STAT6202 Chapter 3 2012/2013 1 ICA WHEN: 14Nov2012

STAT6202 Chapter 3 2012/2013 1 ICA WHEN: 14Nov2012 :45 minutes in in “normal” lecture period WHERE: A -… : : WHAT: Tutorial exercises: 1,2,3,4,5 Course notes: chapters 1,2,3,4,5 and sections 6.1 - 6.5 Lectures: all lectures concerning the above FORMAT: open book (any written material) CALCULATORS: only college approved calculators! NOTE: not only final (numerical) answer important but also clarity of answer! Make sure to include reasoning, and if applicable: formula – substitutions – final numerical outcome IF YOU NEED EXTRA TIME EMAIL [email protected] AND [email protected] ASAP

STAT6202 Chapter 3 2012/2013 2 CHAPTER 3 The Normal Distribution POPULATION VERSUS SAMPLE - NOTATION NORMAL DISTRIBUTION - THE IDEA DEFINITION AND PARAMETERS PROBABILITIES AND VARIABLES THREE PROBABILITY RULES PERCENTAGE POINTS

STAT6202 Chapter 3 2012/2013 3 POPULATION VERSUS SAMPLE Notation WHAT IS THE DIFFERENCE? SO FAR ONLY SAMPLE (DATA) NOW WE WILL START LOOKING AT BOTH

STAT6202 Chapter 3 2012/2013 4 POPULATION VERSUS SAMPLE An example MANAGEMENT OF A STORE ARE INTERESTED IN THE AVERAGE AGE OF THEIR CUSTOMERS. AS SUCH 40 RANDOMLY SELECTED CUSTOMERS OF THEIR STORE ARE ASKED ABOUT THEIR AGE. Population: Sample: μ : : n = N = all customers of the store the 40 randomly selected customers mean age of all customers of the store : mean age of the 40 randomly selected customers 40 total number of customers of the store

STAT6202 Chapter 3 2012/2013 5 CHAPTER 3 The Normal Distribution POPULATION VERSUS SAMPLE - NOTATION NORMAL DISTRIBUTION - THE IDEA DEFINITION AND PARAMETERS PROBABILITIES AND VARIABLES THREE PROBABILITY RULES PERCENTAGE POINTS

STAT6202 Chapter 3 2012/2013 6 NORMAL DISTRIBUTION The idea AN EXAMPLE: WEIGHT OF ALL UK MEN Symmetric: mean = median One peak Bell shaped

STAT6202 Chapter 3 2012/2013 7 NORMAL DISTRIBUTION The definiton Location: mean () Spread: variance (2) Shape: normal distribution RELATIVE FREQUENCY CURVE Or probability density function

STAT6202 Chapter 3 2012/2013 8 The weight of all UK men follows a normal distribution with mean 80 and standard deviation 15 Short notation: NORMAL DISTRIBUTION An example BACK TO THE WEIGHT EXAMPLE

STAT6202 Chapter 3 2012/2013 9 NORMAL DISTRIBUTION The parameters WHAT HAPPENS IF μ CHANGES?

STAT6202 Chapter 3 2012/2013 10 NORMAL DISTRIBUTION The parameters WHAT HAPPENS IF σ CHANGES?

STAT6202 Chapter 3 2012/2013 11 NORMAL DISTRIBUTION Standard normal distribution STANDARD NORMAL DISTRIBUTION Normal distribution with mean = 0 and standard deviaton = 1 Or, equivalently, N(0,1) Table 2 in course notes

STAT6202 Chapter 3 2012/2013 12 CHAPTER 3 The Normal Distribution POPULATION VERSUS SAMPLE - NOTATION NORMAL DISTRIBUTION - THE IDEA DEFINITION AND PARAMETERS PROBABILITIES AND VARIABLES THREE PROBABILITY RULES PERCENTAGE POINTS

STAT6202 Chapter 3 2012/2013 13 NORMAL DISTRIBUTION Probabilities and variables: the theory PROBABILITES IN GENERAL Number between 0 and 1 indicating how likely something is FOR CONTINUOUS DISTRIBUTIONS SPECIFICALLY Shaded area = proportion of the population taking values between a and b x f(x) X: random variable NOTE: P(X=x)=0 i.e. probability of one specific outcome equals 0

