amazon.co.ukM4: Statistical Distributions and Hypothesis Testing
Binomial and Normal distributions, central limit theorem, one- and two-tailed hypothesis tests.
๐ Key Definitions and Core Concepts
Binomial Distribution: X ~ B(n, p) for n independent trials with constant success probability p.
Normal Distribution: X ~ N(ฮผ, ฯยฒ) continuous symmetric distribution standardised via Z = (X - ฮผ)/ฯ.
๐ Key Principles & Specification Requirements
- Binomial P(X = r) = โฟCแตฃ pสณ (1-p)โฟโปสณ.
- Reject Hโ if test statistic falls within critical region or p-value < significance level ฮฑ.
- Normal distribution 68-95-99.7 rule: ~68.3% within 1ฯ, ~95.4% within 2ฯ, ~99.7% within 3ฯ.
๐ก Worked Example Question
Exam-Style Question
Question:
A fair coin is tossed 20 times. 15 heads are observed. Test at 5% level if the coin is biased.
Model Solution & Mark Scheme:
Hโ: p = 0.5, Hโ: p โ 0.5 (two-tailed, ฮฑ = 0.025 per tail). X ~ B(20, 0.5). P(X โฅ 15) = 1 - P(X โค 14) = 0.0207. 0.0207 < 0.025 => Reject Hโ; significant evidence of bias.
โ Practice Questions & Mark Schemes
Q1: If X ~ N(100, 15ยฒ), find P(85 < X < 115).
Show Model Answer
Answer: Z = ยฑ1. P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6826 (68.3%).
Q2: State conditions for binomial modeling.
Show Model Answer
Answer: Fixed number of trials n, two outcomes, constant probability p, independent trials.
๐ฌ Video Resources
- Statistical Distributions and Hypothesis Testing - Overview -- YouTube Search
- Statistical Distributions and Hypothesis Testing - Exam Questions -- YouTube Search
๐ Past Papers & Exam Resources
- AQA A-Level - Past Papers -- AQA
- Edexcel A-Level - Past Papers -- Edexcel
- OCR A-Level - Past Papers -- OCR
๐ Further Reading & Resources
- Physics & Maths Tutor -- Free notes, worksheets, and past paper questions by topic
- Dr Frost Maths -- Free resources, slides, and interactive questions
- BBC Bitesize -- BBC Bitesize revision resources