Empirical rule - Probabilities. Week 5 (1)

Август 3, 2022

Главная
Математика
Empirical rule - Probabilities. Week 5 (1)

Содержание

2. Interpretation of summary statistics A random sample of people attended a recent soccer match. The summary
3. Deviations from the normal distribution - Kurtosis A distribution with positive kurtosis is pointy and a
4. Positively and negatively skewed Positive skewed is when the distribution is skewed to the right Negative
5. Symmetric distribution - Empirical rule Knowing the mean and the standard deviation of a data set
6. Probability as Area Under the Curve DR SUSANNE HANSEN SARAL Ch. 5- f(X) X μ 0.5
7. If the data distribution is symmetric/normal, then the interval: contains about 68% of the values in
8. contains about 95% of the values in the population or the sample contains almost all (about
9. Empirical rule: Application A company produces batteries with a mean lifetime of 1’200 hours and a
10. Empirical rule: Application DR SUSANNE HANSEN SARAL, SUSANNE.SARAL@OKAN.EDU.TR
11. Interpretation of the Empirical rule: Lightbulb lifetime example If the shape of the distribution is normal,
12. Empirical rule exercise
13. Class quizz Empirical rule: (1) Which shape must the distribution have to be able to apply
14. Introduction to Probabilities DR SUSANNE HANSEN SARAL
15. Probability theory “Life would be simpler if we knew for certain what was going to happen
16. Definition of probability Probability is a numerical measure about the likelihood that an event will occur.
17. Probability and time Time Certainty Uncertainty Certainty runs over a short period of time and gradually
18. Probability and its measures: 2 basic rules Rule 1: Probability is measured over a range from
19. Probability and its measures 2 basic rules Certain uncertainty .5 1 0 Dr Susanne Hansen Saral
20. Probability rule 1 and 2 applied - example Rule 1: Probability is measured over a range
21. Probability and definitions Random experiment Sample space Sample point Event 2/28/2017
22. Random experiment In statistics a random experiment is a process that generates two or more possible,
23. All possible experimental outcomes constitute the sample space A sample space (S) of an experiment is
24. Sample space, S - Examples Random experiment: Flip a coin Possible outcomes: Head or tail The
25. Sample space, S - Examples Outcomes of a statistics course: The sample space: S = {AA,
26. Sample space - example The sample space, S = { Google, direct, Yahoo, MSN and all
27. Event An individual outcome of a sample space is called a simple event. An event is
28. Event: – subset of outcomes of a sample space, S Random experiment: Throw a dice (Turkish:
29. Event : Subset of outcomes of a sample space, S Random experiment: Grade marks on an
30. Events Intersection of Events – If A and B are two events in a sample space
31. Union of events Union of Events – If A and B are two events in a
32. Mutually exclusive event A and B are Mutually Exclusive Events if they have no basic outcomes
33. Collectively Exhaustive Events E1, E2, …,Ek are Collectively Exhaustive events if E1 U E2 U….. Ek
34. Complement The Complement of an event A is the set of all basic outcomes in the
35. Examples Let the Sample Space be the collection of all possible outcomes of rolling one dice:
36. Examples – rolling a dice COPYRIGHT © 2013 PEARSON EDUCATION, INC. PUBLISHING AS PRENTICE HALL Ch.
37. Examples Mutually exclusive: A and B are not mutually exclusive The outcomes 4 and 6 are
38. Class exercise DR SUSANNE HANSEN SARAL
40. Скачать презентацию

Слайд 2

Interpretation of summary statistics A random sample of people attended a

recent soccer match. The summary statistics (Excel output) about their ages is here below:

What is the sample size?
What is the mean age?
What is the median?
What shape does the distribution of ages
have? (symmetric or non-symmetric)
What is the measure/s for spread in the data?
Is this a large spread?
What is the Coefficient of variation for
this data?

