Testing a Binomial Distribution as a Model | 二项分布模型检验

📚 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.

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。

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

E(X) = np, Var(X) = np(1 − p)

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.

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。

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。

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

Published by TutorHao | AS Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）