Statistics (Recap)Finance & Management Students
Farzad Javidanrad
October 2013
University of Nottingham-Business School
Probabilityโข Some Preliminary Concepts:
Random: Something that happens (occurs) by chance.
Population: A set of all possible outcome of a random experiment or a collection of all members of a specific group under study. This collection makes an space that all possible samples can be derived from. For that reason it is sometimes called sample space.
Sample: Any subset of population (sample space).
In tossing a die:
Random event is the event of appearing any face of the die.
Population (sample space) is the set of .
Sample is any subset of the set above such as or .
61,2,3,4,5,
3 6,4,2
Probabilityโข Two events are mutually exclusive if they cannot happen together.
The occurrence of one of them prevents the occurrence of another. For example, if the baby is a boy it cannot be a girl and vice versa.
โข Two events are independent if occurrence of one of them has no effect on the chance of occurrence of another. For example, the result of rolling a die has no impact on the outcome of flipping a coin. But in the experiment of taking two cards consecutively from a set of 52 cards (if the cards can be chosen equally likely) the chance of getting the second card is affected by the result of the first card.
โข Two events are exhaustive if they include all possible outcomes together. For example, in rolling a die the possibility of having odd numbers or even numbers.
Probabilityโข If event ๐จ can happen in ๐ different ways out of ๐ equally likely
ways, the probability of event ๐จ can be shown as its relative frequency; i.e. :
๐ ๐ด =๐
๐
U: sample space (population)
A: an event (sample)
Aโ: mutually exclusive event with A
A & Aโ are exhaustive collectively
No. of ways that event ๐ดoccurs
Total of equally likely and possible outcomes
๐ด๐ด
๐ดโฒ
U
Probabilityโข As 0 โค ๐ โค ๐ it can be concluded that
0 โค๐
๐โค 1
Or 0 โค ๐(๐ด) โค 1
โข ๐ ๐ด = 0 means that event ๐ด cannot happen and ๐ ๐ด = 1means that the event will happen with certainty.
โข With the definition of ๐ดโฒ as an event of โnon-occurrenceโ of event ๐ด, we can find that:
๐ ๐ดโฒ =๐ โ๐
๐= 1 โ
๐
๐= 1 โ ๐ ๐ด
Or ๐ ๐ด + ๐ ๐ดโฒ = 1
Probability of Multiple Eventsโข If ๐จ and ๐ฉ are not mutually exclusive events so, the probability of
happening one of them (๐จ ๐๐ ๐ฉ) can be calculated as following:
๐ท ๐จ โช ๐ฉ = ๐ท ๐จ + ๐ท ๐ฉ โ ๐ท(๐จ โฉ ๐ฉ)
๐ ๐ด ๐๐ ๐ต ๐ ๐ด ๐๐๐ ๐ต
๐ ๐ด ๐ ๐ต
๐ ๐ด โฉ ๐ต
Probability of Multiple Events
P(A)
P(B)P(C)
๐ ๐ด โฉ ๐ต โฉ ๐ถ
In case, we are dealing with more events:
๐ท ๐จ โช ๐ฉ โช ๐ช = ๐ท ๐จ + ๐ท ๐ฉ + ๐ท ๐ช โ ๐ท ๐จ โฉ ๐ฉ โ ๐ท ๐จ โฉ ๐ช โ๐ท ๐ฉ โฉ ๐ช + ๐ท(๐จ โฉ ๐ฉ โฉ ๐ช)
Probability of Multiple Eventsโข Considering ๐ท ๐จ โช ๐ฉ = ๐ท ๐จ + ๐ท ๐ฉ โ ๐ท(๐จ โฉ ๐ฉ) we can have the
following situations:
1. If ๐จ and ๐ฉ are mutually exclusive events, then :
๐ท ๐จ โฉ ๐ฉ = ๐
2. If ๐จ and ๐ฉ are two independent events, then:
๐ท ๐จ โฉ ๐ฉ = ๐ท(๐จ) ร ๐ท(๐ฉ)
3. If ๐จ and ๐ฉ are dependent events, then:
๐ท ๐จ โฉ ๐ฉ = ๐ท(๐จ) ร ๐ท(๐ฉ ๐จ) = ๐ท(๐ฉ) ร ๐ท(๐จ ๐ฉ)
Where ๐ท(๐จ ๐ฉ) and ๐ท(๐ฉ ๐จ) are conditional probabilities and in the case of ๐ท(๐จ ๐ฉ) means the probability of event ๐ด provided that event ๐ต has already happened.
Probability of Multiple Eventso The probability of picking at random a Heart or a Queen on a single
experiment from a card deck of 52 is:
๐ ๐ป โช ๐ = ๐ ๐ป + ๐ ๐ โ ๐ ๐ป โฉ ๐ =13
52+
4
52โ
1
52=
4
13
o The probability of getting a 1 or a 4 on a single toss of a fair die is:
๐ 1 โช 4 = ๐ 1 + ๐ 4 =1
6+1
6=
1
3As they cannot happen together they are mutually exclusive events and ๐ 1 โฉ 4 = 0.
o The probability of having two heads in the experiment of tossing two fair coins is: (two independent events)
๐ ๐ป โฉ ๐ป =1
2.1
2=
1
4
Probability of Multiple Eventso The probability of picking two ace without returning the first card
into the batch of 52 playing cards, which represents a conditional probability, is:
๐ 1๐ ๐ก ๐๐๐ โฉ 2๐๐ ๐๐๐ = ๐(1๐ ๐ก ๐๐๐) ร ๐(2๐๐ ๐๐๐ 1๐ ๐ก ๐๐๐)
Or can be written with less words involved:
๐ ๐ด1 โฉ ๐ด2 = ๐(๐ด1) ร ๐(๐ด2 ๐ด1) =4
52ร
3
51=
1
221
โข If two events ๐จ and ๐ฉ are independent from each other then:
๐ท(๐จ ๐ฉ) = ๐ท ๐จ ๐๐๐ ๐ท(๐ฉ ๐จ) = ๐ท(๐ฉ)
Random Variable & Probability Distribution
Some Basic Concepts:
โข Variable: A letter (symbol) which represents the elements of a specific set.
