📚 Selecting a Model | 模型的选择
Selecting a suitable mathematical model is at the heart of applied mathematics. In Edexcel A Level Mathematics, you are expected to recognise when a binomial, normal, discrete uniform or linear regression model is appropriate, and to justify that choice using the context, data and assumptions.
选择合适的数学模型是应用数学的核心。在 Edexcel A Level 数学中，你需要判断何时使用二项分布、正态分布、离散均匀分布或线性回归模型，并能根据背景、数据和假设来证明选择的合理性。
1. The Modelling Cycle | 建模循环
A mathematical model is a simplified representation of a real-world process. The modelling cycle begins with a real problem, makes assumptions, formulates a model, uses mathematics to make predictions, compares these with observed data, and then refines the model if necessary.
数学模型是对现实世界过程的简化表示。建模循环从一个实际问题开始，作出假设，建立模型，用数学进行预测，将预测与观测数据进行比较，如有必要再改进模型。
In an exam, you may be asked to state a reason for selecting a model rather than to perform a long calculation. The key is to link the defining features of the model to the physical situation.
在考试中，你可能会被要求说明选择某个模型的原因，而不是进行冗长的计算。关键是将模型的定义特征与实际情境联系起来。
- Real problem → assumptions → model → predictions → validation → refinement
- 实际问题 → 假设 → 模型 → 预测 → 验证 → 改进
2. Types of Mathematical Model | 数学模型的类型
Models can be deterministic or stochastic. A deterministic model gives the same output for a fixed input, whereas a stochastic model includes random variation. Most probability models in A Level Statistics are stochastic.
模型可以是确定性的或随机的。确定性模型对固定输入给出相同输出，而随机模型包含随机变异。A Level 统计学中的大多数概率模型都是随机模型。
Models may also be discrete or continuous. A discrete random variable takes countable values, such as scores on a die or number of defective items. A continuous random variable can take any value in an interval, such as height or time.
模型也可以是离散的或连续的。离散随机变量取可数值，例如骰子的点数或次品数量。连续随机变量可以取区间内的任意值，例如身高或时间。
Selecting the correct type is the first step. If the data are counts, choose a discrete model. If the data are measurements on a continuous scale, choose a continuous model.
选择正确的类型是第一步。如果数据是计数，就选择离散模型。如果数据是连续尺度上的测量值，就选择连续模型。
3. Key Features of a Binomial Model | 二项分布模型的关键特征
The binomial distribution models the number of successes in a fixed number of independent trials. It is appropriate when there are exactly two possible outcomes, usually called ‘success’ and ‘failure’.
二项分布用于模拟固定次独立试验中成功的次数。当恰好有两种可能结果时，通常称为“成功”和“失败”，它适用。
The conditions for a binomial model are: a fixed number of trials n, each trial is independent, each trial has only two outcomes, and the probability of success p remains constant. If these conditions fail, another model should be selected.
二项分布模型的条件是：试验次数 n 固定，每次试验相互独立，每次试验只有两种结果，成功概率 p 保持不变。如果这些条件不满足，就应选择其他模型。
If X is binomial, then P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, the mean is np, and the variance is np(1 − p).
如果 X 服从二项分布，则 P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ，均值为 np，方差为 np(1 − p)。
4. Key Features of a Normal Model | 正态分布模型的关键特征
The normal distribution models continuous symmetric data that cluster around a mean. It is often used for natural measurements such as heights, weights, or errors in measurement.
正态分布用于模拟围绕均值聚类的连续对称数据。它通常用于自然测量值，例如身高、体重或测量误差。
A normal model is suitable when the data are continuous, roughly symmetric, and have no obvious upper or lower bound. The probability density is highest at the mean and tails off in both directions.
当数据连续、大致对称且没有明显上下界时，正态模型适用。概率密度在均值处最高，并向两侧递减。
If X ~ N(μ, σ²), the parameters are the mean μ and the standard deviation σ. The standardised score is Z = (X − μ) ÷ σ, which allows probabilities to be found using tables or calculators.
如果 X ~ N(μ, σ²)，参数是均值 μ 和标准差 σ。标准化分数为 Z = (X − μ) ÷ σ，可以利用表格或计算器求概率。
5. Matching Context to Distribution | 将情境与分布匹配
To select between a binomial model and a normal model, look at the type of data. Binomial data are integer counts with a fixed maximum n. Normal data are continuous measurements with no fixed maximum.
要在二项分布模型和正态分布模型之间选择，要看数据类型。二项数据是整数计数，有固定上限 n。正态数据是连续测量值，没有固定上限。
For example, the number of heads in 20 coin tosses is binomial because there are 20 independent trials and two outcomes. The mass of apples from an orchard is likely normal because mass is continuous and symmetric.
例如，抛 20 次硬币出现正面的次数服从二项分布，因为有 20 次独立试验和两种结果。果园中苹果的质量很可能服从正态分布，因为质量是连续且对称的。
A discrete uniform model may be chosen when each of a finite number of values is equally likely, such as the score on a fair six-sided die.
