expected number of defectives is 50 × 0.12 = 6, with variance 50 × 0.12 × 0.88 = 5.28 and standard deviation about 2.30.
若一批大量零件中有 12% 存在缺陷，我们随机测试 50 件样品，则缺陷品期望数量为 50 × 0.12 = 6，方差为 50 × 0.12 × 0.88 = 5.28，标准差约为 2.30。
4. The Normal Distribution | 正态分布
The normal distribution models continuous data that cluster around a mean μ with spread determined by standard deviation σ. The curve is bell‑shaped and symmetric, with total area under the curve equal to 1.
正态分布用于建模围绕均值 μ 聚集、分散程度由标准差 σ 决定的连续数据。曲线呈钟形且对称，曲线下总面积等于 1。
f(x) = 1/(σ√(2π)) e⁻⁽ˣ⁻μ⁾²/⁽²σ²⁾
About 68% of the data lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ. These empirical benchmarks help estimate probabilities without integration.
大约 68% 的数据落在 μ ± σ 之间，95% 落在 μ ± 2σ 之间，99.7% 落在 μ ± 3σ 之间。这些经验基准有助于无需积分便能估计概率。
5. Standard Normal and Z‑scores | 标准正态与 Z 分数
Converting a raw score X to a Z‑score standardises the value so that deviance from the mean is measured in units of standard deviation. The standard normal distribution has mean 0 and standard deviation 1.
将原始分数 X 转换为 Z 分数，可使数值标准化，从而用标准差为单位衡量与均值的偏差。标准正态分布的均值为 0，标准差为 1。
Z = (X – μ) / σ
To find P(X < a), compute z = (a – μ)/σ and consult the standard normal table or use technology. For example, if μ = 170 cm and σ = 8 cm, the probability a person is shorter than 180 cm corresponds to Z = 1.25, giving about 0.8944.
要计算 P(X < a)，先求 z = (a – μ)/σ，然后查标准正态表或使用技术工具。例如，若 μ = 170 cm，σ = 8 cm，则身高低于 180 cm 的概率对应 Z = 1.25，约 0.8944。
6. Central Limit Theorem | 中心极限定理
When taking samples of size n from a population with mean μ and standard deviation σ, the distribution of sample means x̄ approaches a normal distribution as n increases, regardless of the population shape, provided n is sufficiently large (usually n ≥ 30).
当从均值为 μ、标准差为 σ 的总体中抽取容量为 n 的样本时，随着 n 增大，样本均值 x̄ 的分布将趋近正态分布，无论总体的形状如何，只要 n 足够大（通常 n ≥ 30）。
x̄ ~ N(μ, σ/√n)
This theorem underpins confidence intervals and hypothesis tests for the mean when the population standard deviation is known. It allows us to use the normal model for inferential statistics on large samples even if the original data are skewed.
该定理为已知总体标准差时均值的置信区间和假设检验奠定了基础。它使得即使原始数据偏斜，我们也能用正态模型对大样本进行推断统计。
7. Confidence Intervals for the Mean | 均值的置信区间
A confidence interval gives a range of plausible values for the population mean μ. A 95% confidence interval is interpreted as: if we took many samples, about 95% of the constructed intervals would contain μ.
置信区间给出了总体均值 μ 的合理取值范围。一个 95% 置信区间的解释为：若多次抽样，大约 95% 构造出的区间会包含 μ。
x̄ ± z* × σ / √n
For a 95% interval, z* = 1.960. For a 99% interval, z* = 2.576. If a sample of 40 bolts has mean diameter 12.1 mm and σ = 0.3 mm, the 95% confidence interval is 12.1 ± 1.96 × 0.3/√40, i.e. (12.01, 12.19) mm.
对于 95% 区间，z* = 1.960；对于 99% 区间，z* = 2.576。若 40 颗螺栓的样本平均直径为 12.1 mm，σ = 0.3 mm，则 95% 置信区间为 12.1 ± 1.96 × 0.3/√40，即 (12.01, 12.19) mm。
8. Hypothesis Testing Logic | 假设检验的逻辑
A hypothesis test quantifies the evidence against a null hypothesis H₀. The alternative hypothesis H₁ states what we suspect is true. A test compares observed data to what is expected under H₀, producing a test statistic.
假设检验量化了反对零假设 H₀ 的证据。备择假设 H₁ 描述我们怀疑为真的情形。检验将观察到的数据与 H₀ 下的期望值进行比较，产生一个检验统计量。
We define a significance level α (often 0.05). If the probability of obtaining a result at least as extreme as the observed one, assuming H₀ true, is less than α, we reject H₀. This probability is the p‑value.