Слайд 3

Deviations from the normal distribution - Kurtosis
A distribution with positive kurtosis

is pointy and a distribution with a negative kurtosis is flatter than a normal distribution

DR SUSANNE HANSEN SARAL, SUSANNE.SARAL@GMAIL.COM

Слайд 4

Positively and negatively skewed
Positive skewed is when the distribution is

skewed to the right
Negative skewed is when the distribution is skewed to the left

Слайд 5

Symmetric distribution - Empirical rule
Knowing the mean and the standard

deviation of a data set we can extract a lot of information about the location of our data.
The information depends on the shape of the histogram (symmetric, skewed, etc.).
If the histogram is symmetric or bell-shaped, we can use the Empirical rule.

DR SUSANNE HANSEN SARAL, SUSANNE.SARAL@OKAN.EDU.TR

Слайд 6

Probability as Area Under the Curve
DR SUSANNE HANSEN SARAL
Ch. 5-
f(X)
X
μ
0.5
0.5
The

total area under the curve is 1.0, and the curve is symmetric, so half (50%) of the data in the data set is above the mean, half (50%) is below

Слайд 7

If the data distribution is symmetric/normal, then the interval:
contains about 68%

of the values in the population or the sample

DR SUSANNE HANSEN SARAL, SUSANNE.SARAL@OKAN.EDU.TR

The Empirical Rule

68%

Слайд 8

contains about 95% of the values in the population or

the sample
contains almost all (about 99.7%) of the values in the population
or the sample

DR SUSANNE HANSEN SARAL, SUSANNE.SARAL@OKAN.EDU.TR

The Empirical Rule

99.7%

95%

(continued)

Слайд 9

Empirical rule: Application
A company produces batteries with a mean lifetime

of 1’200 hours and a standard deviation of 50 hours.
Find the interval for (what values fall into the following interval?):

DR SUSANNE HANSEN SARAL, SUSANNE.SARAL@OKAN.EDU.TR

Слайд 10

Empirical rule: Application

DR SUSANNE HANSEN SARAL, SUSANNE.SARAL@OKAN.EDU.TR

Слайд 11

Interpretation of the Empirical rule: Lightbulb lifetime example
If the shape of

the distribution is normal, then we can conclude :
That approximately 68% of the batteries will last between 1’150 and 1’250 hours
That approximately 95% of the batteries will last between 1’100 and 1’300 hours and
That 99.7% (almost all batteries) will last between 1’050 and 1’350 hours.
It would be very unusual for a battery to loose it’s energy in ex. 600 hours or 1’600 hours. Such values are possible, but not very likely. Their lifetimes would be considered to be outliers

DR SUSANNE HANSEN SARAL, SUSANNE.SARAL@OKAN.EDU.TR

Слайд 12

Empirical rule exercise

Слайд 13

Class quizz
Empirical rule:
(1) Which shape must the distribution have to

be able to apply the Empirical rule?
(2) Which two parameters give information about the shape of a distribution?
(3) What percent approximately of the values in a normal distribution are within one standard deviation above and below the mean ?

Слайд 14

Introduction to Probabilities
DR SUSANNE HANSEN SARAL

Слайд 15

Probability theory
“Life would be simpler if we knew for certain

what was going to happen in the future”
B. Render, R. Stair, Jr. M. Hanna & T. Hale, Quantitative Analysis for Management, 2015
However, risk and uncertainty is a part of our lives

Слайд 16

Definition of probability

Probability is a numerical measure about the

likelihood that an event will occur.

2/28/2017

Слайд 17

Probability and time
Time
Certainty Uncertainty
Certainty runs over a short

period of time and gradually decreases as the time horizon becomes more distant and uncertain.

Слайд 18

Probability and its measures: 2 basic rules
Rule 1:
Probability is measured

over a range from 1 to 0 ( 0 – 100%)
Probability – the chance that an uncertain event will occur (tossing a coin)

0 ≤ P(A) ≤ 1 For any event A

Certain

uncertainty

Dr Susanne Hansen Saral

RISK

Слайд 19

Probability and its measures 2 basic rules

Certain
uncertainty
.5
1
0
Dr Susanne Hansen Saral

Слайд 20

Probability rule 1 and 2 applied - example
Rule 1:
Probability is

measured over a range
from 1 to 0 ( 0 – 100%)

Слайд 21

Probability and definitions

Random experiment
Sample space
Sample point

Event

2/28/2017

Слайд 22

Random experiment
In statistics a random experiment is a process that

generates two or more possible, well defined outcomes. However, we do not know which of the outcomes will occur next.
Examples: Experimental outcomes:
Tossing a coin Head, tail
Throwing a die 1, 2, 3, 4, 5, 6
The outcome of a football match win – lose - equalize – game cancelled

2/28/2017

Слайд 23

All possible experimental outcomes constitute the sample space

A sample

space (S) of an experiment is a list of all possible outcomes.
The outcomes must be collectively exhaustive and mutually exclusive.

