Lecture 7 Dustin Lueker

2STA 291 Fall 2009 Lecture 7

 Sample ◦ Variance ◦ Standard Deviation  Population ◦ Variance ◦ Standard Deviation 3STA 291 Fall 2009 Lecture 7

4 1. Calculate the mean 2. For each observation, calculate the deviation 3. For each observation, calculate the squared deviation 4. Add up all the squared deviations 5. 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

 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 5STA 291 Fall 2009 Lecture 7

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 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 7STA 291 Fall 2009 Lecture 7

8 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 Event

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

 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 10STA 291 Fall 2009 Lecture 7

 Let A and B be two events in a sample space S ◦ P(A∪B)=P(A)+P(B)-P(A∩B) 11STA 291 Fall 2009 Lecture 7 S AB

 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? 12STA 291 Fall 2009 Lecture 7

 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) 13STA 291 Fall 2009 Lecture 7 S AB

 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? 14STA 291 Fall 2009 Lecture 7

 Can be difficult  Different approaches to assigning probabilities to events ◦ Subjective ◦ Objective  Equally likely outcomes (classical approach)  Relative frequency 15STA 291 Fall 2009 Lecture 7

 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 716

 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 717

 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 18STA 291 Fall 2009 Lecture 7

 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 19STA 291 Fall 2009 Lecture 7

 “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 20STA 291 Fall 2009 Lecture 7

 Let A be the event A = {o 1, o 2, …, o k }, where o 1, o 2, …, o k 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? 21STA 291 Fall 2009 Lecture 7

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

 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) 23STA 291 Fall 2009 Lecture 7

 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? 24STA 291 Fall 2009 Lecture 7