JC and Polytechnic Mathematics Material Compilation - Statistics

Title JC and Polytechnic Mathematics Material Compilation – Statistics Editor Lee Jian Lian Date 12/4/2019 Topic [Category: Discrete Probability Distribution] Page Discrete Probability Distribution 2 Binomial Distribution 3 Poisson Distribution – Fundamentals 8 Poisson Distribution as Approximation to Binomial Distribution with large 𝑛 and small 𝑝 11 Topic [Category: Continuous Probability Distribution] Page Continuous Probability Distribution 13 Normal Distribution – Fundamentals 15 Normal Distribution – Distribution of Sample Means 20 Normal Distribution – Central Limit Theorem 22 Confidence Intervals with Normal and 𝑡-distribution: Basics 23 Confidence Intervals with Normal and 𝑡-distribution: Calculation 24 Topic [Category: Hypothesis Testing] Page Hypothesis Testing on One Sample with Normal and 𝑡-distribution 27 Chi-Square-Test for Goodness-of-Fit 32 Chi-Square-Test for Independence 34 Miscellaneous Matters Page Utilizing a Standard Normal Table – Probability from Far-Left (or Negative Infinity) of the Normal Distribution to the 𝑧-score 37 Applicable to Courses of the Following Nature - Information Technology - Engineering - Applied Science/Science - Business/Business Management - GCE ‘A’ Level H1 Mathematics - GCE ‘A’ Level H2 Mathematics & ‘A’ Level H2 Further Mathematics

Title Discrete Probability Distribution Author Liu Hui Ling, Ngee Ann Polytechnic Date 17/10/2018 Discrete Probability Distribution has the following properties. • Takes in discrete variables (Whole number values 𝑘 , where 𝑘 ≥ 0) • Countable number of values involved • Takes in random variables (Sum of all probabilities must be equal to 1) Example Number of Events 0 1 2 3 Probability 0.5 0.25 0.10 0.15 In the case of the Binomial Distribution, as represented by the formula below, 𝑃(𝑋 = 𝑘) = (𝑛 𝑘) 𝑝𝑘(1 − 𝑝)𝑛−𝑘 The following limitation is imposed, as any values that doesn’t comply to the following limitation is undefined. 0 ≤ 𝑘 ≤ 𝑛 In the case of Poisson Distribution, as represented by the formula below, 𝑃(𝑋 = 𝑘) = 𝑒−𝜇 (𝜇𝑘 𝑘!) Where 0 ≤ 𝑘 < ∞ The inequality 0 ≤ 𝑘 < ∞, implies the number of events you are performing the probability calculations for can be any finite whole number greater than or equal to 0. While there is no upper limit to the value of 𝑘, a theorem guarantees the value of all probability will sum up to 1: As the probability within a Binomial Distribution approaches 0 and the number of trials approaches infinity. The Binomial Distribution will converge to the Poisson Distribution. This implies that the Poisson Distribution is just a special case of Binomial Distribution, which means the probability will still sum up to 1 anyway.

Title Polytechnic and A Level H2 Mathematics (Statistics) Binomial Distribution Author Lim Wang Sheng, School of Information Technology, Nanyang Polytechnic [CCA: NYP Mentoring Club] Date 9/