Bridging British Education Virtual Academy

1. Course Basic Information 1. 课程基本信息

Course Name: 0102 Matt Alice 课程名称： 0102 Matt Alice

Topic: Binomial Distribution Calculations (CDF) and Introduction to Hypothesis Testing (H0, H1, Test Statistic) 主题： 二项分布计算 (CDF) 与假设检验导论 (H0, H1, 检验统计量)

Date: Not specified in transcript 日期： 未在文本中指定

Student: Alice 学生： Alice

Teaching Focus 教学重点

Completing complex binomial probability calculations (including 'at least' and working backwards for 'k') and introducing the foundational concepts of binomial hypothesis testing.

完成复杂的二项概率计算（包括“至少”和反向求解‘k’值）以及介绍二项假设检验的基础概念。

Teaching Objectives 教学目标

Accurately calculate cumulative binomial probabilities (P(X ≤ x)) using the calculator. 使用计算器准确计算累积二项概率 (P(X ≤ x))。
Manipulate inequalities (P(X ≥ x) and P(k ≤ X ≤ r)) to utilize the calculator's CDF function effectively. 熟练操作不等式 (P(X ≥ x) 和 P(k ≤ X ≤ r)) 以有效利用计算器的 CDF 功能。
Understand the core components of a hypothesis test: Null Hypothesis (H0), Alternative Hypothesis (H1), Test Statistic, and Significance Level. 理解假设检验的核心组成部分：零假设 (H0)、备择假设 (H1)、检验统计量和显著性水平。

2. Course Content Overview 2. 课程内容概览

Main Teaching Activities and Time Allocation 主要教学活动和时间分配

Reviewing Binomial CDF Application (Example 7): Reviewing P(X ≤ 2) using binomial CDF. Moving on to P(X ≥ 5) requiring the transformation 1 - P(X ≤ 4).

回顾二项 CDF 应用 (例 7)： 使用二项 CDF 回顾 P(X ≤ 2)。接着处理 P(X ≥ 5)，需要转化为 1 - P(X ≤ 4)。

Working Backwards (Finding K): Solving for 'W' (winning threshold) when P(X ≥ W) < 0.05, requiring algebraic manipulation to use CDF: 1 - P(X ≤ W-1) < 0.05 leads to P(X ≤ W-1) > 0.95. Then testing values to find the boundary.

反向求解 (寻找 K)： 当 P(X ≥ W) < 0.05 时求解 'W'（获胜阈值），需要代数操作以使用 CDF：1 - P(X ≤ W-1) < 0.05 转化为 P(X ≤ W-1) > 0.95。然后通过测试数值找到边界。

Compound Probability Calculation (K to R): Briefly discussing how to calculate P(K ≤ X ≤ R) using subtraction of CDFs, emphasizing the need to define the distribution first.

复合概率计算 (K 到 R)： 简要讨论如何使用 CDF 相减来计算 P(K ≤ X ≤ R)，强调首先需要定义分布。

Introduction to Hypothesis Testing: Defining H0 (Null Hypothesis, always '='), H1 (Alternative Hypothesis, '<', '>', or '≠'), Test Statistic (X), and Significance Level (α). Conceptual walkthrough of determining evidence for/against H0 using Example 1 (coin bias).

假设检验介绍： 定义 H0 (零假设，总是 '=')，H1 (备择假设，'<', '>', 或 '≠')，检验统计量 (X)，和显著性水平 (α)。使用例 1（抛硬币偏差）概念性地演示如何判断支持/反对 H0 的证据。

Language Knowledge and Skills 语言知识与技能

Vocabulary:
Cumulative (CDF), Null Hypothesis (H0), Alternative Hypothesis (H1), Test Statistic, Significance Level, One-tail test, Two-tail test, Biased, Unbiased.

词汇：
累积 (CDF), 零假设 (H0), 备择假设 (H1), 检验统计量, 显著性水平, 单尾检验, 双尾检验, 有偏差的, 无偏的。

Concepts:
The application of Binomial CDF for complex probability ranges; the logical framework for hypothesis testing based on comparing observed data probability against a pre-set significance threshold under the assumption that the null hypothesis is true.