โข Random Variable: A variable whose values are randomly appear based on a probability distribution.
โข Probability Distribution: A corresponding rule (function) which corresponds a probability to the values of a random variable.
โข Variables (including random variables) are divided into two general categories:
1) Discrete Variables, and
2) Continuous Variables
Random Variable & Probability Distribution
โข A discrete variable is the variable whose elements (values) can be
corresponded to the values of the natural numbers set or any subset
of that. So, it is possible to put an order and count its elements
(values). The number of elements can be finite or infinite.
โข For a discrete variable it is not possible to define any neighbourhood, whatever small, at any value in its domain. There is a jump from one value to another value.
โข If the elements of the domain of a variable can be corresponded to
the values of the real numbers set or any subset of that, the variable
is called continuous. It is not possible to order and count the
elements of a continuous variable. A variable is continuous if for any
value in its domain a neighbourhood, whatever small, can be defined.
Random Variable & Probability Distribution
โข Probability Distribution: A rule (function) that associates a probability either to all possible elements of a random variable (RV) individually or a set of them in an interval.*
โข For a discrete RV this rule associates a probability to each possible individual outcome. For example, the probability distribution for occurrence of a Head when filliping a fair coin: (Note: ๐๐ = 1)
๐ 0 1
๐(๐ฅ) 0.5 0.5In one trial ๐ป, ๐
๐ 0 1 2
๐(๐ฅ) 0.25 0.5 0.25
In two trials ๐ป๐ป,๐ป๐, ๐๐ป, ๐๐
๐ = ๐ท๐๐๐๐ (+1) --- (0) (-1)
๐(๐ฅ) 0.6 0.1 0.3
Change in the price of a share in one day
o The probability distribution for the price change of a share in stock market
Probability Distributions (Continuous)โข The probability that a continuous random variable chooses
just one of its values in its domain is zero, because the number of all possible outcomes ๐ is infinite and
๐
โโ ๐.
โข For the above reason, the probability of a continuous random variable need to be calculated in an interval.
โข The probability distribution of a continuous random variable is often called a probability density function (PDF) or simply probability function and it is usually shown by ๐(๐) and it has following properties:
I. ๐(๐ฅ) โฅ 0 (similar to ๐ท(๐) โฅ ๐ for discrete RV*)
II. โโ
+โ๐ ๐ฅ ๐๐ฅ = 1 (similar to ๐ท ๐ = ๐ for discrete RV)
III. ๐๐๐ ๐ฅ ๐๐ฅ = ๐ ๐ โค ๐ฅ โค ๐ = ๐น ๐ โ ๐น ๐ (probability
given to set of values in an interval [a,b] )**
Probability Distributions (Continuous)โข where ๐น(๐ฅ) is the integral of the PDF function (๐(๐ฅ)) and it is
called as Cumulative Distribution Function (CDF) and for any real value of ๐ is defined as:
๐น(๐ฅ) โก ๐(๐ โค ๐ฅ)
CDF shows the area under PDF function (๐(๐ฑ)) from โโ to ๐ฑ . For discrete random variable, CDF shows the summation of all probabilities before the value of ๐ฑ .
Adopted from http://beyondbitsandatomsblog.stanford.edu/spring2010/tag/embodied-artifacts/
๐น(๐ฅ)
๐(๐ฅ)
๐น(๐ฅ)โก๐(๐โค๐ฅ)
๐น(๐ฅ)โก๐(๐โค๐ฅ)
Some Characteristics of Probability Distributions
โข Expected Value (Probabilistic Mean Value): It is one of the most important measures which shows the central tendency of the distribution. It is the weighted average of all possible values of random variable ๐ and it is shown by ๐ฌ(๐).
โข For a discreet RV (with n possible outcomes)
๐ฌ ๐ = ๐๐๐ท ๐๐ + ๐๐๐ท ๐๐ +โฏ+ ๐๐๐ท ๐๐ =
๐=๐
๐
๐๐๐ท(๐๐)
โข For a continuous RV
๐ฌ ๐ =
โโ
+โ
๐. ๐ ๐ ๐ ๐
Some Characteristics of Probability Distributions
โข Properties of ๐ฌ(๐):
i. If ๐ is a constant then ๐ฌ ๐ = ๐ .
ii. If ๐ and ๐ are constants then ๐ฌ ๐๐ + ๐ = ๐๐ฌ ๐ + ๐ .
iii. If ๐๐, โฆ , ๐๐ are constants then
๐ฌ ๐๐๐๐ +โฏ+ ๐๐๐๐ = ๐๐๐ฌ ๐๐ +โฏ+ ๐๐๐ฌ(๐๐)
Or
๐ฌ(
๐=๐
๐
๐๐๐๐) =
๐=๐
๐
๐๐๐ฌ(๐๐)
iv. If ๐ and ๐ are independent random variables then
๐ฌ ๐๐ = ๐ฌ ๐ . ๐ฌ ๐
Some Characteristics of Probability Distributions
v. If ๐ ๐ is a function of random variable ๐ then
๐ฌ ๐ ๐ = ๐ ๐ .๐ท(๐)
๐ฌ ๐ ๐ = ๐ ๐ . ๐ ๐ ๐ ๐
โข Variance: To measure how random variable ๐ is dispersed around its expected value, variance can help. If we show ๐ฌ ๐ = ๐ , then
๐๐๐ ๐ = ๐๐ = ๐ฌ[ ๐ โ ๐ฌ ๐๐]
= ๐ฌ[ ๐ โ ๐ ๐]
= ๐ฌ[๐๐ โ ๐๐๐ + ๐๐]
= ๐ฌ ๐๐ โ ๐๐๐ฌ ๐ + ๐๐
= ๐ฌ ๐๐ โ ๐๐
For discreet RV
For continuous RV
Some Characteristics of Probability Distributions
๐๐๐ ๐ =
๐=๐
๐
๐๐ โ ๐ ๐. ๐ท(๐)
๐๐๐ ๐ = โโ+โ
๐๐ โ ๐ ๐. ๐ ๐ ๐ ๐
โข Properties of Variance:
i. if ๐ is a constant then ๐๐๐ ๐ = ๐ .
ii. If ๐ and ๐ are constants then ๐๐๐ ๐๐ + ๐ = ๐๐๐๐๐(๐) .
iii. If ๐ and ๐ are independent random variables then
๐๐๐ ๐ ยฑ ๐ = ๐๐๐ ๐ + ๐๐๐(๐)
can be extended to more variables
For discreet RV
For continuous RV
โข Some of the well-known probability distributions are:
โข The Binomial Distribution:
1. The probability of the occurrence of an event is ๐ and is not changing.
2. The experiment is repeated for ๐ times.
3. The probability that out of ๐ times, the event appears ๐ times is:
๐ ๐ฅ =๐!
