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Statistics

Statistics. Chapter 3: Probability. Where We’ve Been. Making Inferences about a Population Based on a Sample Graphical and Numerical Descriptive Measures for Qualitative and Quantitative Data. Where We’re Going. Probability as a Measure of Uncertainty Basic Rules for Finding Probabilities

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Statistics

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  1. Statistics Chapter 3: Probability

  2. Where We’ve Been • Making Inferences about a Population Based on a Sample • Graphical and Numerical Descriptive Measures for Qualitative and Quantitative Data McClave: Statistics, 11th ed. Chapter 3: Probability

  3. Where We’re Going • Probability as a Measure of Uncertainty • Basic Rules for Finding Probabilities • Probability as a Measure of Reliability for an Inference McClave: Statistics, 11th ed. Chapter 3: Probability

  4. 3.1: Events, Sample Spaces and Probability • An experiment is an act or process of observation that leads to a single outcome that cannot be predicted with certainty. McClave: Statistics, 11th ed. Chapter 3: Probability

  5. 3.1: Events, Sample Spaces and Probability • A sample point is the most basic outcome of an experiment. An Ace A four A Head McClave: Statistics, 11th ed. Chapter 3: Probability

  6. 3.1: Events, Sample Spaces and Probability • A sample space of an experimentis the collection of all sample points. • Roll a single die: S: {1, 2, 3, 4, 5, 6} McClave: Statistics, 11th ed. Chapter 3: Probability

  7. 3.1: Events, Sample Spaces and Probability • Sample points and sample spaces are often represented with Venn diagrams. McClave: Statistics, 11th ed. Chapter 3: Probability

  8. All probabilities must be between 0 and 1. The probabilitiesof all the sample points must sum to 1. Probability Rules for Sample Points 3.1: Events, Sample Spaces and Probability McClave: Statistics, 11th ed. Chapter 3: Probability

  9. All probabilities must be between 0 and 1 The probabilitiesof all the sample points must sum to 1 Probability Rules for Sample Points 3.1: Events, Sample Spaces and Probability 0 indicates an impossible outcome and 1 a certain outcome. Something has to happen. McClave: Statistics, 11th ed. Chapter 3: Probability

  10. 3.1: Events, Sample Spaces and Probability • An event is a specific collection of sample points: • Event A: Observe an even number. McClave: Statistics, 11th ed. Chapter 3: Probability

  11. 3.1: Events, Sample Spaces and Probability • The probability of an event is the sum of the probabilities of the sample points in the sample space for the event. • Event A: Observe an even number. • P(A) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2 McClave: Statistics, 11th ed. Chapter 3: Probability

  12. Calculating Probabilities for Events Define the experiment. List the sample points. Assign probabilities to sample points. Collect all sample points in the event of interest. The sum of the sample point probabilities is the event probability. 3.1: Events, Sample Spaces and Probability McClave: Statistics, 11th ed. Chapter 3: Probability

  13. Calculating Probabilities for Events Define the experiment List the sample points Assign probabilities to sample points Collect all sample points in the event of interest The sum of the sample point probabilities is the event probability 3.1: Events, Sample Spaces and Probability If the number of sample points gets too large, we need a way to keep track of how they can be combined for different events. McClave: Statistics, 11th ed. Chapter 3: Probability

  14. 3.1: Events, Sample Spaces and Probability • If a sample of n elements is drawn from a set of N elements (N ≥ n), the number ofdifferent samples is McClave: Statistics, 11th ed. Chapter 3: Probability

  15. 3.1: Events, Sample Spaces and Probability • If a sample of 5 elements is drawn from a set of 20 elements, the number ofdifferent samples is McClave: Statistics, 11th ed. Chapter 3: Probability

  16. 3.2: Unions and Intersections McClave: Statistics, 11th ed. Chapter 3: Probability

  17. 3.2: Unions and Intersections McClave: Statistics, 11th ed. Chapter 3: Probability

  18. 3.2: Unions and Intersections A A  C A B  C B C McClave: Statistics, 11th ed. Chapter 3: Probability

  19. 3.3: Complementary Events • The complement of any event A is the event that A does not occur, AC. A: {Toss an even number} AC: {Toss an odd number} B: {Toss a number ≤ 3} BC: {Toss a number ≥ 4} A  B = {1,2,3,4,6} [A  B]C = {5} (Neither A nor B occur) McClave: Statistics, 11th ed. Chapter 3: Probability

  20. 3.3: Complementary Events McClave: Statistics, 11th ed. Chapter 3: Probability

  21. A: {At least one head on two coin flips} AC: {No heads} 3.3: Complementary Events McClave: Statistics, 11th ed. Chapter 3: Probability

  22. 3.4: The Additive Rule and Mutually Exclusive Events • The probability of the union of events A and B is the sum of the probabilities of A and B minus the probability of the intersection of A and B: McClave: Statistics, 11th ed. Chapter 3: Probability

  23. 3.4: The Additive Rule and Mutually Exclusive Events At a particular hospital, the probability of a patient having surgery (Event A) is .12, of an obstetric treatment (Event B) .16, and of both .02. What is the probability that a patient will have either treatment? McClave: Statistics, 11th ed. Chapter 3: Probability

  24. 3.4: The Additive Rule and Mutually Exclusive Events • Events A and B are mutually exclusive if AB contains no sample points. McClave: Statistics, 11th ed. Chapter 3: Probability

  25. 3.4: The Additive Rule and Mutually Exclusive Events If A and B are mutually exclusive, McClave: Statistics, 11th ed. Chapter 3: Probability

  26. 3.5: Conditional Probability • Additional information or other events occurring may have an impact on the probability of an event. McClave: Statistics, 11th ed. Chapter 3: Probability

