📚 Testing a Binomial Distribution as a Model | 二项分布模型检验
At AS level, a binomial distribution can be used as a probability model for counting successes in repeated independent trials. Formal testing usually means carrying out a hypothesis test for the success probability p, but the model itself must first satisfy clear conditions, and the conclusion must always be written in the context of the original problem.
在 AS 阶段，二项分布可作为概率模型，用于计算重复独立试验中成功的次数。正式的检验通常是对成功概率 p 进行假设检验，但模型本身必须先满足明确的条件，而且结论必须结合原题情境来书写。
1. What Is a Binomial Model? | 什么是二项分布模型
A binomial model describes the number of successes in a fixed number of independent trials. If X is the number of successes, we write X ~ B(n, p), where n is the number of trials and p is the probability of success in each trial. The distribution is discrete, and each trial has exactly two possible outcomes.
二项分布模型描述固定次独立试验中成功的次数。若 X 表示成功次数，记作 X ~ B(n, p)，其中 n 为试验次数，p 为每次试验成功的概率。该分布是离散的，每次试验只有两种可能结果。
P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ
Here C(n, r) is the binomial coefficient, often written as ⁿCᵣ, which counts the number of orders in which r successes can occur among n trials. The two parameters n and p completely determine the model, so the whole shape of the distribution is fixed once they are known.
其中 C(n, r) 是二项式系数，也常写作 ⁿCᵣ，它统计的是 n 次试验中 r 次成功可能出现的排列方式数目。参数 n 与 p 完全决定了这个模型，因此一旦它们确定，整个分布的形状也就确定了。
E(X) = np, Var(X) = np(1 − p)
standard deviation √(np(1 − p)) tells you the typical distance of the count from its mean.
均值与方差可用于初步判断一组观测数据在某个声称的模型下是否合理，而标准差 √(np(1 − p)) 则刻画了计数值偏离均值的典型幅度。
2. Conditions for a Binomial Model | 二项分布模型的适用条件
A binomial model is only appropriate when four conditions are met. If any condition fails, the binomial distribution may give misleading probabilities, and any hypothesis test based on it will be unreliable.
二项分布模型只有在满足四个条件时才适用。如果任一条件不成立，二项分布可能会给出误导性的概率，而基于它的任何假设检验也都不可靠。
| There is a fixed number of trials, n. | 试验次数 n 是固定的。 |
| Each trial has exactly two outcomes: success or failure. | 每次试验只有两种结果：成功或失败。 |
| The probability of success p is the same for every trial. | 每次试验成功概率 p 保持不变。 |
| The trials are independent. | 各次试验相互独立。 |
The four conditions are easy to remember as BINS: Binary outcomes, Independence, a fixed Number of trials, and the Same probability of success each time. In an exam answer you should state these conditions in the context of the question, not just list them in the abstract.
这四个条件可以用 BINS 来记忆：结果只有两种（Binary）、试验相互独立（Independent）、次数固定（Number）、每次成功概率相同（Same）。在考试作答时，应结合题目情境说明这些条件，而不是抽象地罗列。
For example, if a machine produces items and each item is either defective or not, the trials are the items; independence would require that one defective item does not make the next item more likely to be defective, and a constant p would require the machine's setting not to drift over time.
例如，某机器生产零件，每个零件要么合格要么不合格，此时"试验"就是这些零件：独立性要求一个次品不会让下一个零件更容易成为次品；而 p 保持不变则要求机器的设置在过程中没有漂移。
3. Why 'Testing' Matters | 为什么需要检验
Real data may look as if they come from a binomial model, but the underlying success probability could differ from a claimed value, or the trials may not be truly independent. At AS level, testing a binomial model usually means testing a claim about p, assuming the binomial conditions are satisfied.