๐ฅ! ๐ โ ๐ฅ !๐๐ฅ(1 โ ๐)๐โ๐ฅ
The mean value and standard deviation of the binomial distribution are:
๐ = ๐=0๐ ๐ฅ๐ . ๐ ๐ฅ๐ =๐๐ ๐ = ๐=0
๐ ๐ฅ๐ โ ๐ 2. ๐(๐ฅ๐) = ๐๐(1 โ ๐)
So, to show that the probability distribution of the random variable ๐is binomial we can write: ๐~๐ต๐(๐๐, ๐๐ 1 โ ๐ )
Probability Distributions (Discrete RV)
Probability Distributions (Discrete RV)โข A gambler thinks his chance to get a 1 in rolling a die is high. What
is his chance to have 4 one out of six experiments using a fair die?
The probability of having a one in an individual trial is 1
6and it
remains the same in all 6 experiments. So,
๐ ๐ฅ = 4 =6!
4! 2!
1
6
45
6
2
=375
7776= 0.048 โ 5%
โข The Poisson Distribution:
1. It is used to calculate the probability of number of desired event (no. of successes)in a specific period of time.
2. The average number of desired event (no. of successes) per unit of time remains constant.
โข So, the probability of having ๐ numbers of success is calculated by:
๐ ๐ฅ =๐๐ฅ๐โ๐
๐ฅ!
Where ๐ is the average number of successes in a specific period of time and ๐ = 2.7182 .
โข The mean value and standard deviation of the Poisson distribution are:
๐ =
๐=0
๐
๐ฅ๐ . ๐ ๐ฅ๐ =๐ and ๐ =
๐=0
๐
๐ฅ๐ โ ๐ 2. ๐(๐ฅ๐) = ๐
So, to show that the probability distribution of the random variable ๐ is Poisson we can write: ๐ฟ~Poi(๐, ๐).
o The emergency section in a hospital receives 2 calls per half an hour (4 calls in an hour). The probability of getting just 2 calls in a randomly chosen hour in a random day is:
๐ ๐ฅ = 2 =42๐โ4
2!= 0.146 โ 15%
Probability Distributions (Discrete RV)
The Normal Distribution (Continuous RV)โข The Normal Distribution: It is the best known probability
distribution which reflects the nature of most random variables in the world. The probability density function (PDF) of normal distribution is:
1. Symmetrical around its mean value (๐).
2. Bell-shaped, with two tails approaching the horizontal axis asymptotically as we move further away from the mean.
Adopted from http://www.pdnotebook.com/2010/06/statistical-tolerance-analysis-root-sum-square/
The Normal Distribution (Continuous RV)3. The probability density function (PDF) of normal distribution
can be represented by:
๐ ๐ =๐
๐ ๐๐ ๐โ
๐โ๐ ๐
๐๐๐ (โโ < ๐ < +โ)
Where ๐ and ๐ are mean and standard deviation respectively.
๐ = โโ+โ
๐. ๐ ๐ ๐ ๐ and ๐ = โโ+โ
๐ โ ๐ ๐ . ๐ ๐ ๐ ๐
So, ๐ฟ~๐ต(๐, ๐๐).
โข A linear combination of independent normally distributed random variables is itself normally distributed, that is,
If ๐ฟ~๐ต ๐๐, ๐๐๐ and ๐~๐ต ๐๐, ๐๐
๐ and if ๐ = ๐๐ฟ + ๐๐ then
๐~๐ต(๐๐๐ + ๐๐๐ , ๐๐๐๐
๐ + ๐๐๐๐๐)
โข This can be extended to more than two random variables.
The Normal Distribution (Continuous RV)โข Recalling the last property of PDF (
๐
๐๐ ๐ฅ ๐๐ฅ = ๐(๐ โค ๐ฅ โค ๐)), it is
difficult to calculate the probability using the above PDF with different values of ๐ and ๐. The solution for this problem is to transform the normal variable ๐ to the standardised normal variable (or simply, standard normal
variable) random variable ๐ , by: ๐ =๐โ๐
๐
which its parameters (๐ and ๐2) are independent from the influence of other random variablesโ parameters with normal distribution because we always have:๐ฌ ๐ = ๐ and ๐๐๐ ๐ = ๐ (why?)
โข The probability distribution for the standard normal variable is defined as:
๐ ๐ =๐
๐๐ ๐โ
๐๐
๐ ๐~๐ต(๐, ๐).
Standardised
Adopted and amended from http://www.mathsisfun.com/data/standard-normal-distribution.html
๐ฟ~๐ต(๐, ๐๐) ๐~๐ต(๐, ๐)
The Standard Normal Distribution
0
โข Properties of the standard normal distribution curve:
1. It is symmetrical around y-axis.
2. The area under the curve can be split into two equal areas, that is:
โโ
0
๐ ๐ง ๐๐ง =
0
+โ
๐ ๐ง ๐๐ง = 0.5
โข To find the area under the curve and before ๐๐ = ๐. ๐๐ , using the z-table (next slide), we have:
๐ ๐ง โค ๐ง1 = 1.26 =
โโ
0
๐ ๐ง ๐๐ง +
0
๐ง1
๐ ๐ง ๐๐ง =0.5 + 0.3962 = 0.8962 โ 90%
๐(๐ง)
50%
๐ง
50% 50%
๐๐ = ๐. ๐๐
0.5
0.3
96
2
Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224
0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549
0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852
0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133
0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621
1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830
1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015
1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177
1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319
1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441
1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545
1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633
1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706
1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857
2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890
2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916
2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936
2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952
2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964
2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974
2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981
2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990
Working with the Z-Tableโข To find the probability
๐ 0.89 < ๐ง < 1.5 =
0
๐ง2
๐(๐ง)๐๐ง โ
0
๐ง1
๐ ๐ง ๐๐ง
= ๐น 1.5 โ ๐น 0.89 = 0.4332 โ 0.3133
= 0.119 โ 12%
as both values are positive.
