EconBasicProbability

EconBasicProbability - Introductory Statistics Stats 210...

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1 Introductory Statistics Stats 210 Spring 2007
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2 So what is this all about??? Let’s assume we want to investigate the following relationships What is the relationship between student performance and class size? What is the relationship between education and wages? Do I earn more if I drink a lot? No really !
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3 600 620 640 660 680 700 15 20 25 Test Scores and Student-Teacher Ratio Student-Teacher Ratio Test Scores
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4 0 10 20 30 40 50 5 10 16 20 Highest Grade Attended Hourly Earning and Education Earnings per hour 12
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5 The next slide is VERY, VERY, VERY, VERY, VERY, VERY, ……, VERY IMPORTANT!
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6 Population vs. Sample Population (random) Samples
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7
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8 In a Nutshell…… Probability Theory Describe the whole population in probabilistic terms Finding Population Probabilities Describing Probability Distributions, i.e. Normal Distribution Statistics Use a sample of the population to learn about the population Use a sample of wages to learn about the average wage in the population Use a sample of the population to investigate the relationship between schooling and education
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9 Set Theory: Experiment & Outcome Assume we are performing an experiment Experiment = An action or process whose outcome is subject to uncertainty i.e tossing a coin taking a random persons wage (height) Every experiment will have a number (set) of different outcomes Heads or Tails Virtually any positive number for wage
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10 Set Theory: Sample Space We define the SAMPLE SPACE of an experiment as the set of all possible outcomes Sample space for coin toss S=[Head, Tail] The number of outcomes is finite Outcomes are discrete Sample space for wage S=[0,infinity] Infinite number of outcomes Outcomes are continuous
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11 Set Theory: Event Event = is an outcome OR a set of outcomes of an experiment = subset of the sample space Let’s look at tossing a coin two times Sample Space S = [HH, TH, HT, TT] Event: AT LEAST ONE HEAD Other event: EXACTLY ONE HEAD
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12 S Events and Outcomes = OUTCOMES = EVENTS
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13 The Venn Diagram The rectangle represents the sample space and the forms two events A and B B A S
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14 Relationships between Events Union The union of two events A and B is an event itself It is the event that either A or B or both occur Denoted by: A U B = B A S Note: Can overlap
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15 Relationships between Events Intersection The intersection of two events A and B is an event itself It is the event that both A and B occur Denoted by: A B = B A S
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16 Relationships between Events Complement The complement of an event A is also an event itself It is the event that A does not occur Denoted by: A C = Note: This can be extended to a complement of any event, i.e. (A U B) C A S A C B (A U B) C
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17 Relationships between Events Disjoint Two events that are disjoint have no outcomes in common If A and B are disjoint: A B = Ø where Ø is the empty set B A S
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