Testing a Binomial Distribution as a Model | 检验二项分布作为模型

📚 Testing a Binomial Distribution as a Model | 检验二项分布作为模型

In AS level Statistics, one important skill is to decide whether an observed set of data can be modelled by a binomial distribution. This involves a formal hypothesis test known as the chi‑squared goodness‑of‑fit test. You begin by assuming that the binomial distribution with a specified (or estimated) parameter p is a suitable model, and then you compare observed frequencies with the frequencies you would expect if the model were true. If the difference is too large, you reject the model. This article walks you through the entire process, from setting up hypotheses to interpreting p‑values and critical regions.

在AS统计学中，一项重要的技能是判断一组观测数据是否可以用二项分布来建模。这涉及到一个称为卡方拟合优度检验的正式假设检验。你首先假设带有指定（或估计）参数 p 的二项分布是合适的模型，然后将观测频数与如果模型成立时预期的频数进行比较。如果差异过大，就拒绝该模型。本文将从建立假设到解释 p 值和拒绝域，带你走完整个流程。

1. Why Test a Binomial Model? | 为什么要检验二项模型？

Many real‑world experiments consist of a fixed number of independent trials, each with two possible outcomes. The binomial distribution B(n, p) is the natural candidate to describe the number of successes. However, the assumptions of independence and constant probability of success may not hold. A goodness‑of‑fit test allows you to check whether the data are consistent with the binomial model, or whether there is evidence that the binomial distribution is inappropriate.

许多现实世界的实验包含固定次数的独立试验，每次试验有两种可能的结果。二项分布 B(n, p) 是描述成功次数的自然候选。然而，独立性和每次试验成功概率恒定的假设可能不成立。拟合优度检验可以让你检查数据是否与二项模型一致，或者是否有证据表明二项分布不合适。

2. Chi‑Squared Goodness‑of‑Fit Test: An Overview | 卡方拟合优度检验概述

The chi‑squared (χ²) goodness‑of‑fit test compares the observed frequencies of each category with the expected frequencies calculated from the proposed model. The test statistic measures the total discrepancy, and large values indicate a poor fit. For a binomial model, the categories are the possible numbers of successes (0, 1, 2, …, n).

卡方 (χ²) 拟合优度检验将每一类的观测频数与根据所提模型计算出的期望频数进行比较。检验统计量衡量总差异，数值较大表明拟合较差。对于二项模型，类别就是可能的成功次数 (0, 1, 2, …, n)。

3. Stating the Hypotheses | 陈述假设

The null hypothesis H₀ always states that the binomial distribution is a suitable model for the data. The alternative hypothesis H₁ states that the binomial distribution is not a suitable model. For example, H₀: The data follow a B(10, 0.4) distribution. H₁: The data do not follow a B(10, 0.4) distribution. When p is estimated from the data, the hypotheses refer to a binomial model with that estimated p.

原假设 H₀ 总是陈述二项分布是数据的合适模型。备择假设 H₁ 陈述二项分布不是合适的模型。例如，H₀: 数据服从 B(10, 0.4) 分布。H₁: 数据不服从 B(10, 0.4) 分布。当 p 是从数据中估计出来时，假设指的是带有该估计 p 的二项模型。

4. Calculating Expected Frequencies | 计算期望频数

and p are known or estimated, you find the probability for each possible outcome x = 0, 1, …, n using the binomial formula: P(X = x) = ⁿCₓ pˣ (1−p)ⁿ⁻ˣ es you need to combine adjacent categories so that every expected frequency is at least 5.

一旦二项参数 n 和 p 已知或已估计，你用二项公式求每个可能结果 x = 0, 1, …, n 的概率：P(X = x) = ⁿCₓ pˣ (1−p)ⁿ⁻ˣ。将每个概率乘以观测总数 N 得到期望频数。有时你需要合并相邻的类别，使得每个期望频数至少为 5。

5. Observed Frequencies from Data | 数据中的观测频数

pts each, you might record the number of successes per trial. Tally these into the categories x = 0, 1, 2, …, n to form the observed frequencies Oᵢ.

你会得到一个频数表，显示实验中每个成功次数出现的次数。例如，在 50 次每次 10 次尝试的试验中，你可能记录每次试验的成功次数。将这些统计到类别 x = 0, 1, 2, …, n 中，形成观测频数 Oᵢ。

6. The Test Statistic Formula | 检验统计量公式

Eᵢ, where the sum runs over all categories after any necessary combining. Oᵢ are the observed frequencies and Eᵢ are the expected frequencies. This statistic follows approximately a χ² distribution, provided the expected frequencies are sufficiently large (at least 5 for each category).

