Elementary Probability Concepts

CHAPTER FIVE
ELEMENTARY PROBABILITY

Introduction
• Probability: can be defined as a measure of the likelihood that a particular event
will occur or it is a science of decision making with calculated risk in face of
uncertainty.
• It is the measure of how likely an outcome is to occur.
• Probability theory: is the branch of mathematics that studies the possible.
• Most people express probabilities in terms of percentages.
• But, it is more convenient to express probabilities as fractions.
• It is always found between 0 and 1,inclusively.
 Closer the number is to one =>The more likely the event
 A probability of zero => An event that can't occur
 A probability of one => an event that is certain to occur.
Thursday, November 16,
2023
2

Definitions of some probability terms
Experiment: is any activity that generates
outcomes.
Outcome :The result of a single trial of a random
experiment
Example:
• The outcome of the sex of a newborn from a
mother in delivery room is either Male or female
• Tossing a coin two times
• Producing a new products
• Rolling a die once
2023
3

Definition of terms …Cont’d
Sample Space: The collection of all possible
outcomes of an experiment.
Example:
The sample space for the sex of newborns when two
mothers are in the gynecology ward to give birth
is:{MM, MF, FM, FF}
Tossing coin (Head, Tail)
Event: It is a subset of sample space.
Eg. From the above experiment, an event
consisting of at least one female is
E = {MF, FM, FF}
2023
4

2023
5
 Elementary event (simple event): is a single possible
outcome of an experiment.
 Composite (compound) event: is an event having
two or more elementary events in it
 Equally Likely Events: Events which have the same
equal chance of occurrence.
 Mutually Exclusive Events: Two events which cannot
happen at the same time.
 Independent Events: Two events are independent if
the occurrence of one does not affect the probability of
Definitions terms…Cont’d

Basic notations of probability
2023
6

2023
7
Counting rules
• In order to determine the number of outcomes, one
can use several rules of counting.
Addition rule
The multiplication rule
Permutation rule
Combination rule

Addition rule
2023
8

2023
9
The Multiplication rule
 If a choice consists of k steps of which the first can be made in n1
ways, the second can be made in n2 ways…, the kth can be made
in nk ways. Suppose also that each way of doing procedure 2
may be followed by any way of doing procedure 1, then the
whole choice can be made in n1 * n2*…*nk ways
Example:
Distribution of Blood Types There are four blood types, A, B,
AB, and O. Blood can also be Rh+ and Rh-. Finally, a blood
donor can be classified as either male or female. How many
different ways can a donor have his or her blood labeled?

2023
10

Solution
Since there are 4 possibilities for blood type, 2 possibilities
for Rh factor, and 2 possibilities for the gender of the
donor, there are 4 2 2, or 16, different classification
categories, as shown.
2023
11

2023
12
Permutation
• An arrangement of n objects in a specified order
Permutation Rules:
1. The number of permutations of n distinct objects taken all
together is n!
2. The arrangement of n objects in a specified order using r
objects at a time
3. The number of permutations of n objects in which k1 are alike
k2 are alike ---- etc is
n
k
n
k
k
k
n
P
*
...
*
!*
!
2
1

1
*
2
*
3
*
.....
*
)
2
(
*
)
1
(
*
! 

 n
n
n
n
 
1
=
1!
=
0!
definition
In
n!
=
!
0
!
!
! n
n
n
n
Pn
n 

)!
(
!
r
n
n
Pr
n



2023
13
Example:
1. Suppose we have a letters A,B, C, D
a. How many permutations are there taking all the four?
b. How many permutations are there two letters at a time?
2. How many different permutations can be made from the
letters in the word “STATISTICS ”?

