Business Statistics. Lecture 3: Random Variables and the Normal Distribution

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1 Business Statistics Lecture 3: Random Variables and the Normal Distribution 1

2 Goals for this Lecture A little bit of probability Random variables The normal distribution 2

3 Probability vs. Statistics Probability: You assume a mechanism that generates particular outcomes and then calculate the chance of other outcomes E.g., Given a fair coin, what is the chance of flipping four heads and two tails out of six flips? Statistics: After seeing some outcomes, you try to say something about the mechanism generating the outcomes E.g., After flipping four heads and two tails you ask, What are the chances this coin is fair? Two sides of the same coin (pun intended!) Language of probability common to both 3

4 Definitions Sample space (S) Set of all possible outcomes of an experiment Event Collection of one or more outcomes Probability Function assigning a number from 0 to 1 to events, subject to rules Venn Diagrams S S A 4

5 Examples: Sample Spaces and Events Roll a fair die: S = {1,2,3,4,5,6} Simple events Roll is a 1 Roll is a 6 Compound event Roll is even: {2, 4, 6} Roll is less than 4: {1, 2, 3} Fair: each simple event is equally likely Other sample space examples: Flip of one coin: S = {H, T} Flip two coins: S = {(H,H), (H,T), (T,H), (T,T)} 5

6 Set Theory Terminology Venn Diagrams Union: A B outcomes in event A or event B or both S A B Intersection: A B outcomes in both event A and event B S A B Complement: A c outcomes in S not in event A S A A c Mutually exclusive or disjoint events events with no outcomes in common S A B 6

7 Some Notation Pr(A) or P(A) is shorthand for the probability that event A occurs For a coin, we might write Pr(H) to mean the probability that a head occurs, for example If we define N(A) as the number of A events in the sample space, then Pr( H) N( H) totalnumber of outcomes in under the assumption that all simple events are equally likely S 1 2 7

8 Probability of the Union of Disjoint Events Disjoint (or mutually exclusive) events: Both events cannot happen at the same time A B Either A or B (or not A or B ) will happen Probability of the union of two disjoint events: Pr(A or B) = Pr(A U B) = Pr(A) + Pr(B) Ex: Probability of rolling a 1 or a 2 on a die: Pr(roll 1 or 2) = Pr(roll 1) + Pr(roll 2) = 1/6 + 1/6 = 1/3 8

9 General Rule for Probability of the Union of Two Events Both events can happen at the same time Yellow/green striped region A B In general, probability of the union of two events: Pr (A and b) = Pr(A U B) = Pr(A) + Pr(B) Pr(A B) U Pr(A B) is the intersection of A and B Basically, the striped area is counted twice in Pr(A) + Pr(B), so one must be subtracted off When events are disjoint Pr(A B) = 0 U U 9

10 Probability An Event Will Not Happen Complementary events: Either one or the other will happen, but not both A Not A Either A will happen or not A will happen Pr(not A) = Pr(A c ) = 1 - Pr(A) Ex: The probability that you do not roll a 3 is 1 minus the probability that you roll a 3 Pr(not roll 3) = 1 Pr(roll 3) = 1 1/6 = 5/6 10

11 Probability of the Intersection of Independent Events Independent events: Two observations are independent if knowing A B the value of one doesn t help you guess the value of the other Rule: Pr(A and B) = Pr(A B) = Pr(A) x Pr(B) Example: In two rolls, the probability you roll a 1 both times Pr( roll a 1 both times) = Pr(roll 1 on first roll) x Pr(roll 1 on second roll) = 1/6 x 1/6 = 1/36 U 11

12 Dependence Opposite of independence Knowing the value of one observation helps you guess the value of another Example: The average price of GM s stock was $59.50 in September. What will the average price be for October? Your best guess uses the September information, so the average monthly stock prices are dependent 12

13 Variables vs. Random Variables A variable is simply a notational placeholder for a measured or observed value E.g., let the variable A equal your age For me, A=47 (years) A random variable is a variable for a random observation E.g., let the random variable X be the age of a random person in the class For a specific person, X has a value For a collection of people, X has a distribution, which gives the frequency of occurrence of ages in class 13

14 Random Variables From the first class: Let X be the outcome of a dice roll X is a random variable X can be equal to 1, 2, 3, 4, 5, or 6 depending on what occurs on the roll of a dice X has a distribution: Probability X=x is 1/6, for x=1,2,3,4,5, or 6 Notation: Pr(X=x)=1/6, for x=1,2,3,4,5, or 6 14

