Statistical Reasoning & Practice, Week 5 Notes
Statistical Reasoning & Practice, Week 5 Notes 36-201
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This 4 page Class Notes was uploaded by Monica Chang on Saturday October 1, 2016. The Class Notes belongs to 36-201 at Carnegie Mellon University taught by Gordon Weinberg in Fall 2016. Since its upload, it has received 12 views. For similar materials see Statistical Reasoning and Practice in Statistics at Carnegie Mellon University.
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Date Created: 10/01/16
Week 5 (start of exam 2 material) Introduction to probability: - Random variation is natural, and we use probability to measure it and minimize random error by increasing sample size - Random selection makes studies as representative as possible - Random assignment to explanatory variable allows us to determine causation Intuition of probability: - Probability is a measures likelihood numerically - Probability is between 0 and 1 - Probability is unitless - Probability ≤ 0.05 relatively surprising - Probability ≤ 0.01 very surprising Axioms of probability: 1. Probability has to be between 0 and 1 2. If outcomes are disjoint, the probability of either of two disjoint outcomes is the sum of their individual probabilities 3. Sample space must have probability of 1 Probability notation and terminology: - Random Sampling: Simple random sample (SRS): 1. Equal likelihood: every individual has the same probability of being selected as any other individual 2. Independence: likelihood of an individual being selected must be independent of any selections beforehand o The two above conditions are met when we select with replacement. o *However, when we select WITHOUT replacement, the independence condition is destroyed, so a rule of thumb is that the population must be at least 10-20 times the sample size so that replacement doesn’t have a large affect on independence. - Trial – each run of a probability experiment - If A is an event P(A) is the probability of the event A. - Complement rule: P(not A)=1−P(A)=P(A ) c - total ¿of waysoutcomeoccurs P outcomeofinteres= ¿ of possibleoutcomes¿ - Disjoint (mutually exclusive) – when outcomes cannot have together (imagine no mutual intersection on venn diagram) - Elementary outcomes – outcomes in a probability experiment that have irreducible probabilities and are mutually disjoint - Sample space – set of all elementary outcomes in a probability experiment, denoted by S - Intersection of events (“and” situation) o Gives proportion of population that share two qualities at the same time o P A∩B ) o The conjunction between two events cannot be calculated by an operation between probabilities (the probability gets smaller) - Union of events (“or” situation) o Gives proportion of elements in either (or both) of two qualities o P A∪B ) o General addition rule: P A∨B =P A +P (B)–P (A∧B ) P A∪B )=P(A)+P(B)– P(A∩B) Conditional Probability: - The probability of event A given event B is the probability of both happening divided the probability of B. P(A∧B ) - In words: P Agiven B)= P ( ) o General multiplication rule in words: P(A∧B =P (AgivenB ∗P(B) P(A∩B) - In notation: P(A∨B )= P(B) o General multiplication rule in notation: P(A∩B =P A∨B ∗P(B) - Ex. Rolling a die: o Probability of getting a 4 = 1/6 o Conditional probability of 4, given even: P(4 given even) = P(4|even) = (1/6)/(1/2) = 1/3 - For other examples/problems, look at lecture 11 Statistical Independence: - Independence - when the probability of one outcome does not change probability of other outcomes - Statistical independence w/ variables is when association between A and be are not significantly different than what you’d expect from random chance - Statistical independence w/ sampling is when any individual has an equal probability of being selected regardless of who has been selected already (statistical dependence indicates sampling error) - When conditional probabilities and the marginal probability of an event are equal, then events A and B are independent: c For independent events P (B|A=P B|A )=P (B) - Special Multiplication Rule for Independent Events: For independent events, P A∩B )=P (A∗P(B) o Proof: From the general multiplication rule, we know that P(A∩B =P A∨B ∗P(B) , so when A and B are independent, P A∨B ) is the same as P(A) , so for independent events, P A∩B )=P(A)∗P(B) - You can use the above formulas to test for independence
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