โข To find the probability in the negative area we
need to find the equivalent area in the positive side:
๐ โ1.32 < ๐ง < โ1.25 = ๐ 1.25 < ๐ง < 1.32
= ๐น 1.32 โ ๐น 1.25
= 0.4066 โ 0.3944 = 0.0122 โ 1%
1.50.89
Working with the Z-Tableโข To find ๐(โ2.15 < ๐ง) we can write:
โโ
โ2.15
๐. ๐๐ง =
โโ
0
๐. ๐๐ง โ
โ2.15
0
๐. ๐๐ง
= 0.5 โ 0.4842 = 0.0158 โ 2%
โข And finally, to find ๐(๐ง โฅ 1.93) , we have:
1.93
+โ
๐. ๐๐ง =
0
+โ
๐. ๐๐ง โ
0
1.93
๐. ๐๐ง
= 0.5 โ 0.4732 = 0.0268
0-2.15 =โก
0
2.15
๐. ๐๐ง
0 =1.93
An Exampleo If the income of employees in a big company normally distributed
with ๐ = ยฃ๐๐๐๐๐ and ๐ = ยฃ๐๐๐๐, what is the probability of an employee picked randomly have an income
a) above ยฃ22000, b) between ยฃ16000 and ยฃ24000.
a) We need to transform ๐ to ๐ firstly:
๐ ๐ฅ > 22000 = ๐๐ฅ โ 20000
4000>
22000 โ 20000
4000
= ๐ ๐ง > 0.5 = 0.5 โ 01915 = 0.3085 โ 31%
b) ๐ 16000 < ๐ฅ < 24000 = ๐(16000โ20000
4000<
๐ฅโ20000
4000<
24000โ20000
4000)
= ๐ โ1 < ๐ง < 1
= 0.3413 + 0.3413
= 0.6826 โ 68%
The 2(Chi-Squared)Distributionโข The ๐(Chi-Squared)Distribution:
Let ๐๐, ๐๐, โฆ , ๐๐be ๐ independent standardised normal distributed random variables, then the sum of the squares of them
๐ =
๐=1
๐
๐๐2
have a Chi-Square distribution with a degree of freedom equal to the number of random variables (๐ ๐ = ๐). So, ๐ฟ~ .
The mean value and standard
deviation of the RV with a Chi-Squared
distribution are ๐ ๐๐๐ ๐๐
Respectively. So we can write:
๐ฟ~
2
k
Probability Density Function (PDF) of 2 Distribution
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๐ ๐ฅ2 = 32 ๐๐ = 16 = 0.01 or ๐ฅ20.01 ,16 = 32
The t-Distributionโข If ๐~๐ต ๐, ๐ and ๐ฟ~ and two random variables
๐ and ๐ฟ are independent then the random variable
๐ =๐
๐ฟ ๐
=๐. ๐
๐ฟ
follows studentโs t-distribution (t-distribution) with ๐ degree of freedom. For a sample size ๐ we have ๐ ๐ = ๐ = ๐ โ ๐.
โข The mean value and standard deviation of this distribution are
๐ = ๐ ๐ > ๐
๐๐๐ ๐๐๐๐๐๐ ๐ = ๐, ๐๐ =
๐โ๐
๐โ๐๐ > ๐
โ ๐ = ๐๐๐๐ ๐๐๐๐๐๐ ๐ = ๐, ๐
)2,(2 kkk
The t-Distributionโข The t-distribution like the standard normal distribution is a bell-
shaped and symmetrical distribution with zero mean (n>2) but it is flatter but as the degree of freedom increases (or ๐ increases)it approaches the standard normal distribution and for ๐โฅ๐๐ their behaviours are similar.