15 Pr (X=x) Plotting a Probability Distribution Let X denote the outcome of a fair die i.e., Pr(X=x)=1/6, for x=1,2,3,4,5, or 6 We can draw the probability function: 2/6 1/ x 15

16 Probability Distributions Can be for either discrete or continuous variables (data) Gives the probability of an event or set of events Sum over all possible events equals 1 Means one of the possible events must happen E.g., Rolled die must give a 1, 2, 3, 4, 5, or 6 16

17 Normal Distributions Normal Distribution is an important continuous distribution Symmetric, bell-shaped For population, described by its Mean: Standard deviation: Notation: N(, Being non-normal does not mean abnormal 2 ) Greek letter mu Greek letter sigma 17

18 Properties of the Normal Curve Symmetric Bell shaped Unimodal Thin tails The normal curve is a model relating the mean and variance to the quantiles 18

19 Why Focus on the Normal Distribution? Normal distribution describes many natural phenomenon well Central Limit Theorem explains why Statistical theorem: Distribution of sums of random variables tends toward the normal The more things that are summed, the more like the normal Result is that averages tend to have a normal distribution 19

20 Highly skewed process Central Limit Theorem in Action Mean of Mean of Mean of

21 Statistics and Parameters A statistic is a one-number summary of data Statistics can be for samples or populations x-bar and s are examples of sample statistics and are parameters of the normal distribution We often estimate parameters with statistics Estimate with X Estimate with s 21

22 Parameters vs. Statistics Every population summary Mean ( ) Standard Deviation ( ) Proportion (p) Parameters has a corresponding sample summary Mean Standard deviation s Proportion Statistics x pˆ Sample statistics are good guesses for population parameters, but they re not the same 22

23 The Empirical Rule If the normal curve fits well then: 68% of the data is within 1 SD of the mean 95% within 2 SD 99% within 3 SD % % 99% Z 23

24 Standardizing a Normal Distribution Standardizing means turning an observation from a N(, 2 ) into a N(0,1) observation If X comes from a N(, Z X has a N(0,1) distribution 2 ) then If and are estimated, then use X x Z s 24

25 Finding the Probability for a Normal Distribution (1) See Table A3-2 on page 491 in Business Statistics Enter with a value of a Read across to the p column to get probability of being between a and a Example: a=1 Probability is of being between 1 and 1 Empirical rule! Note: Can also go in the other direction to find the a-value corresponding to a probability 25

26 Finding the Probability for a Normal Distribution (2) See Table A3-1 on page 492 in Business Statistics Enter with a value of a Read across to the p column to get probability of being less than a Example: a=1: Probability is Example: a=-1: Probability is So, Pr(-1<z<1) = Pr(z < 1) Pr(z < -1) = = Empirical rule again! 26

27 Finding the Probability for a Normal Distribution (3) In Excel, use NORMSDIST function =NORMSDIST(a) = Pr(Z<a) Just like Table A3-1 Can also use NORMDIST function Gives probability for any normal distribution Form: =NORMDIST(a,,,1) So, NORMDIST(a,0,1,1) = NORMSDIST(a) 27

28 Exercises in finding the Probability for a Normal Distribution For a standard normal distribution: What is the probability of being outside of the interval (3, -3)? What is the probability of getting an observation less than 2? For a N(1,3 2 ) distribution: What is the probability of being within one standard deviation of the mean? What is the probability of getting an observation greater than 7? 28

29 Solutions 29

30 Using Normal Probabilities to Test Assertions You are production manager of a widget manufacturing facility Defective widget: quality characteristic < 7 Line supervisor says not to worry: Distribution of quality characteristic is N(16,9) Should you worry or not? You re a careful production manager Visit the line and pick a random widget Widget s quality characteristic measures 5 Do you believe the supervisor s distribution assertion? 30

31 The Calculations 31

32 GMAT Score RelChange Testing Normality Normal Quantile Plot, or Q-Q Plot X-axis: observed data Y-axis: expected data if normal model were true Close to straight line means close to normal JMP: After Analyze > Distribution > red triangle > Normal Quantile Pl Normal Quantile Plot GMAT Case Count Axis Normal Quantile Plot GM Stock Case 32

33 Highly skewed process Evaluating Normality Mean of Mean of Mean of

34 In the Readings don t worry too much about: Sampling with and without replacement Permutations and combinations Uniform, t, chi-square, and F distributions If we had more time, we d cover these topics 34

35 What we have learned so far Types of data and types of variation Descriptive statistics Statistical plots and graphs Random variables and a little probability And, Or, Not rules The normal distribution Standardizing Calculating probabilities 35

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