真实数据可能看起来来自二项分布模型，但真实的成功概率可能与声称值不同，或者各次试验可能并不真正独立。在 AS 阶段，检验二项分布模型通常是指检验关于 p 的声明，并假设二项条件已经满足。
A claim such as "the coin is fair" or "the new drug cures 70% of patients" is a claim about a parameter, and a sample of observations will almost never match the claim exactly. Hypothesis testing gives a formal rule for deciding whether the difference between what we observed and what the model predicts is small enough to be explained by chance.
诸如"这枚硬币是均匀的"或"新药治愈率为 70%"这样的说法，本质上都是关于参数的断言，而观测样本几乎不可能与断言完全一致。假设检验给出一条形式化的规则，用来判断观测值与模型预测值之间的差异，小到能否仅用随机性来解释。
A full goodness-of-fit test comparing observed frequencies with expected frequencies across all values of X belongs to a later course, but the logic is the same: choose a test statistic, find its distribution when the model is true, and see whether the observed value is extreme.
把各取值的观测频数与期望频数做整体比较的完整拟合优度检验属于后续课程内容，但其逻辑是相同的：选定一个检验统计量，求出模型成立时该统计量的分布，再看观测值是否极端。
4. Setting Up a Hypothesis Test | 建立假设检验
We use hypotheses to make a formal decision about p. The null hypothesis H₀ states that p has a specified value, while the alternative hypothesis H₁ states that p is different in some direction. The test is always carried out on the assumption that H₀ is true.
我们用假设来对 p 作出形式化判断。原假设 H₀ 声明 p 等于某个指定值，而备择假设 H₁ 则声明 p 在某个方向上与之不同。检验始终是在假定 H₀ 成立的前提下进行的。
The three possible forms of H₁ are: p < p₀, which is a lower-tail test; p > p₀, which is an upper-tail test; and p ≠ p₀, which is a two-tailed test. The value p₀ is the claimed probability from the null hypothesis.
H₁ 有三种可能形式：p < p₀ 为左尾检验；p > p₀ 为右尾检验；p ≠ p₀ 为双尾检验。其中 p₀ 是原假设中声称的概率值。
You must also choose a significance level, usually 5% or 10% at AS level, sometimes 1%. The significance level is the probability of rejecting H₀ when it is actually true, so a smaller level means stronger evidence is needed before we reject.
你还必须选择显著性水平，在 AS 阶段通常取 5% 或 10%，有时取 1%。显著性水平是 H₀ 实际为真时却拒绝它的概率，因此水平越小，拒绝所需的证据就越强。
5. One-Tailed and Two-Tailed Tests | 单尾与双尾检验
The wording of the question determines the form of H₁. Words such as "increased", "more than", "at least" or "improved" suggest an upper-tail test, while "decreased", "less than" or "reduced" suggest a lower-tail test.
题目的措辞决定了 H₁ 的形式。"增加""多于""至少""提高"等词提示右尾检验，而"减少""少于""降低"等词提示左尾检验。
Words such as "different from", "has changed" or "is not the same as" suggest a two-tailed test, because a departure in either direction is interesting. In a two-tailed test at the 5% level, you compare the tail probability with 0.025 on each side rather than with 0.05 on one side.
"不同于""发生了变化""与……不一样"等措辞提示双尾检验，因为偏离任何一个方向都值得关注。在 5% 水平的双尾检验中，你要把单侧尾部概率与 0.025 比较，而不是把一侧概率与 0.05 比较。
Lower tail: H₁: p < p₀, reject if the observed count is small.
左尾：H₁: p < p₀，观测计数值偏小时拒绝 H₀。
Upper tail: H₁: p > p₀, reject if the observed count is large.
右尾：H₁: p > p₀，观测计数值偏大时拒绝 H₀。
Two tails: H₁: p ≠ p₀, reject if the observed count is extreme in either direction.
双尾：H₁: p ≠ p₀，观测计数值偏向任一端极端时拒绝 H₀。
A common error is to use a two-tailed test when the question clearly implies a direction, which doubles the p-value and can change the decision. Read the final sentence of the question carefully before writing H₁.
