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(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture
Lecture 16
Binomial Random Variables
Definition What if Events are Dependent? Center, Spread, Shape of Counts, Proportions Normal Approximation (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 2
Looking Back: Review
4 Stages of Statistics Data Production (discussed in Lectures 1-4) Displaying and Summarizing (Lectures 5-12) Probability Finding Probabilities (discussed in Lectures 13-14) Random Variables (introduced in Lecture 15) Binomial Normal Sampling Distributions Statistical Inference (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 3
Definition (Review)
Discrete Random Variable: one whose possible values are finite or countably infinite (like the numbers 1, 2, 3, …) Looking Ahead: To perform inference about categorical variables, need to understand behavior of sample proportion. A first step is to understand behavior of sample counts. We will eventually shift from discrete counts to a normal approximation, which is continuous. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 4
Definition
Binomial Random Variable counts sampled individuals falling into particular category; Sample size n is fixed Each selection independent of others Just 2 possible values for each individual Each has same probability p of falling in category of interest
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 5 Example: A Simple Binomial Random Variable Background : The random variable X is the count of tails in two flips of a coin. Questions: Why is X binomial? What are n and p? How do we display X? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 7 Example: A Simple Binomial Random Variable Background : The random variable X is the count of tails in two flips of a coin. Responses: Sample size n fixed? Each selection independent of others? Just 2 possible values for each? Each has same probability p? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 10 Example: A Simple Binomial Random Variable Looking Back: We already discussed this random variable when learning about probability distributions. Responses: Display with ___________ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 11
Example: Determining if R.V. is Binomial
Background : Consider following R.V.: Pick card from deck of 52, replace, pick another. X =no. of cards picked until you get ace. Question: Is X binomial?
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 19
Example: Determining if R.V. is Binomial
Background : Consider following R.V.: Pick 16 cards with replacement from deck of 52. W =no. of clubs, X =no. of diamonds, Y =no. of hearts, Z =no. of spades Response: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 20
Example: Determining if R.V. is Binomial
Background : Consider following R.V.: Pick with replacement from German deck of 32 (doesn’t include numbers 2-6), then from deck of 52, back to deck of 32, etc. for 16 selections altogether. X =no. of aces picked. Question: Is X binomial? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 23
Example: Determining if R.V. is Binomial
Background : Consider following R.V.: Pick with replacement from German deck of 32 (doesn’t include numbers 2-6), then from deck of 52 , back to deck of 32, etc. for 16 selections altogether. X =no. of aces picked. Response: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 24
Example: Determining if R.V. is Binomial
Background : Consider following R.V.: Pick 16 cards with replacement from deck of 52. X =no. of hearts picked. Question: Is X binomial?
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 26
Example: Determining if R.V. is Binomial
Background : Consider following R.V.: Pick 16 cards with replacement from deck of 52. X =no. of hearts picked. Response: fixed n = 16 selections independent (with replacement) just 2 possible values (heart or not) same p = 0.25 for all selections (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 27
Requirement of Independence
Snag: Binomial theory requires independence Actual sampling done without replacement so selections are dependent Resolution: When sampling without replacement, selections are approximately independent if population is at least 10n. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 28
Example: A Binomial Probability Problem
Background : The proportion of Americans who are left-handed is 0.1. Of 44 presidents, 7 have been left-handed (proportion 0.16). Question: How can we establish if being left-handed predisposes someone to be president? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 30
Example: A Binomial Probability Problem
Background : The proportion of Americans who are left-handed is 0.10. Of 44 presidents, 7 have been left-handed (proportion 0.16). Response: Determine if 7 out of 44 (0.16) is ________________ when sampling at random from a population where 0.10 fall in the category of interest.
