ICA ü WHEN: 14 Nov 2012 : 45 minutes in in “normal” lecture period ü WHERE: A -… ü WHAT: o Tutorial exercises: o Course notes: o Lectures: 1, 2, 3, 4, 5 chapters 1, 2, 3, 4, 5 and sections 6. 1 - 6. 5 all lectures concerning the above ü FORMAT: open book (any written material) ü CALCULATORS: only college approved calculators! : : ü NOTE: not only final (numerical) answer important but also clarity of answer! o Make sure to include reasoning, and o if applicable: formula – substitutions – final numerical outcome ü IF YOU NEED EXTRA TIME EMAIL VERA@STATS. UCL. AC. UK AND F. GILLOWAY@UCL. AC. UK ASAP STAT 6202 Chapter 3 2012/2013 1
CHAPTER 3 The Normal Distribution ü POPULATION VERSUS SAMPLE - NOTATION ü NORMAL DISTRIBUTION - THE IDEA ü DEFINITION AND PARAMETERS ü PROBABILITIES AND VARIABLES ü THREE PROBABILITY RULES ü PERCENTAGE POINTS 2 STAT 6202 Chapter 3 2012/2013
POPULATION VERSUS SAMPLE Notation ü WHAT IS THE DIFFERENCE? ü SO FAR ONLY SAMPLE (DATA) ü NOW WE WILL START LOOKING AT BOTH Sample Population μ Mean Standard deviation sx σ Variance sx 2 σ2 n N Size 3 STAT 6202 Chapter 3 2012/2013
POPULATION VERSUS SAMPLE An example ü MANAGEMENT OF A STORE ARE INTERESTED IN THE AVERAGE OF THEIR CUSTOMERS. AS SUCH 40 RANDOMLY SELECTED CUSTOMERS OF THEIR STORE ASKED ABOUT THEIR AGE. o Population: all customers of the store o Sample: o the 40 randomly selected customers o μ: mean age of all customers of the store o : : o n = 40 mean age of the 40 randomly selected customers N = total number of customers of the store 4 STAT 6202 Chapter 3 2012/2013
CHAPTER 3 The Normal Distribution ü POPULATION VERSUS SAMPLE - NOTATION ü NORMAL DISTRIBUTION - THE IDEA ü DEFINITION AND PARAMETERS ü PROBABILITIES AND VARIABLES ü THREE PROBABILITY RULES ü PERCENTAGE POINTS 5 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION The idea ü AN EXAMPLE: WEIGHT OF ALL UK MEN o Symmetric: mean = median o One peak o Bell shaped 6 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION The definiton ü RELATIVE FREQUENCY CURVE o Or probability density function STAT 6202 Chapter 3 2012/2013 o Location: mean ( ) o Spread: variance ( 2) o Shape: normal distribution 7
NORMAL DISTRIBUTION An example ü BACK TO THE WEIGHT EXAMPLE ü The weight of all UK men follows a normal distribution with mean 80 and standard deviation 15 ü Short notation: STAT 6202 Chapter 3 2012/2013 8
NORMAL DISTRIBUTION The parameters ü WHAT HAPPENS IF μ CHANGES? 9 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION The parameters ü WHAT HAPPENS IF σ CHANGES? 10 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION Standard normal distribution ü STANDARD NORMAL DISTRIBUTION o Normal distribution with mean = 0 and standard deviaton = 1 o Or, equivalently, N(0, 1) o Table 2 in course notes 11 STAT 6202 Chapter 3 2012/2013
CHAPTER 3 The Normal Distribution ü POPULATION VERSUS SAMPLE - NOTATION ü NORMAL DISTRIBUTION - THE IDEA ü DEFINITION AND PARAMETERS ü PROBABILITIES AND VARIABLES ü THREE PROBABILITY RULES ü PERCENTAGE POINTS 12 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION Probabilities and variables: theory ü PROBABILITES IN GENERAL o Number between 0 and 1 indicating how likely something is ü FOR CONTINUOUS DISTRIBUTIONS SPECIFICALLY f(x) Shaded area = proportion of the population taking values between a and b X: random variable x NOTE: P(X=x)=0 i. e. probability of one specific outcome equals 0 STAT 6202 Chapter 3 2012/2013 13
