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STA 291 Fall 2009

STA 291 Fall 2009. Lecture 7 Dustin Lueker. Symbols. Variance and Standard Deviation. Sample Variance Standard Deviation Population Variance Standard Deviation. Variance Step By Step. Calculate the mean For each observation, calculate the deviation

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STA 291 Fall 2009

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  1. STA 291Fall 2009 Lecture 7 Dustin Lueker

  2. Symbols STA 291 Fall 2009 Lecture 7

  3. Variance and Standard Deviation • Sample • Variance • Standard Deviation • Population • Variance • Standard Deviation STA 291 Fall 2009 Lecture 7

  4. Variance Step By Step • Calculate the mean • For each observation, calculate the deviation • For each observation, calculate the squared deviation • Add up all the squared deviations • Divide the result by (n-1) Or N if you are finding the population variance (To get the standard deviation, take the square root of the result) STA 291 Fall 2009 Lecture 7

  5. Empirical Rule • If the data is approximately symmetric and bell-shaped then • About 68% of the observations are within one standard deviation from the mean • About 95% of the observations are within two standard deviations from the mean • About 99.7% of the observations are within three standard deviations from the mean STA 291 Fall 2009 Lecture 7

  6. Empirical Rule STA 291 Fall 2009 Lecture 7

  7. Probability Terminology • Experiment • Any activity from which an outcome, measurement, or other such result is obtained • Random (or Chance) Experiment • An experiment with the property that the outcome cannot be predicted with certainty • Outcome • Any possible result of an experiment • Sample Space • Collection of all possible outcomes of an experiment • Event • A specific collection of outcomes • Simple Event • An event consisting of exactly one outcome STA 291 Fall 2009 Lecture 7

  8. Experiment, Sample Space, Event Examples: Experiment 1. Flip a coin 2. Flip a coin 3 times 3. Roll a die 4. Draw a SRS of size 50 from a population Sample Space 1. 2. 3. 4. Event 1. 2. 3. 4. STA 291 Fall 2009 Lecture 7

  9. Complement S A • Let A denote an event • Complement of an event A • Denoted by AC, all the outcomes in the sample space S that do not belong to the event A • P(AC)=1-P(A) • Example • If someone completes 64% of his passes, then what percentage is incomplete? STA 291 Fall 2009 Lecture 7

  10. Union and Intersection • Let A and B denote two events • Union of A and B • A ∪ B • All the outcomes in S that belong to at least one of A or B • Intersection of A and B • A ∩ B • All the outcomes in S that belong to both A and B STA 291 Fall 2009 Lecture 7

  11. Additive Law of Probability • Let A and B be two events in a sample space S • P(A∪B)=P(A)+P(B)-P(A∩B) S A B STA 291 Fall 2009 Lecture 7

  12. Additive Law of Probability • Let A and B be two events in a sample space S • P(A∪B)=P(A)+P(B)-P(A∩B) • At State U, all first-year students must take chemistry and math. Suppose 15% fail chemistry, 12% fail math, and 5% fail both. Suppose a first-year student is selected at random, what is the probability that the student failed at least one course? STA 291 Fall 2009 Lecture 7

  13. Disjoint Events (Mutually Exclusive) • Let A and B denote two events • A and B are Disjoint (mutually exclusive) events if there are no outcomes common to both A and B • A∩B=Ø • Ø = empty set or null set • Let A and B be two disjoint events in a sample space S • P(A∪B)=P(A)+P(B) S A B STA 291 Fall 2009 Lecture 7

  14. Assigning Probabilities to Events • The probability of an event occurring is nothing more than a value between 0 and 1 • 0 implies the event will never occur • 1 implies the event will always occur • How do we go about figuring out probabilities? STA 291 Fall 2009 Lecture 7

  15. Assigning Probabilities to Events • Can be difficult • Different approaches to assigning probabilities to events • Subjective • Objective • Equally likely outcomes (classical approach) • Relative frequency STA 291 Fall 2009 Lecture 7

  16. Subjective Probability Approach • Relies on a person to make a judgment on how likely an event is to occur • Events of interest are usually events that cannot be replicated easily or cannot be modeled with the equally likely outcomes approach • As such, these values will most likely vary from person to person • The only rule for a subjective probability is that the probability of the event must be a value in the interval [0,1] STA 291 Fall 2009 Lecture 7

  17. Equally Likely (Laplace) • The equally likely approach usually relies on symmetry to assign probabilities to events • As such, previous research or experiments are not needed to determine the probabilities • Suppose that an experiment has only n outcomes • The equally likely approach to probability assigns a probability of 1/n to each of the outcomes • Further, if an event A is made up of m outcomes then P(A) = m/n STA 291 Fall 2009 Lecture 7

  18. Examples • Selecting a simple random sample of 2 individuals • Each pair has an equal probability of being selected • Rolling a fair die • Probability of rolling a “4” is 1/6 • This does not mean that whenever you roll the die 6 times, you always get exactly one “4” • Probability of rolling an even number • 2,4, & 6 are all even so we have 3 possibly outcomes in the event we want to examine • Thus the probability of rolling an even number is 3/6 = 1/2 STA 291 Fall 2009 Lecture 7

  19. Relative Frequency (von Mises) • Borrows from calculus’ concept of the limit • We cannot repeat an experiment infinitely many times so instead we use a ‘large’ n • Process • Repeat an experiment n times • Record the number of times an event A occurs, denote this value by a • Calculate the value of a/n STA 291 Fall 2009 Lecture 7

  20. Relative Frequency Approach • “large” n? • Law of Large Numbers • As the number of repetitions of a random experiment increases, the chance that the relative frequency of occurrence for an event will differ from the true probability of the even by more than any small number approaches 0 • Doing a large number of repetitions allows us to accurately approximate the true probabilities using the results of our repetitions STA 291 Fall 2009 Lecture 7

  21. Probabilities of Events • Let A be the event A = {o1, o2, …, ok}, where o1, o2, …, ok are k different outcomes • Suppose the first digit of a license plate is randomly selected between 0 and 9 • What is the probability that the digit 3? • What is the probability that the digit is less than 4? STA 291 Fall 2009 Lecture 7

  22. Conditional Probability • Note: P(A|B) is read as “the probability that A occurs given that B has occurred” STA 291 Fall 2009 Lecture 7

  23. Independence • If events A and B are independent, then the events have no influence on each other • P(A) is unaffected by whether or not B has occurred • Mathematically, if A is independent of B • P(A|B)=P(A) • Multiplication rule for independent events A and B • P(A∩B)=P(A)P(B) STA 291 Fall 2009 Lecture 7

  24. Example • Flip a coin twice, what is the probability of observing two heads? • Flip a coin twice, what is the probability of observing a head then a tail? A tail then a head? One head and one tail? • A 78% free throw shooter is fouled while shooting a three pointer, what is the probability he makes all 3 free throws? None? STA 291 Fall 2009 Lecture 7

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