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 36 Example: Standard Deviation of Sample Count Background : Probability of being left-handed is approx. 0.1. Randomly sample 100 people. Sample count has mean 100(0.1)= 10, standard deviation Question: How do we interpret these? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 38 Example: Standard Deviation of Sample Count Background : Probability of being left-handed is approx. 0.1. Randomly sample 100 people. Sample count has mean 100(0.1)= 10, standard deviation Response: On average, expect sample count =___ lefties. Counts vary; typical distance from ___ is ___. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 39 Example: S.D. of Sample Proportion Background : Probability of being left-handed is approx. 0.1. Randomly sample 100 people. Sample proportion has mean 0.l, standard deviation Question: How do we interpret these? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 41 Example: S.D. of Sample Proportion Background : Probability of being left-handed is approx. 0.1. Randomly sample 100 people. Sample proportion has mean 0.l, standard deviation Response: On average, expect sample proportion ____ lefties. Proportions vary; typical distance from____ is ____.
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 42
Example: Role of Sample Size in Spread
Background : Consider proportion of tails in various sample sizes n of coinflips. Questions: What is the standard deviation for n =1? n =4? n =16? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 44
Example: Role of Sample Size in Spread
Background : Consider proportion of tails in various sample sizes n of coinflips. Responses: n =1: s.d.= n =4: s.d.= n =16: s.d.= Because of n in the denominator of the formula for standard deviation, spread of sample proportion _________as n increases. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 45
Shape of Distribution of Count, Proportion
Binomial count X or proportion for repeated random samples has shape approximately normal if samples are large enough to offset underlying skewness. (Central Limit Theorem) For a given sample size n , shapes are identical for count and proportion. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 46
Example: Underlying Coinflip Distribution
Background : Distribution of count or proportion of tails in n =1 coinflip ( p =0.5): Question: What are the distributions’ shapes?
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Example: Distribution of for 16 Coinflips
Background : Distribution of proportion of tails in n =16 coinflips ( p =0.5): Question: What is the shape? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 55
Example: Distribution for 16 Coinflips
Background : Distribution of proportion of tails in n =16 coinflips ( p =0.5): Response: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 56
Example: Underlying Distribution of Lefties
Background : Distribution of proportion of lefties ( p =0.1) for samples of n =1: Question: What is the shape? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 58
Example: Underlying Distribution of Lefties
Background : Distribution of proportion of lefties ( p =0.1) for samples of n =1: Response:
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 59
Example: Dist of of Lefties for n= 16
Background : Distribution of proportion of lefties ( p =0.1) for n =16: Question: What is the shape? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 61
Example: Dist of of Lefties for n= 16
Background : Distribution of proportion of lefties ( p =0.1) for n =16: Response: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 62
Example: Dist of of Lefties for n= 100
Background : Distribution of proportion of lefties ( p =0.1) for n =100: Question: What is the shape? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L16. 64
Example: Dist of of Lefties for n= 100
Background : Distribution of proportion of lefties ( p =0.1) for n =100: Response:
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Example: Solving the Left-handed Problem
Background : The proportion of Americans who are left-handed is 0.1. We consider P( p ≥7/44=0.16) for a sample of 44 presidents. Response: ________________, approx. is poor. Probability 0_._ 0_. Actual probability is 0. Approximated probability is 0.10._ 0.16 0.1^ 0. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L14. 72 Example: From Count to Proportion and Vice Versa Background : Consider these reports: In a sample of 87 assaults on police, 23 used weapons. 0.44 in sample of 25 bankruptcies were due to med. bills Question: In each case, what are n , X , and? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L14. 74 Example: From Count to Proportion and Vice Versa Background : Consider these reports: In a sample of 87 assaults on police, 23 used weapons. 0.44 in sample of 25 bankruptcies were due to med. bills Response: First has n =___, X =____, _____ Second has n =____, ____, X =________ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L14. 86 Lecture Summary (Binomial Random Variables) Definition; 4 requirements for binomial R.V.s that do or don’t conform to requirements Relaxing requirement of independence Binomial counts, proportions Mean Standard deviation Shape Normal approximation to binomial