  27. 3.5: Conditional Probability • Additional information may have an impact on the probability of an event. • P(Rolling a 6) is one-sixth (unconditionally). • If we know an even number was rolled, the probability of a 6 goes up to one-third. McClave: Statistics, 11th ed. Chapter 3: Probability

  28. 3.5: Conditional Probability • The sample space is reduced to only the conditioning event. • To find P(A), once we know B has occurred (i.e., given B), we ignore BC (including the A region within BC). B BC A A McClave: Statistics, 11th ed. Chapter 3: Probability

  29. 3.5: Conditional Probability B BC A A McClave: Statistics, 11th ed. Chapter 3: Probability

  30. 3.5: Conditional Probability B A McClave: Statistics, 11th ed. Chapter 3: Probability

  31. 3.5: Conditional Probability • 55% of sampled executives had cheated at golf (event A). • P(A) = .55 • 20% of sampled executives had cheated at golf and lied in business (event B). • P(A  B) = .20 • What is the probability that an executive had lied in business, given s/he had cheated in golf, P(B|A)? McClave: Statistics, 11th ed. Chapter 3: Probability

  32. 3.5: Conditional Probability • P(A) = .55 • P(A  B) = .20 • What is P(B|A)? McClave: Statistics, 11th ed. Chapter 3: Probability

  33. 3.6: The Multiplicative Rule and Independent Events • The conditional probability formula can be rearranged into the Multiplicative Rule of Probability to find joint probability. McClave: Statistics, 11th ed. Chapter 3: Probability

  34. Assume three of ten workers give illegal deductions Event A: {First worker selected gives an illegal deduction} Event B: {Second worker selected gives an illegal deduction} P(A) = P(B) = .1 + .1 + .1 = .3 P(B|A) has only nine sample points, and two targeted workers, since we selected one of the targeted workers in the first round: P(B|A) = 1/9 + 1/9 = .11 + .11 = .22 The probability that both of the first two workers selected will have given illegal deductions P(AB) = P(B|A)P(A) = .(3) (.22) = .066 3.6: The Multiplicative Rule and Independent Events McClave: Statistics, 11th ed. Chapter 3: Probability

  35. 3.6: The Multiplicative Rule and Independent Events McClave: Statistics, 11th ed. Chapter 3: Probability

  36. 3.6: The Multiplicative Rule and Independent Events A Tree Diagram First selected worker Second selected worker McClave: Statistics, 11th ed. Chapter 3: Probability

  37. Dependent Events P(A|B) ≠ P(A) P(B|A) ≠ P(B) Independent Events P(A|B) = P(A) P(B|A) = P(B)  Studying stats 40 hours per week Working 40 hours per week Having blue eyes 3.6: The Multiplicative Rule and Independent Events McClave: Statistics, 11th ed. Chapter 3: Probability

  38. Dependent Events P(A|B) ≠ P(A) P(B|A) ≠ P(B) Independent Events P(A|B) = P(A) P(B|A) = P(B) Mutually exclusive events are dependent: P(B|A) = 0 Since P(B|A) = P(B), P(AB) = P(A)P(B|A) = P(A)P(B) 3.6: The Multiplicative Rule and Independent Events McClave: Statistics, 11th ed. Chapter 3: Probability

  39. 3.7: Random Sampling • If n elements are selected from a population in such a way that every set of n elements in the population has an equal probability of being selected, the n elements are said to be a (simple) random sample. McClave: Statistics, 11th ed. Chapter 3: Probability

  40. How many five-card poker hands can be dealt from a standard 52-card deck? Use the combination rule: 3.7: Random Sampling McClave: Statistics, 11th ed. Chapter 3: Probability

  41. 3.7: Random Sampling • Random samples can be generated by • Mixing up the elements and drawing by hand, say, out of a hat (for small populations) • Random number generators • Random number tables • Table I, App. B • Random sample/number commands on software McClave: Statistics, 11th ed. Chapter 3: Probability

  42. 3.8: Some Additional Counting Rules Situation Number of Different Results Multiplicative Rule • Draw one element from each of k sets, sized n1, n2, n3, … nk Permutations Rule • Draw n elements, arranged in a distinct order, from a set of N elements Partitions Rule • Partition N elements into k groups, sized n1, n2, n3, … nk(ni=N) McClave: Statistics, 11th ed. Chapter 3: Probability

  43. 3.8: Some Additional Counting Rules • Multiplicative Rule Assume one professor from each of the six departments in a division will be selected for a special committee. The various departments have four, seven, six, eight, six and five professors eligible. How many different committees, X*, could be formed? McClave: Statistics, 11th ed. Chapter 3: Probability

  44. 3.8: Some Additional Counting Rules • Permutations If there are eight horses entered in a race, how many different win, place and show possibilities are there? McClave: Statistics, 11th ed. Chapter 3: Probability

  45. 3.8: Some Additional Counting Rules • Partitions Rule The technical director of a theatre has twenty stagehands. She needs eight electricians, ten carpenters and two props people. How many different allocations of stagehands, Y*, can there be? McClave: Statistics, 11th ed. Chapter 3: Probability

  46. 3.8: Bayes’s Rule • Given k mutually exclusive and exhaustive events B1,B2,… Bk, and an observed event A, then McClave: Statistics, 11th ed. Chapter 3: Probability

  47. 3.8: Bayes’s Rule • Suppose the events B1, B2, and B3, are mutually exclusive and complementary events with P(B1) = .2, P(B2) = .15 and P(B3) = .65. Another event A has these conditional probabilities: P(A|B1) = .4, P(A|B2) = .25 and P(A|B3) = .6. • What is P(B1|A)? McClave: Statistics, 11th ed. Chapter 3: Probability

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