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STA 291 Summer 2010

STA 291 Summer 2010. Lecture 6 Dustin Lueker. Coefficient of Variation. Standardized measure of variation Idea A standard deviation of 10 may indicate great variability or small variability, depending on the magnitude of the observations in the data set

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STA 291 Summer 2010

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  1. STA 291Summer 2010 Lecture 6 Dustin Lueker

  2. Coefficient of Variation • Standardized measure of variation • Idea • A standard deviation of 10 may indicate great variability or small variability, depending on the magnitude of the observations in the data set • CV = Ratio of standard deviation divided by mean • Population and sample version STA 291 Summer 2010 Lecture 6

  3. Example • Which sample has higher relative variability? (a higher coefficient of variation) • Sample A • mean = 62 • standard deviation = 12 • CV = • Sample B • mean = 31 • standard deviation = 7 • CV = STA 291 Summer 2010 Lecture 6

  4. 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 Summer 2010 Lecture 6

  5. 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 Summer 2010 Lecture 6

  6. 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 Summer 2010 Lecture 6

  7. 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 Summer 2010 Lecture 6

  8. 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 Summer 2010 Lecture 6

  9. 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 Summer 2010 Lecture 6

  10. 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 Summer 2010 Lecture 6

  11. Conditional Probability • Note: P(A|B) is read as “the probability that A occurs given that B has occurred” STA 291 Summer 2010 Lecture 6

  12. 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 Summer 2010 Lecture 6

  13. 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 Summer 2010 Lecture 6

  14. 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 Summer 2010 Lecture 6

  15. 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 Summer 2010 Lecture 6

  16. 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 possible outcomes in the event we want to examine • Thus the probability of rolling an even number is 3/6 = 1/2 STA 291 Summer 2010 Lecture 6

  17. 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 Summer 2010 Lecture 6

  18. 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 event by more than any small number approaches zero • Doing a large number of repetitions allows us to accurately approximate the true probabilities using the results of our repetitions STA 291 Summer 2010 Lecture 6

  19. Random Variables • X is a random variable if the value that X will assume cannot be predicted with certainty • That’s why its called random • Two types of random variables • Discrete • Can only assume a finite or countably infinite number of different values • Continuous • Can assume all the values in some interval STA 291 Summer 2010 Lecture 6

  20. Examples • Are the following random variables discrete or continuous? • X = number of houses sold by a real estate developer per week • X = weight of a child at birth • X = time required to run 800 meters • X = number of heads in ten tosses of a coin STA 291 Summer 2010 Lecture 6

  21. Discrete Probability Distribution • A list of the possible values of a random variable X, say (xi) and the probability associated with each, P(X=xi) • All probabilities must be nonnegative • Probabilities sum to 1 STA 291 Summer 2010 Lecture 6

  22. Example • The table above gives the proportion of employees who use X number of sick days in a year • An employee is to be selected at random • Let X = # of days of leave • P(X=2) = • P(X≥4) = • P(X<4) = • P(1≤X≤6) = STA 291 Summer 2010 Lecture 6

  23. Expected Value of a Discrete Random Variable • Expected Value (or mean) of a random variable X • Mean = E(X) = μ = ΣxiP(X=xi) • Example • E(X) = STA 291 Summer 2010 Lecture 6

  24. Variance of a Discrete Random Variable • Variance • Var(X) = E(X-μ)2 = σ2 = Σ(xi-μ)2P(X=xi) • Example • Var(X) = STA 291 Summer 2010 Lecture 6

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