常见的错误是：题目明显暗示了方向，考生却用了双尾检验，这会使 p 值翻倍，甚至改变结论。因此在写 H₁ 之前，务必仔细阅读题目最后一句。
6. The Distribution Under H₀ | H₀ 成立时的分布
Once H₀ fixes the value of p, the distribution of the count X is completely known: X ~ B(n, p₀). The whole test follows from this single distribution, so identifying it correctly is the most important technical step.
一旦 H₀ 确定了 p 的取值，计数值 X 的分布就完全已知：X ~ B(n, p₀)。整个检验都源自这一分布，因此正确地写出它是技术上最关键的一步。
Note that n comes from the study design and is not tested; it is assumed fixed and known. Only p is the subject of the hypotheses, and the observed count r is the test statistic.
注意 n 来自研究设计，并不是被检验的对象，它被假定为固定且已知。只有 p 是假设的主题，而观测到的成功次数 r 就是检验统计量。
When you state the model, it is good practice to write a sentence such as: "Assuming H₀ is true, X ~ B(20, 0.3), provided the trials are independent and p is constant." This shows the examiner that you understand the assumptions behind the calculation.
在写出模型时，最好加上一句说明，例如："假定 H₀ 成立，且各次试验相互独立、p 保持不变，则 X ~ B(20, 0.3)。"这能向阅卷人表明你理解计算背后的假定。
7. Critical Regions and Significance Levels | 临界域与显著性水平
h that P(X ≤ c) ≤ α, where α is the significance level; the critical region is then X ≤ c.
临界域是使 H₀ 被拒绝的检验统计量取值的集合。对于左尾检验，你要找出满足 P(X ≤ c) ≤ α 的最大整数 c（α 为显著性水平），此时临界域为 X ≤ c。
tical region X ≥ c. For a two-tailed test you apply the same rule with α ÷ 2 in each tail.
对于右尾检验，你要找出满足 P(X ≥ c) ≤ α 的最小整数 c，临界域为 X ≥ c。对于双尾检验，则对每一侧尾部使用 α ÷ 2 的相同规则。
te, the actual significance level is usually less than the nominal one. For instance, if X ~ B(20, 0.3) and we test at the 5% t 0.1070 > 0.05, the critical region is X ≤ 2 and the actual significance level is 3.54%, not exactly 5%.
由于 X 是离散的，实际显著性水平通常小于名义水平。例如当 X ~ B(20, 0.3)、在 5% 水平上检验 H₁: p < 0.3 时，P(X ≤ 2) = 0.0354，而 P(X ≤ 3) = 0.1070。因为 0.0354 ≤ 0.05 但 0.1070 > 0.05，所以临界域为 X ≤ 2，实际显著性水平是 3.54%，而不是恰好 5%。
P(X ≤ 2) = 0.0354 ≤ 0.05, P(X ≤ 3) = 0.1070 > 0.05 → Critical region: X ≤ 2
Always state the critical region using the inequality that matches the direction of H₁, and remember that a value equal to the boundary c lies inside the critical region.
写临界域时，务必使用与 H₁ 方向一致的不等号，并记住恰好等于边界值 c 的观测值属于临界域。
8. The p-Value Approach | p 值方法
The p-value is the probability of obtaining a result at least as extreme as the one observed, assuming H₀ is true. It is compared directly with the significance level: if the p-value is less than or equal to α, we reject H₀.
p 值是在假定 H₀ 成立时，得到至少与观测结果同样极端的结果的概率。它可直接与显著性水平比较：若 p 值小于或等于 α，则拒绝 H₀。
-tailed test you double the smaller of the two tail probabilities, which is equivalent to working with α ÷ 2 in each tail.
对于观测值为 r 的左尾检验，p 值等于 P(X ≤ r)；右尾检验则为 P(X ≥ r)。对于双尾检验，要把两个尾部概率中较小的那个翻倍，这等价于在每侧尾部使用 α ÷ 2。
The critical region method and the p-value method always give the same decision, so you may use whichever the question prefers. The p-value has the advantage of showing how strong the evidence is, while the critical region is quicker to apply once it has been found.