Permutation…con’d
• Example 3: Jimma University Registrar Office wants to give identity
number for students by using 4 digits. The number should be
considered by the following numbers only: {0, 1, 2, 3, 4, 5, and 6}.
Hence, how many different ID Numbers could be given by the
Registrar?
a. Without repeating the number
b. With repetition of numbers
• Solution:- We have 7 possible numbers for 4 digits. But the
required number of digits for ID
number is 4. Hence n = 7 & r = 4
The possible number of id No. given for student without repeating
the number is
2023
14

Permutation…con’d
• a. The possible number of id No. given for student without repeating
the number is
7*6*5*4=840
b. The possible number of id No. given for student with repeating the
number is
7*7*7*7=2401
2023
15
1st digit 2nd digit 3rd digit 4th digit
7 6 5 4
1st digit 2nd digit 3rd digit 4th digit
7 7 7 7

2023
16
Combination rule
Selection of objects without regard to order is denoted by
Example:
1. In how many ways 5 patients be chosen out of 9 patients?
2. How many ways can a 5 injured persons be selected from 10
injured people in a certain car accident.
3. Among 15 pack drugs two of them are defectives. In how
many ways can a pharmacologist chose three of the pack drug for
inspection so that:
a) There is no restriction,
b) None of defective drug is included,
c) Only one of the defective drug is included,
d) Two of the defective drug is included.








r
n
or
Cr
n and is given by the formula: !
)!*
(
!
r
r
n
n
r
n











Approaches to measuring Probability
There are different conceptual approaches to the
study of probability theory. These are:
• The classical approach.
• Empirical probability approach.
• The axiomatic approach.
2023
17

The classical approach
This approach is used when:
Each outcome in a sample space is equally likely to
occur.
That is if an experiment has n equally likely
outcomes, then each possible outcome must have
probability of 1/n to occur.
Definition: If a random experiment with N equally likely
outcomes is conducted and out of these NA outcomes are
favorable to the event A, then the probability that event A
occur denoted is defined as:
2023
18
)
(
)
(
outcomes
.
)
(
S
n
A
n
of
number
Total
A
to
favourable
outcomes
of
No
N
N
A
P A




Example
1. A fair die is tossed once. What is the probability of
getting
a. Number 4? b) An odd number?
c. An even number? D. Number 8?
2. What is the probability of getting at least one female birth
from two pregnant mothers?
2023
19

Empirical probability
Is based on observations obtained from
experiments/a large number of trials or from
historical data
Example:
A medical doctor realized that out of 100,000
patients visited the hospital, there are 50 cancer
cases. What is the probability that a patient to be
examined will be positive for cancer?
P(+ve for cancer) = 50/100,000 = 0.0005
Thursday, November 16, 2023 20

Axiomatic Approach:
Axiomatic Approach:
2023
21

Conditional probability
Conditional Events: If the occurrence of one event has an
effect on the next occurrence of the other event then the
two events are conditional or dependent events.
The conditional probability of an event A given that B has
already occurred, denoted
0
)
(
,
)
(
)
(
)
|
( 

 B
P
B
P
B
A
P
B
A
P
2023
22

Example
1. Suppose in country X the chance that an infant lives to age
25 is 0.95, whereas the chance that he lives to age 65 is
0.75. For the latter, it is understood that to survive to age
65 means to survive both from birth to age 25 and from
age 25 to 65. What is the chance that a person 25 years of
age survives to age 65?
2. For a student enrolling at freshman at certain university
the probability is 0.25 that he/she will get scholarship and
0.75 that he/she will graduate. If the probability is 0.2 that
he/she will get scholarship and will also graduate. What is
the probability that a student who got a scholarship
graduate?
2023
23

1. What is the probability of left-handed given that
it is a male?
2. What is the probability of female given that they
were right-handed?
3. What is the probability of being left-handed?
P(LH | M) = 12/50 = 0.24
P(F| RH) = 42/80 = 0.525
P(LH) = 20/100 = 0.20
.
2023
24
Male Female Total
Right handed 38 42 80
Left handed 12 8 20
Total 50 50 100

