SAJC 2021 JC1 H1 Math Normal Distribution (Teacher)

SAJC 2021 JC 1 H1 Mathematics Normal Distribution Page 1 of 35 Chapter 4 (Statistics) Normal Distribution (Teacher’s copy) Objectives At the end of the chapter, students should be able to: (a) Understand the nature of the normal distribution which include  a normal distribution is used to model a continuous random variable  use normal curves to illustrate normal distributions with different  and   the shape and location of a normal curve are determined by the values of  and   the area under a normal curve between ax and bx is the probability  bxa P  the total area under a normal curve is 1 (b) Understand X and Y have independent normal distributions, then bYaX  has a normal distribution with mean yx ba   and variance 2222 yx ba   Contents 4.1 Introduction to Normal Distribution 4.2 Continuous Random Variable 4.3 Normal Distribution Curve 4.3.1 Properties of the Normal Distribution Curve 4.4 Finding Probabilities 4.5 Use of Inverse Normal 4.6 Standard Normal Distribution 4.7 Linear Combinations of Independent Normal Distributions 4.7.1 Properties of Expectation & Variance 4.7.2 Sum and Difference of Independent Normal Distributions 4.7.3 Sum of n independent observations from the same Normal Distribution 4.8 Summary and Checklist 4.9 Learning Experience

SAJC 2021 JC 1 H1 Mathematics Normal Distribution Page 2 of 35 4.1 Introduction to Normal Distribution If you sketch a histogram corresponding to the heights of 17 years old male students in a SAJC, what would you observe? You will realized that majority of the students have about the same height with a few students who are taller than the average height and another few who are shorter than the rest. This is logical since there are fewer very short or very tall boys, and even fewer very very short or very very tall boys. The outline of the histogram generally shapes like a bell-shaped curve. Sample size = 400 Height Frequency Relative Frequency 1.650  1.675 112 112/400 = 0.28 1.675  1.700 128 128/400 = 0.32 1.700  1.725 116 116/400 = 0.29 In fact, such a ‘pattern’ of distribution can be observed for many other physical phenomena as well, such as weight of watermelons in a particular market , I.Q. scores of a population, time taken by students to run 100m, examination results of students in SAJC, household income of Singaporeans etc. Since it is so common, we shall name the distribution a normal distribution (and the bell-shape curve a normal curve) and study it with some depth. The normal variable is an example of a continuous random variable. Note: FrequencyRelative Frequency Total