STA 291 Spring 2008 Lecture 6 Dustin Lueker

Symbols STA 291 Spring 2008 Lecture 6

Variance and Standard Deviation Sample Variance Standard Deviation Population STA 291 Spring 2008 Lecture 6

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 Spring 2008 Lecture 6

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 Spring 2008 Lecture 6

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 Spring 2008 Lecture 6

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 STA 291 Spring 2008 Lecture 6

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 Spring 2008 Lecture 6

Complement Let A denote an even Complement of an event A Example Denoted by AC, all the outcomes in the sample space S that do not belong to the even A P(AC)=1-P(A) Example If someone completes 64% of his passes, then what percentage is incomplete? STA 291 Spring 2008 Lecture 6

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 Spring 2008 Lecture 6

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 Spring 2008 Lecture 6

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 (mutually exclusive) events in a sample space S P(A∪B)=P(A)+P(B) STA 291 Spring 2008 Lecture 6

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 Spring 2008 Lecture 6

Assigning Probabilities to Events Can be difficult Different approaches to assigning probabilities to events Objective Equally likely outcomes (classical approach) Relative frequency Subjective STA 291 Spring 2008 Lecture 6

Equally Likely Approach 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 Spring 2008 Lecture 6

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 Spring 2008 Lecture 6

Relative Frequency Approach 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 Spring 2008 Lecture 6

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 Spring 2008 Lecture 6

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 Spring 2008 Lecture 6