Sensitivity
2023
By Yasin N.
25
 Sensitivity is the probability that the test says
a person has the disease when in fact they
do have the disease.
 If you knew someone was SICK, what would
you want their test result to be?
 We call this sensitivity; The probability that a
sick individual will have a positive test.
 It is a measure of how likely it is for a test to
pick up the presence of a disease in a person

Data lay out.
2023
By Yasin N.
26

True Status of
Nature (S)
Test Result (T)
Positive (+) Negative (-)
Disease (+) a b
No disease(-) c d

Specificity
2023
By Yasin N.
27


Positive predictive value (PPV)
2023
By Yasin N.
28


Negative predictive value (NPV)
2023
By Yasin N.
29


Example:
2023
By Yasin N.
30
 The following example is take from Statistical Methods
for the Social Sciences, 3rd edition. The data are
provided for the results from a screening test for HIV
that was performed on a group of 100,000 people.
 Then:
a. Compute Sensitivity and Specificity
b. Compute Positive and Negative predictive value
HIV Status Test Result
Positive (+) Negative (-)
Positive (+) 475 25
Negative (-) 4975 94525

Solution
2023
By Yasin N.
31


Solution(Cont.)
2023
By Yasin N.
32


Exercise
A test is used in 50 people with disease and 50 people without. These
are the results:
 Then:
a. Compute Sensitivity and Specificity
b. Compute Positive and Negative predictive value
Disease
+ -
Test
+ 48 3
- 2 47
2023
By Yasin N.
33

Total probability rule
 Let A1, …AN be N mutually exclusive events, whose
union gives the sample space S. Hence the events A
constitute a partition of S. For any event B, a subset of S,
we have

 If A1,A2,…,An are mutually disjoint events with
P(Ai)≠0,(i=1,2,…,n), the for any arbitrary event B which is a
subset of S such that P(B)>0, we have
Bayes’ theorem

Example
2023
By Yasin N.
36
 Suppose that it is known that a fraction 0.03 of
the people in a town have tuberculosis(TB). A
tuberculosis test is given with the following
properties : If the person does have TB, the test
will indicate it with a probability of 0.99. If he
does not have TB, then there is a probability
0.002 that the test will erroneously indicate that
he does. For one random selected person, the
test shows that he has TB. What is the probability

Example…Cont’d
 An insurance company issues life insurance policies in
three separate categories: Standard, Preferred, and Ultra-
preferred. Of the company’s policyholders, 50% are
standard,40% are preferred, and 10% are ultra-preferred.
Each standard policyholder has probability of 0 .010 of
dying in the next year, each preferred policyholder has
probability 0 .005 of dying in the next year, and each
ultra-preferred policyholder has probability 0 .001 of
dying in the next year.
a. What is the probability of “death in the next year”?
b. If a policyholder dies in the next year, What is the
probability that the deceased policyholder was ultra-
preferred?
11/16/2023 Introduction to Biostatistics 37

Solution: Let S , P , U denote the standard, preferred,
and ultra-preferred policyholders, and le t D denote the
event “dies in the next year”. We need to compute
 
( ) 0.1, ( ) 0.5, ( ) 0.4
=0 .001 (D|S) 0 .01, ( | ) 0 .00
D|U 5
P U P S P
P
P
and P P D P
  
 
a.              
| | |
P D P D U P U P D S P S P D P P P
  
0 .001*0.1 0.01*0.5 0.005*0.4 0.0071
   
b.
   