概念：
二项 CDF 在复杂概率范围中的应用；基于在零假设为真的前提下，将观测数据的概率与预设的显著性阈值进行比较的假设检验的逻辑框架。

Skills Practiced:
Advanced probability calculation, inequality manipulation, formal hypothesis testing setup.

练习技能：
高级概率计算，不等式操作，正式的假设检验设置。

Teaching Resources and Materials 教学资源与材料

Textbook examples (specifically Example 7 and parts of 8). 教科书例题（特别是例 7 和 8 的部分内容）。
Graphing/Statistical calculator (used for CDF). 图形/统计计算器 (用于 CDF)。

3. Student Performance Assessment (Alice) 3. 学生表现评估 (Alice)

Participation and Activeness 参与度和积极性

Alice showed strong active participation, especially during the complex problem-solving sections (like working backwards for 'W' and testing boundary values for 'R'). 爱丽丝表现出积极的参与度，尤其是在复杂的解题部分（如反向求解‘W’和测试‘R’的边界值）。

Language Comprehension and Mastery 语言理解和掌握

High comprehension demonstrated in applying the 1 - CDF trick for 'at least' problems and correctly identifying the need to manipulate inequalities to fit the calculator's input requirements. 在应用‘至少’问题的 1 - CDF 技巧和正确识别需要操作不等式以适应计算器输入要求方面表现出高理解力。
Initial conceptual clarity needed for the formal wording of H0 and H1 in hypothesis testing, but the student grasped the core idea quickly. 在假设检验中对 H0 和 H1 的正式措辞需要初步的概念清晰度，但学生很快掌握了核心思想。

Language Output Ability 语言输出能力

Oral: 口语：

Generally fluent, although some hesitation when articulating complex statistical procedures (e.g., converting P(X ≥ 5) to 1 - P(X ≤ 4)). 整体流利，但在阐述复杂的统计过程时有些犹豫（例如，将 P(X ≥ 5) 转换为 1 - P(X ≤ 4)）。

Written: 书面：

N/A - No written work was explicitly reviewed, but performance in guided problem-solving suggests accuracy.

不适用 - 未明确审查书面作业，但指导下的解题表现表明准确性较高。

Student's Strengths 学生的优势

Strong procedural fluency in using the binomial CDF function once the correct inequality form is achieved. 一旦达到正确的不等式形式，在应用二项 CDF 函数方面表现出很强的程序流畅性。
Good intuition for testing boundary values (iterative testing for K and R in Q8). 对测试边界值有很好的直觉（在 Q8 中对 K 和 R 进行迭代测试）。
Quickly grasped the conceptual difference between H0 and H1 in the hypothesis testing introduction. 在假设检验介绍中，快速掌握了 H0 和 H1 之间的概念差异。

Areas for Improvement 需要改进的方面

Ensuring the initial step of defining the distribution (N and P) is always explicitly stated, even when implied. 确保总是明确说明定义分布 (N 和 P) 的初始步骤，即使是在隐含的情况下。
Solidifying the formal language and expected structure for hypothesis test conclusions (i.e., using 'sufficient evidence' language). 巩固假设检验结论的正式语言和预期结构（即使用‘有足够证据’的措辞）。

4. Teaching Reflection 4. 教学反思

Effectiveness of Teaching Methods 教学方法的有效性

The teacher effectively used the student's attempt to guide them through complex boundary testing, reinforcing why checking the values immediately above and below the boundary is crucial. 教师有效地利用了学生的尝试，引导他们完成复杂的边界测试，强化了检查边界值上下数值的重要性。
The transition from the highly computational binomial chapter to the theoretical introduction of hypothesis testing was managed well by linking the probability calculations directly to the concept of the test statistic. 将高度计算性的二项分布章节过渡到假设检验的理论介绍处理得当，方法是将概率计算直接与检验统计量的概念联系起来。

Teaching Pace and Time Management 教学节奏和时间管理

The pace was fast during the review of binomial distribution (Q7/Q8) as the student demonstrated prior knowledge, allowing ample time for the new topic (Hypothesis Testing). 在二项分布复习部分（Q7/Q8），节奏较快，因为学生表现出先前的知识，为新主题（假设检验）留出了充足的时间。

Classroom Interaction and Atmosphere 课堂互动和氛围

Engaging, collaborative, and inquisitive, with the teacher encouraging the student to articulate their calculator inputs and reasoning processes.