STAT6202 Chapter 3 2012/2013 14 NORMAL DISTRIBUTION Probabilities and variables: example BACK TO THE WEIGHT EXAMPLE What proportion of UK men weigh less than 90kg? Or equivalently, What is the probability that the weight of a randomly selected UK man is less than 90kg? X is the weight of a randomly selected UK man (X ~ N(80,152))

STAT6202 Chapter 3 2012/2013 15 NORMAL DISTRIBUTION Standardised normal variable STANDARD NORMAL DISTRIBUTIONS: N(0,1) REMEMBER LINEAR TRANSFORMATIONS? a + b xi Means? Variances? Standardised variables? LINEAR TRANSFORMATIONS STANDARDISED NORMAL VARIABLE

STAT6202 Chapter 3 2012/2013 16 NORMAL DISTRIBUTION Probabilities: calculations FOR NORMAL DISTRIBUTIONS Translate probability in terms of standard normally distributed variable Then use statistical table 2

STAT6202 Chapter 3 2012/2013 17 NORMAL DISTRIBUTION Probabilities: An Illustration (1) BACK TO THE WEIGHT EXAMPLE X is the weight of a randomly selected UK man (X ~ N(80,152)) What proportion of UK men weigh less than 90kg? Or equivalently, What is the probability that the weight of a randomly selected UK man is less than 90kg?

STAT6202 Chapter 3 2012/2013 18 NORMAL DISTRIBUTION Probabilities: An Illustration (2) THE WEIGHT EXAMPLE CONTINUED What proportion of UK men weigh less than 70kg? Or equivalently, What is the probability that the weight of a randomly select UK man is less than 70kg?

STAT6202 Chapter 3 2012/2013 19 CHAPTER 3 The Normal Distribution POPULATION VERSUS SAMPLE - NOTATION NORMAL DISTRIBUTION - THE IDEA DEFINITION AND PARAMETERS PROBABILITIES AND VARIABLES THREE PROBABILITY RULES PERCENTAGE POINTS

STAT6202 Chapter 3 2012/2013 20 NORMAL DISTRIBUTION Three probability rules (Where z2 > z1)

STAT6202 Chapter 3 2012/2013 21 NORMAL DISTRIBUTION Probabilities: An Illustration (3) THE WEIGHT EXAMPLE CONTINUED What proportion of UK men weigh less than 70kg?

STAT6202 Chapter 3 2012/2013 22 CHAPTER 3 The Normal Distribution POPULATION VERSUS SAMPLE - NOTATION NORMAL DISTRIBUTION - THE IDEA DEFINITION AND PARAMETERS PROBABILITIES AND VARIABLES THREE PROBABILITY RULES PERCENTAGE POINTS

STAT6202 Chapter 3 2012/2013 23 NORMAL DISTRIBUTION Percentage points: what are they? SO FAR WE HAVE LOOKED AT: P(X<70) = ? NOW THE OPPOSITE PROBLEM: P(X < ?) = 0.2514 HOW TO GO ABOUT THIS? Calculate the z-value for which: P(Z < ?) = 0.2514 Find the corresponding x-value:

STAT6202 Chapter 3 2012/2013 24 NORMAL DISTRIBUTION Percentage points: an illustration BACK TO THE WEIGHT EXAMPLE: N(80,152) Weight c such that 5% of men weigh more P(Z > z)=0.05  P(Z  z)= 0.95  z = 1.645

STAT6202 Chapter 3 2012/2013 25 NORMAL DISTRIBUTION Percentage points: often used ones SOME UPPER PERCENTILE POINTS OF N(0,1) STANDARD DEVIATION RANGES: X~ N(,2) What proportions of the population are within , 2, 3 bounds of :

STAT6202 Chapter 3 2012/2013 26 NORMAL DISTRIBUTION Percentage points: often used ones SOME UPPER PERCENTILE POINTS OF N(0,1) STANDARD DEVIATION RANGES: X~ N(,2) What proportions of the population are within , 2, 3 bounds of :