2/28/2017

Слайд 24

Sample space, S - Examples
Random experiment: Flip a coin
Possible outcomes:

Head or tail
The sample space: S= {head, tail}
There are no other possible outcomes, therefore they are collectively exhaustive.
When head occurs, tail cannot occur – meaning the outcomes are mutually exclusive.
The sample points in this example are head and tail.

DR SUSANNE HANSEN SARAL

Слайд 25

Sample space, S - Examples
Outcomes of a statistics course:
The sample

space: S = {AA, BA, BB, CB, CC, DC, DD, FD, FF, VF)}.
There are no other possible outcomes, therefore they are collectively exhaustive.
When one of the outcomes occur, no other outcome can occur, therefore they are mutually exclusive.
The sample points are the individual outcomes of the sample space, S = {AA, BA, BB, CB, CC, DC, DD, FD, FF, VF}.

DR SUSANNE HANSEN SARAL

Слайд 26

Sample space - example
The sample space, S = { Google,

direct, Yahoo, MSN and all other}
Mutually exclusive: When a person visits Google it can not visit Yahoo at the same time
Collectively exhaustive: There are no other possible search engines
Sample points: Google, Direct, Yahoo, MSN, all others

Слайд 27

Event
An individual outcome of a sample space is called a

simple event.
An event is a collection or set of one or more simple events in a sample space.

2/28/2017

Слайд 28

Event: – subset of outcomes of a sample space, S
Random

experiment: Throw a dice (Turkish: zar).
Possible outcomes, sample space, S is: {1, 2, 3, 4, 5, 6}
We can define the event “toss only even numbers”. Let A be the event «toss only
even numbers»:
We use the letter A to denote the event: A: {2, 4, 6}
If the experimental outcome are 2, 4, or 6, we would say that the
event A has occurred.

2/28/2017

Слайд 29

Event : Subset of outcomes of a sample space, S
Random

experiment: Grade marks on an exam
Possible outcomes (Sample space): Numbers between 0 and 100
We can define an event, «achieve an A», as the set of numbers that
lie between 80 and 100. Let A be the event «achieve an A»:
A = (80, 81, 82 …….98, 99,100)
If the outcome is a number between 80 and 100, we would say that the event A has occurred.

2/28/2017

Слайд 30

Events
Intersection of Events – If A and B are two

events in a sample space S, then the intersection, A ∩ B, is the set of all outcomes in S that belong to both A and B. We also call this a joint event.
They are not mutually exclusive since they have values in common

Ch. 3-

(continued)

A∩B

Слайд 31

Union of events
Union of Events – If A and B are

two events in a sample space S, then the union, A U B, is the set of all outcomes in S that belong to at least one of the two events. Therefore the union of A U B occurs if and only if either A or B or both occur.

Ch. 3-

(continued)

The entire shaded area represents
A U B

Слайд 32

Mutually exclusive event
A and B are Mutually Exclusive Events if

they have no basic outcomes in common
i.e., the set A ∩ B is empty, indicating that A ∩ B have no values
in common
Example: Tossing a coin: A is the event of tossing a head. B is the event of tossing a tail. They cannot occur at the same time.

Ch. 3-

(continued)

Слайд 33

Collectively Exhaustive
Events E1, E2, …,Ek are Collectively Exhaustive events if E1

U E2 U….. Ek = S
i.e., the events completely cover the sample space
Example: Tossing a coin - possible events: head and tail
Events, head and tail are collectively exhaustive because they make up the entire
sample space, S

Ch. 3-

Слайд 34

Complement
The Complement of an event A is the set of all

basic outcomes in the sample space that do not belong to A. The complement is denoted
Example: Roll a die
A = (all possible even numbers)
= (all possible uneven numbers

Ch. 3-

(continued)

Слайд 35