参与度高、协作性强、充满探究精神，教师鼓励学生清晰表达他们的计算器输入和推理过程。

Achievement of Teaching Objectives 教学目标的达成

Objectives related to complex binomial calculations were largely met, demonstrated by successful navigation of Q7/Q8 boundary conditions. 与复杂二项计算相关的目标基本达成，通过成功驾驭 Q7/Q8 的边界条件得以证明。
Hypothesis testing objectives were introduced effectively, setting a strong foundation for the next lesson. 假设检验目标得到了有效介绍，为下一课打下了坚实的基础。

5. Subsequent Teaching Suggestions 5. 后续教学建议

Teaching Strengths 教学优势

Identified Strengths: 识别的优势：

Exceptional ability to troubleshoot calculator inputs in real-time during complex procedures. 在复杂程序中实时解决计算器输入问题的能力非凡。
Clear explanation of the necessity to manipulate inequalities (e.g., converting greater than or equal to into 1 minus the less than or equal to form). 清晰解释了操作不等式的必要性（例如，将大于等于转换为 1 减去小于等于的形式）。

Effective Methods: 有效方法：

Using iterative guessing/checking (bracketing) to find critical values (K and R) when analytical solutions are complex. 在解析解复杂时，使用迭代猜测/检查（分界法）来寻找临界值 (K 和 R)。
Connecting statistical terms (H0, H1, Test Statistic) directly to context (coin bias example). 将统计术语（H0, H1, 检验统计量）直接与具体情境（抛硬币偏差的例子）联系起来。

Positive Feedback: 正面反馈：

The teacher confirmed the student's grasp of why specific calculations (like P(X ≤ 6) vs P(X ≤ 7) for boundary crossing) are necessary. 教师确认了学生对为什么特定计算（如边界穿越的 P(X ≤ 6) 与 P(X ≤ 7)）是必要性的理解。

Next Teaching Focus 下一步教学重点

Formal structure, wording, and calculation of one-tailed and two-tailed Binomial Hypothesis Tests. 二项假设检验（单尾和双尾）的正式结构、措辞和计算。

Specific Suggestions for Student's Needs 针对学生需求的具体建议

Calculation Procedure: 计算流程：

Always explicitly write the distribution definition (X ~ Bin(n, p)) at the start of complex binomial problems, as this habit is crucial for Year 2 continuity. 在复杂的二项式问题开始时，务必明确写出分布定义 (X ~ Bin(n, p))，因为这个习惯对第二年的课程衔接至关重要。
When solving for a boundary like P(X ≥ W) < α, practice writing out the full algebraic conversion steps clearly (e.g., 1 - P(X ≤ W-1) < α) before testing values in the calculator. 当求解 P(X ≥ W) < α 这样的边界值时，练习清楚地写出完整的代数转换步骤（例如，1 - P(X ≤ W-1) < α），然后再在计算器中测试数值。

Hypothesis Testing Setup: 假设检验设置：

For hypothesis testing, memorize the precise concluding statement structure: 'There is sufficient/insufficient evidence to reject H0 in favour of H1 at the X% significance level.' 对于假设检验，请记住精确的结论陈述结构：‘在 X% 的显著性水平下，没有/有足够证据拒绝 H0 而支持 H1。’
Practice identifying the 'more extreme' outcome in context (e.g., if H1 is P > 0.5, then 6 heads is more extreme than 5 heads in 8 tosses). 练习根据语境识别‘更极端’的结果（例如，如果 H1 是 P > 0.5，则在 8 次投掷中，6 次正面比 5 次正面更极端）。

Recommended Supplementary Learning Resources or Homework 推荐的补充学习资源或家庭作业

Review textbook questions on working backwards (finding K/R) in the binomial distribution section. 复习二项分布章节中关于反向求解（寻找 K/R）的教科书题目。
Attempt Question 6 from the textbook to practice setting up H0 and H1 based on the wording and calculating the test statistic probability against the 5% significance level. 尝试教科书上的第 6 题，练习根据措辞设置 H0 和 H1，并计算检验统计量概率与 5% 显著性水平的比较。