โข From the table (next slide)
๐ ๐ก = 1.706 ๐๐ =26 = 0.05 โ 5% or ๐ก0.05,26 = 1.706
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= ๐. ๐๐๐
5%
df 0.20 0.15 0.10 0.05 0.025 0.01 0.005 0.0025 0.001 0.0005
1 1.376 1.963 3.078 6.314 12.706 31.821 63.656 127.321 318.289 636.578
2 1.061 1.386 1.886 2.920 4.303 6.965 9.925 14.089 22.328 31.600
3 0.978 1.250 1.638 2.353 3.182 4.541 5.841 7.453 10.214 12.924
4 0.941 1.190 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.610
5 0.920 1.156 1.476 2.015 2.571 3.365 4.032 4.773 5.894 6.869
6 0.906 1.134 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5.959
7 0.896 1.119 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.408
8 0.889 1.108 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5.041
9 0.883 1.100 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.781
10 0.879 1.093 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.587
11 0.876 1.088 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.437
12 0.873 1.083 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4.318
13 0.870 1.079 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4.221
14 0.868 1.076 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4.140
15 0.866 1.074 1.341 1.753 2.131 2.602 2.947 3.286 3.733 4.073
16 0.865 1.071 1.337 1.746 2.120 2.583 2.921 3.252 3.686 4.015
17 0.863 1.069 1.333 1.740 2.110 2.567 2.898 3.222 3.646 3.965
18 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.197 3.610 3.922
19 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.174 3.579 3.883
20 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.153 3.552 3.850
21 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.135 3.527 3.819
22 0.858 1.061 1.321 1.717 2.074 2.508 2.819 3.119 3.505 3.792
23 0.858 1.060 1.319 1.714 2.069 2.500 2.807 3.104 3.485 3.768
24 0.857 1.059 1.318 1.711 2.064 2.492 2.797 3.091 3.467 3.745
25 0.856 1.058 1.316 1.708 2.060 2.485 2.787 3.078 3.450 3.725
26 0.856 1.058 1.315 1.706 2.056 2.479 2.779 3.067 3.435 3.707
27 0.855 1.057 1.314 1.703 2.052 2.473 2.771 3.057 3.421 3.689
28 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.047 3.408 3.674
29 0.854 1.055 1.311 1.699 2.045 2.462 2.756 3.038 3.396 3.660
30 0.854 1.055 1.310 1.697 2.042 2.457 2.750 3.030 3.385 3.646
31 0.853 1.054 1.309 1.696 2.040 2.453 2.744 3.022 3.375 3.633
32 0.853 1.054 1.309 1.694 2.037 2.449 2.738 3.015 3.365 3.622
33 0.853 1.053 1.308 1.692 2.035 2.445 2.733 3.008 3.356 3.611
34 0.852 1.052 1.307 1.691 2.032 2.441 2.728 3.002 3.348 3.601
35 0.852 1.052 1.306 1.690 2.030 2.438 2.724 2.996 3.340 3.591
36 0.852 1.052 1.306 1.688 2.028 2.434 2.719 2.990 3.333 3.582
37 0.851 1.051 1.305 1.687 2.026 2.431 2.715 2.985 3.326 3.574
38 0.851 1.051 1.304 1.686 2.024 2.429 2.712 2.980 3.319 3.566
39 0.851 1.050 1.304 1.685 2.023 2.426 2.708 2.976 3.313 3.558
40 0.851 1.050 1.303 1.684 2.021 2.423 2.704 2.971 3.307 3.551
50 0.849 1.047 1.299 1.676 2.009 2.403 2.678 2.937 3.261 3.496
60 0.848 1.045 1.296 1.671 2.000 2.390 2.660 2.915 3.232 3.460
80 0.846 1.043 1.292 1.664 1.990 2.374 2.639 2.887 3.195 3.416
100 0.845 1.042 1.290 1.660 1.984 2.364 2.626 2.871 3.174 3.390
150 0.844 1.040 1.287 1.655 1.976 2.351 2.609 2.849 3.145 3.357
Infinity 0.842 1.036 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.290
The F Distributionโข If ๐1~ and ๐2~ and ๐1 and ๐2 are independent then the
random variable
๐น =
๐1๐1
๐2
๐2
follows F distribution with ๐1 and ๐2 degrees of freedom, i.e.:
๐น~๐น๐1,๐2 or ๐น~๐น(๐1, ๐2)
โข This distribution is skewed to
the right as the Chi-Square
distribution but as ๐1 and ๐2increase (๐ โ โ) it approaches
to normal distribution.
2
2k2
1k
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The F Distributionโข The mean and standard deviation of the F distribution are:
๐ =๐2
๐2โ2๐๐๐ (๐2 > 2) and
๐ =๐2
๐2โ2
2(๐1+๐2โ2)
๐1(๐2โ4)๐๐๐ (๐2 > 4)
โข Relation between t & Chi-Square Distributions with F distribution:
โข For a random variable ๐~๐ก๐it can be shown that ๐2~๐น1,๐. This can also be written as
๐ก๐2 = ๐น1,๐
โข If ๐2 is large enough, then ๐1. ๐น๐1,๐2~
2
1k
๐ผ = 0.25All adopted from http://www.stat.purdue.edu/~yuzhu/stat514s05/tables.html
๐ผ = 0.10
๐ผ = 0.05
๐ผ = 0.025
๐ผ = 0.01
Statistical Inference (Estimation)โข Statistical inference or statistical induction is one of the most
important aspect of decision making and it refers to the process of drawing a conclusion about the unknown parameters of the population from a sample of randomly chosen data.
โข So, the idea is that a sample of randomly chosen data provides the best information about parameters of the population and it can be considered as a representative of the population when its size reasonably (appropriately) large.
โข The first step in statistical inference (induction) is estimation which is the process of finding an estimate or approximation for the population parameters (such as mean value and standard deviation) using the data in the sample.
Statistical Inference (Estimation)โข The value of ๐ฟ (sample mean) in a randomly chosen and
appropriately large sample is a good estimator of the population mean ๐ . The value of ๐๐(sample variance) is also a good estimator of the population variance ๐๐.
โข Before taking any sample from population (when the sample is not realised or observed) we can talk about the probability distribution of a hypothetical sample. The probability distribution of a random variable ๐ in a hypothetical sample follows the probability distribution of the population even if the sampling process is repeated for many times.
โข But the probability distribution of the sample mean ๐ฟ in repeated sampling does not necessarily follow the probability distribution of its population when number of sampling increases.
Central Limit Theoremโข Central Limit Theorem:
Imagine random variable ๐ฟ with any probability distribution is defined in a population with the mean ๐ and the variance ๐๐. If we get ๐ independent samples ๐ฟ๐, ๐ฟ๐, โฆ , ๐ฟ๐ and for each sample we
calculate the mean values ๐ฟ๐, ๐ฟ๐, โฆ , ๐ฟ๐(see figure below)
๐ฟ~๐. ๐. ๐ (๐, ๐๐)
๐ฟ๐
๐ฟ๐
โฎ
๐ฟ๐
๐. ๐. ๐ โกIndependent & Identically Distributed RVs
Central Limit TheoremAs the number of sampling increases infinitely, the random variable ๐ฟhas a normal distribution (regardless of the population distribution) and we have
๐ฟ~๐ต ๐,๐๐
๐when ๐ โ +โ
And in the standard form:
๐ = ๐ฟ โ ๐ ๐ฟ
๐ ๐ฟ=
๐ฟ โ ๐๐
๐
=๐ ๐ฟ โ ๐
๐~๐ต(๐, ๐)
o Taking sample of 36 elements from a population with the mean of 20 and standard deviation of 12, what is the probability that the sample mean falls between 18 and 24?