临界域法与 p 值法给出的结论总是一致的，因此可选用题目偏好的方法。p 值的优点是能显示证据的强度，而临界域一旦求出，使用起来更快。
9. Writing the Conclusion in Context | 结合情境书写结论
A conclusion must contain three elements: the decision about H₀, the significance level used, and the meaning of the decision in the context of the problem. A bare statement such as "reject H₀" scores poorly because it says nothing about the real-world claim.
结论必须包含三个要素：对 H₀ 的判断、所用的显著性水平、以及该判断在题目情境中的含义。只写"拒绝 H₀"这样的短语得分很低，因为它没有涉及现实中的断言。
A good conclusion reads: "There is sufficient evidence, at the 5% level, to reject H₀ and conclude that the proportion of defective items has decreased." Note the use of "sufficient evidence" rather than "proof", because a hypothesis test never proves anything with certainty.
一个良好的结论是："在 5% 显著性水平下，有充分证据拒绝 H₀，从而认为次品比例已经下降。"注意这里用"有充分证据"而不是"证明了"，因为假设检验从来不能确定性地证明任何事。
When we do not reject H₀, the correct wording is "there is insufficient evidence to reject H₀", not "H₀ is true" or "we accept H₀". Failing to reject simply means the observed data are compatible with the claimed value, not that the claim has been verified.
当我们不拒绝 H₀ 时，正确的表述是"没有足够证据拒绝 H₀"，而不是"H₀ 为真"或"我们接受 H₀"。不拒绝只意味着观测数据与声称值相容，并不意味着该断言得到了证实。
Two types of error are possible. A Type I error occurs when H₀ is true but is rejected, and its probability is the significance level. A Type II error occurs when H₀ is false but is not rejected, and its probability depends on the true value of p.
可能出现两类错误。第一类错误指 H₀ 为真却被拒绝，其概率就是显著性水平；第二类错误指 H₀ 为假却未被拒绝，其概率取决于 p 的真实取值。
10. Worked Example: A One-Tailed Test | 例题精讲：单尾检验
A machine is supposed to produce 30% defective items. After a repair, a sample of 20 items contains only 1 defective item. Test, at the 5% level, whether the proportion of defective items has decreased.
某机器本应生产 30% 的次品。维修后抽取 20 个零件，其中仅有 1 个次品。试在 5% 显著性水平下检验次品比例是否下降。
hat an item is defective. Then H₀: p = 0.3 and H₁: p < 0.3, which is a lower-tail test.
第一步：定义参数与假设。设 p 为零件为次品的概率，则 H₀: p = 0.3，H₁: p < 0.3，属于左尾检验。
Step 2: State the model and its assumptions. Assuming H₀ and that items are independent with constant p, X ~ B(20, 0.3), where X is the number of defective items in the sample.
第二步：写出模型及其假设。假定 H₀ 成立，且各零件相互独立、p 保持不变，则 X ~ B(20, 0.3)，其中 X 为样本中的次品数。
.1070 > 0.05, so the critical region is X ≤ 2 and the actual significance level is 3.54%.
第三步：求临界域。由前面的计算，P(X ≤ 2) = 0.0354 ≤ 0.05，P(X ≤ 3) = 0.1070 > 0.05，因此临界域为 X ≤ 2，实际显著性水平为 3.54%。
cal region X ≤ 2. So we reject H₀ and conclude that there is sufficient evidence, at the 5% level, that the proportion of defective items has decreased.
第四步：比较并下结论。观测值 X = 1，落在临界域 X ≤ 2 之内，因此拒绝 H₀，并得出结论：在 5% 水平下有充分证据表明次品比例已经下降。
X ≤ 1) = 0.0008 + 0.0068 = 0.0076, which is also less than 0.05, giving the same decision.
若改用 p 值法，P(X ≤ 1) = 0.0008 + 0.
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