NORMAL DISTRIBUTION Probabilities and variables: example ü BACK TO THE WEIGHT EXAMPLE o What proportion of UK men weigh less than 90 kg? Or equivalently, o What is the probability that the weight of a randomly selected UK man is less than 90 kg? o X is the weight of a randomly selected UK man (X ~ N(80, 152)) 14 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION Standardised normal variable ü STANDARD NORMAL DISTRIBUTIONS: N(0, 1) ü REMEMBER LINEAR TRANSFORMATIONS? a + b xi § Means? § Variances? § Standardised variables? ü LINEAR TRANSFORMATIONS ü STANDARDISED NORMAL VARIABLE 15 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION Probabilities: calculations ü FOR NORMAL DISTRIBUTIONS o Translate probability in terms of standard normally distributed variable o Then use statistical table 2 STAT 6202 Chapter 3 2012/2013 16
NORMAL DISTRIBUTION Probabilities: An Illustration (1) ü BACK TO THE WEIGHT EXAMPLE o X is the weight of a randomly selected UK man (X ~ N(80, 152)) o What proportion of UK men weigh less than 90 kg? Or equivalently, o What is the probability that the weight of a randomly selected UK man is less than 90 kg? 17 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION Probabilities: An Illustration (2) ü THE WEIGHT EXAMPLE CONTINUED o What proportion of UK men weigh less than 70 kg? Or equivalently, o What is the probability that the weight of a randomly select UK man is less than 70 kg? 18 STAT 6202 Chapter 3 2012/2013
CHAPTER 3 The Normal Distribution ü POPULATION VERSUS SAMPLE - NOTATION ü NORMAL DISTRIBUTION - THE IDEA ü DEFINITION AND PARAMETERS ü PROBABILITIES AND VARIABLES ü THREE PROBABILITY RULES ü PERCENTAGE POINTS 19 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION Three probability rules (Where z 2 > z 1) 20 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION Probabilities: An Illustration (3) ü THE WEIGHT EXAMPLE CONTINUED o What proportion of UK men weigh less than 70 kg? 21 STAT 6202 Chapter 3 2012/2013
CHAPTER 3 The Normal Distribution ü POPULATION VERSUS SAMPLE - NOTATION ü NORMAL DISTRIBUTION - THE IDEA ü DEFINITION AND PARAMETERS ü PROBABILITIES AND VARIABLES ü THREE PROBABILITY RULES ü PERCENTAGE POINTS 22 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION Percentage points: what are they? ü SO FAR WE HAVE LOOKED AT: P(X<70) = ? ü NOW THE OPPOSITE PROBLEM: P(X < ? ) = 0. 2514 ü HOW TO GO ABOUT THIS? o Calculate the z-value for which: o Find the corresponding x-value: P(Z < ? ) = 0. 2514 23 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION Percentage points: an illustration ü BACK TO THE WEIGHT EXAMPLE: N(80, 152) o Weight c such that 5% of men weigh more P(X>c)=0. 05 P(Z > z)=0. 05 P(Z z)= 0. 95 z = 1. 645 24 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION Percentage points: often used ones ü SOME UPPER PERCENTILE POINTS OF N(0, 1) p zp 0. 5 0. 25 0. 1 0. 05 0. 025 0. 01 0. 000 0. 674 1. 282 1. 645 1. 960 2. 326 3. 090 ü STANDARD DEVIATION RANGES: X~ N( , 2) o What proportions of the population are within , 2 , 3 bounds of : 25 STAT 6202 Chapter 3 2012/2013
NORMAL DISTRIBUTION Percentage points: often used ones ü SOME UPPER PERCENTILE POINTS OF N(0, 1) p zp 0. 5 0. 25 0. 1 0. 05 0. 025 0. 01 0. 000 0. 674 1. 282 1. 645 1. 960 2. 326 3. 090 ü STANDARD DEVIATION RANGES: X~ N( , 2) o What proportions of the population are within , 2 , 3 bounds of : 26 STAT 6202 Chapter 3 2012/2013