๐ 18 < ๐ฅ < 24 = ๐ โ1 < ๐ฅ โ 20
12
36
< 2 = 0.3413 + 0.4772 โ 82%
Estimationโข In previous slides we introduced some of the most important
probability distributions for discrete & continuous random variables.
โข In many cases we know the nature of the probability distribution of a random variable, defined in a population, but have no idea about its parameters such as mean value or/and standard deviation.
โข Point Estimation:
โข To estimate the unknown parameters of a probability distribution of a random variable we can either have a point estimation or an interval estimation using an estimator.
โข The estimator is a function of the sample values ๐๐, ๐๐, โฆ , ๐๐ and it
is often called a statistic. If ๐ฝ represent that estimator we have: ๐ฝ = ๐(๐๐, ๐๐, โฆ , ๐๐)
Estimationโข ๐ฝ is said to be an unbiased estimator of true ๐ฝ (parameter of the
population) if ๐ฌ ๐ฝ = ๐ฝ. Because the bias itself is defined as
๐ฉ๐๐๐ = ๐ฌ ๐ฝ โ ๐ฝ
o For example, the sample mean ๐ฟ is a point and unbiased estimator for the unknown parameter ๐ (population mean):
๐ฝ = ๐ฟ = ๐ ๐๐, ๐๐, โฆ , ๐๐ =๐
๐๐๐ + ๐๐ +โฏ+ ๐๐
It is unbiased because ๐ฌ ๐ฟ = ๐.
โข The sample variance in the form of ๐๐ = ๐๐โ ๐
๐
๐is a point but a
biased estimator of the population variance ๐๐ in a small sample:
๐ฌ ๐๐ = ๐๐(๐ โ๐
๐) โ ๐๐
But it is a consistent estimator because it will approaches to ๐๐when the sample size ๐ increases indefinitely (๐ โ โ)
โข With Besselโs correction (changing ๐ to (๐ โ ๐)) we can define another sample variance which is unbiased even for small sample size.
๐๐ = ๐๐ โ ๐ ๐
๐ โ ๐
โข The methods of finding point estimators are mostly least-square method and maximum likelihood method which among them the first method will be discussed later.
Estimation
Interval Estimationโข Interval Estimation:
โข Interval estimation, in contrary, provides an interval or a range of possible estimates at a specific level of probability, which is called level of confidence, within which the true value of the population parameter may lie.
โข If ๐ฝ๐ and ๐ฝ๐ are respectively the lowest and highest estimates of ๐ฝ
the probability that ๐ฝ is covered by the interval ๐ฝ๐, ๐ฝ๐ is:
๐๐ซ ๐ฝ๐ โค ๐ฝ โค ๐ฝ๐ = ๐ โ ๐ถ (0 < ๐ผ < 1)
Where ๐ โ ๐ถ is the level of confidence and ๐ถ itself is called level of
significance. The interval ๐ฝ๐, ๐ฝ๐ is called confidence interval.
Interval Estimation How to find ๐ฝ๐ ๐๐๐ ๐ฝ๐? In order to find the lower and upper limits of a confidence interval we need to have a prior knowledge about the nature of distribution of the random variable in the population. If random variable ๐ is normally distributed in the population and the
population standard deviation (๐) is known, the 95% confidence interval for the unknown population mean (๐) can be constructed by finding the symmetric z-values associated to 95% area under the standard normal curve:
๐ โ ๐ถ = ๐๐% โ ๐ถ = ๐% โ๐ถ
๐= ๐. ๐%
So, ยฑ๐๐.๐๐๐ = ยฑ๐. ๐๐
We know that: ๐ = ๐ฟโ๐ ๐ฟ
๐ ๐ฟ=
๐ฟโ๐๐
๐
, so:
๐ท(โ๐ ๐ถ ๐โค ๐ โค ๐ ๐ถ ๐
) = ๐๐%
Adopted & altered from http://upload.wikimedia.org/wikipedia/en/b/bf/NormalDist1.96.png
=1โ๐ผ
๐ถ
๐= ๐. ๐๐๐
๐ถ
๐= ๐. ๐๐๐
โ๐ ๐ถ ๐= = ๐ ๐ถ ๐
Interval Estimationโข So we can write:
๐ท ๐ โ ๐. ๐๐๐ ๐ โค ๐ โค ๐ + ๐. ๐๐๐ ๐ = ๐. ๐๐
Or
๐ท ๐ โ ๐. ๐๐๐
๐โค ๐ โค ๐ + ๐. ๐๐
๐
๐= ๐. ๐๐
Therefore, the interval ๐ โ ๐. ๐๐๐
๐, ๐ + ๐. ๐๐
๐
๐represents a 95%
confidence interval (๐ถ๐ผ95%)of the unknown value of ๐.
It means in repeated random sampling (for 100 times) we
expect 95 out of 100 intervals, such as the above, cover the
unknown value of the population mean ๐ .
๐ ฬ โ๐.๐๐ ๐/โ๐ = = ๐ ฬ โ๐.๐๐ ๐/โ๐Adopted and altered from http://forums.anarchy-online.com/showthread.php?t=604728
Interval Estimation for population Proportion
A confidence interval can be constructed for the population proportion (see the graph below)
๐~๐ต๐(๐๐, ๐๐ 1 โ ๐ )
๐๐
๐ ๐2
๐๐
โฎ
๐๐
๐ ๐ = ๐ฌ ๐ = ๐ =๐
๐
๐๐ ๐= ๐๐๐ ๐ =
๐๐
๐๐=
๐(๐ โ ๐)
๐
๐ in each sample represents a
sample proportion. In repeated random sampling ๐ has its own probability distribution with mean value and
variance
Interval Estimation for population Proportionโข The 90% confidence interval for the population proportion ๐ when
sample size is bigger than 30 (n>30) and there is no information about the population variance will be constructed as following:
ยฑ๐ ๐ถ ๐=
๐ โ ๐
๐(๐ โ ๐)๐
๐ท(โ๐ ๐ถ ๐โค ๐ โค +๐ ๐ถ ๐
) = ๐ โ ๐ถ
๐ท( ๐ โ ๐ ๐ถ ๐.