检验统计量为 χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ，其中求和涵盖所有必要合并后的类别。Oᵢ 是观测频数，Eᵢ 是期望频数。只要期望频数足够大（每个类别至少为 5），该统计量近似服从 χ² 分布。

7. Degrees of Freedom | 自由度

The degrees of freedom ν for the chi‑ ining) − 1. If p was estimated from the data, one degree of freedom is lost for each estimated parameter, so ν = (number of categories) − 2. Recognizing this is crucial for finding the correct critical value.

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卡方检验的自由度 ν 取决于参数 p 是预先指定的还是从数据中估计的。如果 p 是给定的，ν = (合并后的类别数) − 1。如果 p 是从数据中估计的，每估计一个参数就损失一个自由度，因此 ν = (类别数) − 2。认识到这一点对于找到正确的临界值至关重要。

8. Conditions for the Test | 检验的条件

For the chi‑squared approximation to be valid, no expected frequency should be smaller than 1, and no more than 20% of the expected frequencies should be smaller than 5. If this is not met, you must combine adjacent categories until the condition holds. Also, the data must consist of independent observations; this is usually satisfied if each trial is distinct.

为了使卡方近似有效，任何期望频数都不应小于 1，且期望频数中小于 5 的比例不应超过 20%。如果不满足这一点，你必须合并相邻的类别直到条件成立。此外，数据必须由独立的观测组成；如果每次试验是独立的，这通常满足。

9. Finding the Critical Value | 查找临界值

on table with the correct degrees of freedom and the chosen significance level (commonly 5% e, you reject H₀. For example, with ν = 4 and α = 0.05, the critical value is χ²₄ (0.05) = 9.488.

使用卡方分布表，根据正确的自由度和选定的显著性水平（通常为 5%）。临界值是将拒绝域与接受域分开的数值。如果检验统计量超过临界值，则拒绝 H₀。例如，当 ν = 4 且 α = 0.05 时，临界值为 χ²₄ (0.05) = 9.488。

10. Interpreting the Result | 解释结果

If χ² > critical value, there is sufficient evidence at the chosen significance level to conclude that the binomial distribution does not provide an adequate fit. Otherwise, there is insufficient evidence to reject the binomial model. Always write your conclusion in the context of the original problem, mentioning both the test statistic and the critical value.

如果 χ² > 临界值，则在该显著水平下有足够证据断定二项分布不能提供足够的拟合。否则，没有足够证据拒绝二项模型。始终在原始问题的背景下写出你的结论，并提及检验统计量和临界值。

11. Worked Example: Seed Germination | 示例：种子发芽

er of germinating seeds follows a B(5, 0.6) distribution. The observed frequencies are: 0 successes: 2, 1: 6, 2: 14, 3: 12, 4: 5, 5: 1. Compute expected probabilities and multiply by 40. Combine categories to ensure E ≥ 5. The computed χ² might be compared against χ² with appropriate degrees of freedom. Conclude whether the binomial model is plausible.

假设将 100 粒种子以 5 粒为一组种植，记录 40 组中每组发芽的种子数。你想检验发芽种子数是否服从 B(5, 0.6) 分布。观测频数为：0 次成功：2, 1: 6, 2: 14, 3: 12, 4: 5, 5: 1。计算期望概率并乘以 40。合并类别以确保 E ≥ 5。计算出的 χ² 与适当自由度的 χ² 相比。得出结论该二项模型是否合理。

P(X=0) = ⁵C₀ (0.6)⁰ (0.4)⁵ = 0.01024, expected = 0.41

观测频数：0 成功 2，期望 0.41；合并使期望≥5。

12. Common Pitfalls and Tips | 常见陷阱与技巧

Students often forget to adjust the degrees of freedom when p is estimated. Another mistake is using the wrong total number of observations N. Always double‑check that the sum of Oᵢ equals N and that the sum of Eᵢ also equals N. If categories are combined, recalculate both observed and expected frequencies for the new combined groups. Finally, never state that you 'accept' H₀; you can only 'fail to reject' it.

学生们经常忘记在估计 p 时调整自由度。另一个错误是使用错误的观测总数 N。务必再次核对 Oᵢ 的总和等于 N，且 Eᵢ 的总和也等于 N。如果合并了类别，为新合并的组重新计算观测频数和期望频数。最后，永远不要说“接受” H₀；你只能说“未能拒绝”它。

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