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DR SUSANNE HANSEN SARAL
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Empirical rule - Probabilities. Week 5 (1) презентация
Содержание
- 1. Empirical rule - Probabilities. Week 5 (1)
- 2. Interpretation of
- 3. Deviations from the normal distribution - Kurtosis
- 5. Symmetric
- 7. If the data distribution is symmetric/normal, then
- 8. contains about 95% of
- 9. Empirical rule: Application A company
- 10. Empirical rule: Application DR SUSANNE HANSEN SARAL, SUSANNE.SARAL@OKAN.EDU.TR
- 11. Interpretation of the Empirical rule: Lightbulb lifetime
- 12. Empirical rule exercise
- 14. Introduction to Probabilities DR SUSANNE HANSEN SARAL
- 15. Probability theory
- 16. Definition
- 18. Probability and its measures: 2 basic rules
- 19. Probability and its measures
- 20. Probability
- 21. Probability
- 22. Random experiment In statistics
- 23. All possible experimental outcomes
- 24. Sample space, S - Examples
- 25. Sample space, S - Examples
- 27. Event An individual
- 28. Event: – subset of outcomes
- 29. Event :
- 30. Events Intersection of Events – If
- 31. Union of events Union of Events –
- 32. Mutually exclusive event A and B
- 38. Class exercise DR SUSANNE HANSEN SARAL
- 39. Class exercise DR SUSANNE HANSEN SARAL
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:
Слайд 1BBA182 Applied Statistics
Week 5 (1) Empirical rule - Probabilities
DR SUSANNE HANSEN
SARAL
EMAIL: SUSANNE.SARAL@OKAN.EDU.TR
HTTPS://PIAZZA.COM/CLASS/IXRJ5MMOX1U2T8?CID=4#
WWW.KHANACADEMY.ORG
EMAIL: SUSANNE.SARAL@OKAN.EDU.TR
HTTPS://PIAZZA.COM/CLASS/IXRJ5MMOX1U2T8?CID=4#
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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
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.
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 7If 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
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?):
Find the interval for (what values fall into the following interval?):
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
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 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 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
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.
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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)
Probability – the chance that an uncertain event will occur (tossing a coin)
0 ≤ P(A) ≤ 1 For any event A
Certain
uncertainty
.5
1
0
Dr Susanne Hansen Saral
RISK
Слайд 20 Probability rule 1 and 2
applied - example
Rule 1:
Probability is measured over a range
from 1 to 0 ( 0 – 100%)
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
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.
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}.
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.
An event is a collection or set of one or more simple events in a sample space.
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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.
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.
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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.
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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
They are not mutually exclusive since they have values in common
COPYRIGHT © 2013 PEARSON EDUCATION, INC. PUBLISHING AS PRENTICE HALL
Ch. 3-
(continued)
A
B
A∩B
S
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.
COPYRIGHT © 2013 PEARSON EDUCATION, INC. PUBLISHING AS PRENTICE HALL
Ch. 3-
(continued)
A
B
The entire shaded area represents
A U B
S
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.
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.
COPYRIGHT © 2013 PEARSON EDUCATION, INC. PUBLISHING AS PRENTICE HALL
Ch. 3-
(continued)
A
B
S