๐(๐โ ๐)
๐โค ๐ โค ๐+๐ ๐ถ ๐
. ๐(๐โ ๐)
๐) = ๐. ๐
So, the confidence interval can be simply written as:
๐ช๐ฐ๐๐% = ๐ โ ๐. ๐๐๐ ๐(๐ โ ๐)
๐ =90% ๐ถ ๐ = ๐. ๐๐ ๐ถ ๐ = ๐. ๐๐
โ๐ ๐ถ ๐= โ๐. ๐๐๐ ๐ ๐ถ ๐
= ๐. ๐๐๐
Obviously, if we had knowledge about the
population variance we were be able to estimate
the population proportion ๐ directly.
Why?
Adopted and altered fromhttp://www.stat.wmich.edu/s216/book/node83.html
Exampleso Imagine the weight of people in a society distributed normally. A
random sample of 25 with the sample mean 72 kg is taken from this society. If the standard deviation of the population is 6 kg find a)the 90% b)95% and c) 99% confidence interval for the unknown population mean.
a) 1 โ ๐ผ = 0.9 โ๐ผ
2= 0.05 โ ๐ ๐ผ 2
= 1.645
So, ๐ถ๐ผ90% = 72 ยฑ 1.645 ร6
25= 70.03 , 73.97
b) 1 โ ๐ผ = 0.95 โ๐ผ
2= 0.025 โ ๐ ๐ผ 2
= 1.96
So, ๐ถ๐ผ95% = 72 ยฑ 1.96 ร6
25= 69.65 , 74.35
c) 1 โ ๐ผ = 0.99 โ๐ผ
2= 0.005 โ ๐ ๐ผ 2
= 2.58
So, ๐ถ๐ผ99% = 72 ยฑ 2.58 ร6
25= 68.9 , 75.1
Exampleso Samples from one of the lines of production in a factory suggests
that 10% of products are defective. If the range of 1% difference between sample and population proportion is acceptable what sample size we need to construct a 95% confidence interval for the population proportion? What about if the acceptable gap between sample & population proportion increased to 3%?
1 โ ๐ผ = 0.95 โ๐ผ
2= 0.025 โ ๐ ๐ผ 2
= 1.96
๐ ๐ผ 2=
๐ โ ๐
๐(1 โ ๐)๐
โ 1.96 =0.01
0.1 ร 0.9๐
โ ๐ = 196 ร 0.3 2 โ 3458
If the gap increases to 3% then:
1.96 =0.03
0.1ร0.9
๐
โ ๐ = 196 ร 0.1 2 โ 385
Interval Estimation (Using t-distribution) โข If the population standard deviation ๐ is unknown and we use
sample standard deviation ๐ instead, and the size of the sample is less than 30 (๐ < ๐๐) then the random variable
๐ โ ๐๐
๐
~๐๐โ๐
has t-distribution with ๐ ๐ = ๐ โ ๐.
This means a confidence interval for the population mean ๐ will be in the form of:
๐ช๐ฐ(๐โ๐ถ) = ๐ โ ๐ ๐ถ ๐,๐โ๐
๐
๐, ๐ + ๐ ๐ถ ๐,๐โ๐
๐
๐
โ๐ ๐ถ๐,๐โ๐
๐ ๐ถ๐,๐โ๐
1 โ ๐ผ % ๐ถ
๐
๐ถ
๐
Adopted and altered from http://cnx.org/content/m46278/latest/?collection=col11521/latest
Interval Estimationโข The following flowchart can help to choose between Z and t-
distributions when the interval estimation is constructed for ๐ in the population.
Use nonparametric
methods
Adopted from http://www.expertsmind.com/questions/flow-chart-for-confidence-interval-30112489.aspx
Interval Estimationโข Here there is a list of confidence intervals for the subject parameters
in the population.
Adopted from http://www.bls-stats.org/uploads/1/7/6/7/1767713/250709.image0.jpg
Hypothesis Testing โข Hypothesis testing is one of the important aspects of statistical inference.
The main idea is to find out if some claims/statements (in the form of hypothesis) about population parameters can be statistically rejected by the evidence from the sample using a test statistic (a function of sample).
โข Claims can be made in the form of null hypothesis (๐ป0) against the alternative hypothesis (๐ป1) and they are just rejectable. These two hypotheses should be mutually exclusive and collectively exhaustive. For example:
๐ป0: ๐ = 0.8 ๐๐๐๐๐๐ ๐ก ๐ป1: ๐ โ 0.8
๐ป0: ๐ โฅ 2.1 ๐๐๐๐๐๐ ๐ก ๐ป1: ๐ < 2.1
๐ป0: ๐2 โค 0.4 ๐๐๐๐๐๐ ๐ก ๐ป1: ๐
2 > 0.4
Always remember that the equality sign comes with ๐ป0.
โข If the value of the test statistic lies in the rejection area(s) the null hypothesis must be rejected, otherwise the sample does not provide sufficient evidence to reject the null hypothesis.
Hypothesis Testing โข Assuming we know the distribution of the random variable in the
population and also having statistical independence between different random variables, in hypothesis testing we need to follow the following steps:
1. Stating the relevant null & alternative hypotheses. The state of the null hypothesis (being =,โฅ,โค something)indicates how many rejection regions we will have (for = sign we will have two regions and for others just one region; depending on the difference between the value of estimator and the claimed value for the population parameter the rejection area could be on the right or left of the distribution curve).
๐ป0: ๐ = 0.5
๐ป1: ๐ โ 0.5
๐ป0: ๐ โฅ 0.5 (๐๐ ๐ โค 0.5)
๐ป1: ๐ < 0.5 (๐๐ ๐ > 0.5)Graphs Adopted from http://www.soc.napier.ac.uk/~cs181/Modules/CM/Statistics/Statistics%203.html
Hypothesis Testing 2. Identifying the level of significance of the test (๐ถ) and it is usually
considered to be 5% or 1%, depending on the nature of the test and the goals of researcher. When ๐ถ is known with the prior knowledge about the sample distribution, the critical region(s) (or rejection area(s)) can be identified.