当有限个数值中每个值发生的可能性相等时，可以选择离散均匀分布模型，例如公平六面骰子的点数。
6. Normal Approximation to the Binomial | 二项分布的正态近似
Sometimes a binomial model is selected, but calculations become impractical when n is large. In such cases, a normal model can be used as an approximation, provided the success probability p is not too close to 0 or 1.
有时选择了二项分布模型，但当 n 很大时计算变得不现实。在这种情况下，只要成功概率 p 不太接近 0 或 1，就可以用正态模型作为近似。
The common rule is that the normal approximation is valid if np > 5 and n(1 − p) > 5. The approximating normal distribution has mean np and variance np(1 − p).
常用规则是，如果 np > 5 且 n(1 − p) > 5，正态近似有效。近似的正态分布均值为 np，方差为 np(1 − p)。
When approximating a discrete distribution with a continuous one, use a continuity correction. For example, P(X ≤ r) becomes P(Y < r + ½) for the normal variable Y.
当用连续分布近似离散分布时，要使用连续性校正。例如，P(X ≤ r) 变为 P(Y < r + ½)，其中 Y 是正态变量。
Selecting this approximation is itself a modelling decision: the binomial model is preferred when exact calculations are possible, while the normal approximation is selected for large n.
选择这种近似本身就是一种建模决策：当可以精确计算时优先使用二项分布模型，而 n 很大时选择正态近似。
7. Linear Regression Models | 线性回归模型
A linear regression model is selected when a scatter diagram shows a roughly straight-line relationship between two variables. The model is written as y = a + bx, where a is the intercept and b is the gradient.
当散点图显示两个变量之间大致呈直线关系时，选择线性回归模型。模型写作 y = a + bx，其中 a 是截距，b 是斜率。
The least squares regression line minimises the sum of the squared vertical distances from the data points to the line. The gradient is calculated as b = Sxy ÷ Sxx, and the intercept is a = ȳ − b x̄.
最小二乘回归线使得数据点到直线的垂直距离平方和最小。斜率计算公式为 b = Sxy ÷ Sxx，截距为 a = ȳ − b x̄。
You should only select a linear model if the scatter diagram suggests linearity. A curved pattern would indicate that a linear model is not appropriate.
只有当散点图显示线性关系时，才应选择线性模型。曲线模式则表明线性模型不适用。
Extrapolation beyond the observed data range should be avoided because the linear trend may not continue.
应避免在观测数据范围之外进行外推，因为线性趋势可能不会延续。
8. Correlation and Model Suitability | 相关性与模型适用性
The product moment correlation coefficient, r, measures the strength and direction of a linear relationship. It ranges from −1 to 1, where values close to 1 or −1 indicate a strong linear association.
积矩相关系数 r 衡量线性关系的强度和方向。其取值范围为 −1 到 1，接近 1 或 −1 表示强线性相关。
A value of r close to 0 suggests no linear correlation, so a linear regression model may not be suitable. However, even a high r does not prove causation; it only supports a linear model.
r 接近 0 表明没有线性相关，因此线性回归模型可能不适用。然而，即使 r 很高也不能证明因果关系，它只支持线性模型。
A hypothesis test for correlation can be used to decide whether the sample correlation is strong enough to suggest a linear relationship in the population. The null hypothesis is usually H₀: ρ = 0.
相关性的假设检验可用于判断样本相关是否足够强，以表明总体中存在线性关系。零假设通常为 H₀: ρ = 0。
9. Goodness of Fit and Residuals | 拟合优度与残差
A residual is the difference between an observed value and the value predicted by the model: residual = observed y − predicted y. For a well-fitting linear model, residuals should be small and randomly scattered.
残差是观测值与模型预测值之差：残差 = 观测 y 值 − 预测 y 值。对于拟合良好的线性模型，残差应当较小且随机分布。
Plotting residuals against the explanatory variable helps to check model suitability. A random scatter supports the linear model; a curved or fan-shaped pattern suggests the model should be revised.
将残差对解释变量作图有助于检验模型适用性。随机散点支持线性模型；弯曲或扇形模式表明模型需要修正。
Outliers can distort a regression line. You should identify outliers and consider whether they should be removed or whether a different model is needed.
异常值会使回归线失真。你应当识别异常值，并考虑是将其剔除还是需要选择不同的模型。
10. Assumptions, Limitations and Refinement | 假设、局限性与模型改进
Every model relies on assumptions. A binomial model assumes independence and a constant probability of success. A normal model assumes symmetry, continuity and no extreme skew. A linear regression model assumes a straight-line relationship and roughly constant spread of residuals.
每个模型都依赖于假设。二项分布模型假设独立性和恒定的成功概率。正态模型假设对称性、连续性和无极端偏斜。线性回归模型假设直线关系以及残差大致恒定散布。
When the assumptions are not met, the model should be refined. For example, if trials are not independent, the binomial model is invalid. If residuals show a curve, a quadratic or other model may be more appropriate.
当假设不满足时，应改进模型。例如，如果试验不独立，二项分布模型就无效。如果残差显示曲线，二次模型或其他模型可能更合适。
Model selection is not just about fitting data; it is about choosing a model that is simple enough to use, yet complex enough to capture the important features of the real situation.
模型选择不仅仅是对数据进行拟合，而是选择一个既足够简单以便使用，又足够复杂以捕捉实际情况重要特征的模型。
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