1. STA 291 Spring 2008
Lecture 6
Dustin Lueker

2. Symbols
STA 291 Spring 2008 Lecture 6

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

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 Spring 2008 Lecture 6

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 Spring 2008 Lecture 6

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

7. 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

8. 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

9. 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

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 Spring 2008 Lecture 6

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)
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

12. 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

13. 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

14. 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

15. 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

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 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

17. 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

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 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

19. 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