Here we have two critical values for standard normal
distributions associated to the level
of significance ๐ผ =5% and ๐ผ = 1%
Adopted from http://www.psychstat.missouristate.edu/introbook/sbk26.htm
๐๐ผ=1.65
๐๐ผ=2.33
Hypothesis Testing 3. Constructing a test statistic (a function based on the sample distribution &
sample size). This function is used to decide whther or not to reject ๐ฏ๐.
Table
Ad
op
ted
from
http
://ww
w.b
ls-stats.org/u
plo
ads/1/7/6/7/1
76771
3/250714.im
age0.jp
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Here we have a list of some of
the test statistics
for testing different
hypotheses
Hypothesis Testing 4. Taking a random sample from the population and calculating the value of the test statistic. If the value is in the rejection area the null hypothesis ๐ฏ๐
will be rejected in favour of the alternative ๐ฏ๐at the predetermined significance level ๐ถ, otherwise the sample does not provide sufficient evidence to reject ๐ฏ๐ (this does not mean that we accept ๐ฏ๐)
Adopted from http://www.onekobo.com/Articles/Statistics/03-Hypotheses/Stats3%20-%2010%20-%20Rejection%20Region.htm
โ๐๐ถ ๐๐ โ ๐๐ถ,๐ ๐ if there is a left-tail test
โ๐๐ถ
๐๐๐ โ ๐๐ถ
๐,๐ ๐ if there is a two-tail test
+๐๐ถ ๐๐ + ๐๐ถ,๐ ๐ if there is a right-tail test
+๐๐ถ
๐๐๐ + ๐๐ถ
๐,๐ ๐ if there is a two-tail test
Exampleo A chocolate factory claims that its new tin of cocoa powder contains at
least 500 gr of the powder. A standard checking agency takes a random sample of ๐ = 25 of the tins and found out that sample mean weight of tins is ๐ = 520 ๐๐ and the sample standard deviation is ๐ = 75 ๐๐. If we assume the weight of cocoa powder in tins has a normal distribution, does the sample provide enough evidence to support the claim at 95% level of confidence?
1. ๐ป0: ๐ โฅ 500
๐ป1: ๐ < 500 (so, it is a one-tail test)
2. Level of significance ๐ผ = 5% โ ๐ก๐ผ2,(๐โ1) = ๐ก0.05,24 = 1.711 (it is t-
distribution because ๐ < 30 and we do not have a prior knowledge about the population standard deviation)
3. The value of the test statistics is : ๐ก =๐โ๐
๐
๐
=520โ500
75
25
= 1.33
4. As 1.33 < 1.711 we are not in the rejection area so, the claim cannot be rejected at 5% level of significance.
Type I & Type II Errorsโข Two types of errors can occur in hypothesis testing:
A. Type I error; when based on our sample we reject a true null hypothesis.
B. Type II error; when based on our sample we cannot reject a false null hypothesis.
โข By reducing the level of significance ๐ถ we can reduce the probability of making type I error (why?) however, at the same time, we increase the probability of making type II error.
โข What would happen to type I and type II errors if we increase the sample size? (Hint: look at the confidence intervals)
Adopted from http://whatilearned.wikia.com/wiki/Hypothesis_Testing?file=Type_I_and_Type_II_Error_Table.jpg
Type I & Type II Errorsโข The following graph shows how a change of the critical line (critical
value) changes the probability of making type I and type II errors:
๐ท ๐ป๐๐๐ ๐ฐ ๐๐๐๐๐ = ๐ถ
And
๐ท ๐ป๐๐๐ ๐ฐ๐ฐ ๐๐๐๐๐ = ๐ท
Adopted from http://www.weibull.com/hotwire/issue88/relbasics88.htm
The Power Of a Test:
The power of a test is the probability that the test will correctly reject the null hypothesis. It is
the probability of not committing type II error. The power is
equal to ๐ โ ๐ท which means by reducing ๐ทthe power of the test
will increase.
The P-Valueโข It is not unusual to reject ๐ป0 at some level of significance, for
example ๐ผ = 5% , but being unable to reject it at some other levels, e.g. ๐ผ = 1% . The dependence of the final decision to the value of ๐ผ is the weak point of the classical approach.
โข In the new approach, we try to find p-value which is the lowest significance level at which ๐ป0 can be rejected. If the level of significance is determined at 5% and the lowest significance level at which ๐ป0 can be rejected (p-value) is 2% so the null hypothesis should be rejected; i.e.
๐ โ ๐๐๐๐๐ < ๐ถ
To understand this concept better letโs look at an example:
โข Suppose we believe that the mean life expectancy of the people in a city is 75 years (๐ป0: ๐ = 75). But our observation shows a sample mean of 76 years for a sample size of 100 with a sample variance of 4 years.
Reject ๐ป0
The P-Valueโข The Z-score (test statistic) can be calculated as following:
โข At 5% level of significance the critical Z-value is 1.96 so we must reject ๐ฏ๐. But, we should not have had this result (or should not have had those observations in our random sample) from the beginning if our assumption about the population mean ๐ was correct.
โข The p-value is the probability of
having these type of results
or even worse than that (i.e. a Z-score
bigger than 2.5) considering the null
hypothesis was correct,
๐ท(๐ โฅ ๐. ๐ ๐ = ๐๐) = ๐ โ ๐๐๐๐๐ โ ๐. ๐๐๐ (it means in 1000 samples this type of results can happen theoretically 6 times; but it has happened in our first random sampling).
๐ =๐ โ ๐๐ ๐
=76 โ 75
4
100
= 2.5
Z=2.5
๐ท ๐ โฅ ๐. ๐โ ๐. ๐๐๐
http
://faculty.elgin
.edu
/dke
rnler/statistics/ch
10/10
-2.htm
l
The P-Valueโข As we cannot deny what we have observed and obtained from the
sample, eventually we need to change our belief about the population mean and reject our assumption about that.
โข The smaller the p-value, the stronger evidence against ๐ป0.