我们定义显著性水平 α（通常为 0.05）。若在 H₀ 为真的假设下，得到至少与观察结果一样极端的概率小于 α，则拒绝 H₀。这个概率就是 p 值。
9. Testing a Proportion | 检验一个比例
Suppose a manufacturer claims that 90% of its microchips are fault‑free. A sample of 120 chips finds 100 fault‑free. We set H₀: p = 0.9, H₁: p < 0.9 and choose α = 0.05.
假设一家制造商声称其微芯片的 90% 没故障。120 片样本中发现 100 片无故障。设定 H₀: p = 0.9，H₁: p < 0.9，并选取 α = 0.05。
Under H₀ the number of fault‑free chips X ~ B(120, 0.9) has mean 108 and variance 10.8. The observed value 100 is lower. The Z‑statistic is Z = (100 – 108)/√10.8 ≈ –2.43, giving a p‑value ≈ 0.0075. Since p < 0.05, we reject the claim and conclude the true proportion is lower than 90%.
在 H₀ 下，无故障芯片数量 X ~ B(120, 0.9)，均值为 108，方差为 10.8。观察值 100 偏低。Z 统计量为 Z = (100 – 108)/√10.8 ≈ –2.43，对应 p 值约 0.0075。由于 p < 0.05，我们拒绝声明，认为真实比例低于 90%。
10. Chi‑squared Goodness‑of‑fit Test | 卡方拟合优度检验
To test whether observed frequencies fit an expected distribution, we use the chi‑squared statistic. This is especially useful for categorical data such as genetic ratios or survey responses.
为了检验观察频数是否符合某一期望分布，我们使用卡方统计量。这对于分类数据（如遗传比例或问卷调查）特别有用。
χ² = Σ (O − E)² / E
Degrees of freedom = number of categories – 1. A large χ² value indicates poor fit. Compare the computed χ² to a critical value from the χ² distribution at a given α to decide whether to reject H₀.
自由度 = 类别数 – 1。较大的 χ² 值表示拟合不佳。将计算得到的 χ² 与给定 α 下 χ² 分布的临界值比较，决定是否拒绝 H₀。
| Category | Observed (O) | Expected (E) | (O–E)²/E |
| Red | 52 | 50 | 0.08 |
| Blue | 30 | 50 | 8.00 |
| Green | 68 | 50 | 6.48 |
| Total | 150 | 150 | χ² = 14.56 |
With 2 degrees of freedom, the critical value at α = 0.05 is 5.991. Since 14.56 > 5.991, we reject H₀ and conclude the observed distribution differs significantly from the expected uniform distribution.
自由度为 2，α = 0.05 下的临界值为 5.991。由于 14.56 > 5.991，我们拒绝 H₀，认为观察分布与期望的均匀分布存在显著差异。
11. Two‑sample t‑test Concept | 双样本 t 检验概念
When comparing the means of two independent groups with unknown population standard deviations, the two‑sample t‑test is appropriate. It checks whether the observed difference in sample means is large enough to indicate a real population difference.
当比较两个独立组的均值且总体标准差未知时，使用双样本 t 检验。它检验样本均值的观察差异是否足够大，以至于表明总体确实存在差异。
The test statistic uses the pooled or unpooled estimate of standard error. For equal variances, the pooled variance s²ₚ is calculated. The degrees of freedom are n₁ + n₂ – 2. Technology or t‑tables deliver the p‑value for the decision.
检验统计量使用合并或未合并的标准误差估计。在等方差情形下，计算合并方差 s²ₚ。自由度为 n₁ + n₂ – 2。通过技术工具或 t 表获得 p 值以做出决策。
12. Connecting Probability to Decision Making | 概率与决策制定的衔接
All the topics in this review set converge on the principle of making decisions under uncertainty. Whether calculating a probability, constructing an interval, or testing a hypothesis, we rely on probability models to quantify risk and error.
本复习集的所有主题都汇聚于在不确定情况下做出决策的原则。无论是计算概率、构造区间还是检验假设，我们都依赖概率模型来量化风险和误差。
Critically, a p‑value is not the probability that H₀ is true. A non‑significant result does not prove H₀ true; it simply indicates insufficient evidence against it. Careful interpretation of statistical conclusions is a key skill for IB students and beyond.
至关重要的是，p 值并非 H₀ 为真的概率。不显著的结果并不能证明 H₀ 为真，仅表明反对它的证据不足。审慎解读统计结论是 IB 学生及以后所需的关键技能。
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