           
0 .001*0.1
0 .001*0.1+0.01*0.5+0.005
|
( | )
| | | *0.4
0.0001
0.01408451
0.0
=
071
P D U P U
P U D
P D U P U P D S P S P D P P P
 
 

11/16/2023 Introduction to Biostatistics 38

Random variable
and
Probability distribution
Chapter Six

Random Variable
• A random variable (r.v): is a variable whose value are is
determined by chance.
Example:
1. In the experiment of tossing a coin three times, let we define
the random variable X as number of heads. What is the possible
values of r. v X?
2. Let a pair of fair dice be tossed and let X denotes the sum of the
points obtained. What is the possible value of X?
2023
40

Types of Random Variable
Discrete random variable: are variables which can
assume only a specific number of values. They
have values that can be counted.
• Examples:
1.Dead/alive
2.Number of car accidents per week.
3.Number of patients.
4.Number of bacteria per two cubic centimeter of water.
2023
41

Continuous random variable: are variables that
can assume all values between any two given
values.
Examples:
1.weight of patients at hospital.
2. blood pressure
3.Life time of light bulbs.
2023
42

Probability distribution
2023
43

Probability distribution of discrete random variable
2023
44

Expected and variance of discrete r.v.
2023
45

Probability distribution of continuous r.v.
2023
46

Expected and variance of discrete r.v
2023
47

Example
2023
48

Common Discrete probability distributions
•Binomial Distribution
•Poisson Distribution
Common Continuous Probability Distributions
•Normal Distribution
2023
49

Binomial Distribution
A binomial experiment is a probability experiment
that satisfies the following requirements called
assumptions of a binomial distribution.
• There is only two outcomes in Bernoulli trials
(success or failure)
• Fixed number of trials (n) i.e. n should be discrete
• At each trial the probability of success (p) remains
the same
• n trials are independent.
2023
50

Binomial Distribution……
• The binomial distribution is given by the probability mass
function ( pmf)
• In the formula, n= number of trials
x= number of successes in a trial
n-x = number of failures in a trial
p = probability of success (= x/n)
q = 1 - p = probability of failure
The parameters of the binomial distribution are n and p
2023
n
x
q
p
x
n
x
X
P x
n
x
,....,
2
,
1
,
0
,
)
( 









 
( )
E X np
  
2
var( ) (1 )
X np p
   
51

Binomial Distribution…..
• Examples
• Tossing a coin 20 times to see how many tails
occur.
• Asking 200 people if they watch BBC news.
• Asking 100 people if they favor the ruling party.
• Rolling a die to see if a 5 appears.
2023
52

Example (Binomial ..)
Example 1: Five fair coins are tossed. Find the probability of
obtaining
a. No heads
b. At least four heads
c. At most 2 heads
d. Exactly 2 heads
Example 2:A given mid-exam contains 10 multiple choice
questions, and each question has four alternatives with one exact
answer. Find the probability that the student exactly answered
a. 3 questions c. At least 3 questions
b. 8 questions
2023
53

2023
Example (Binomial ..)
Suppose that in a certain malarias area past experience
indicates that the probability of a person with a high
fever will be positive for malaria is 0.7. Consider 3
randomly selected patients (with high fever) in that same
area.
a) What is the probability that no patient will be positive
for malaria?
b) What is the probability that exactly one patient will be
positive for malaria?
c) What is the probability that exactly two of the patients
will be positive for malaria?
d) What is the probability that all patients will be positive
for malaria?
54

The Poisson Probability
Distribution
• The passion distribution is also
used to represent the probability
distribution of a discrete random
variable.
• It is employed in describing random
events that occurs rarely over a
continuum of time or space

• Example
- Number of misprinting
- Natural disasters like earth quake , accident
- No of telephone calls per hour
- No of car accident occurs per week
2023
56

The Poisson Distribution of the random variable X.
!
)
(
x
e
x
P
x 
 

 = mean number of occurrences in the
given unit of time, area, volume, etc.
 e = 2.71828….
 Mean µ = , variance: 2 =
2023
57

Example
1. In a hospital, the average number of new born female
baby in every 24 hours is 7. what is the probability that
a. No female babies are born in a day
b. Only three females babies are born per day
c. 2 female babies are born in 12 hours
2023
58

2023
Continuous Probability Distributions
The Normal distribution
♣ The Normal Distribution is by far the most important
probability distribution in statistics.
♣ The normal distribution is a theoretical, continuous
probability distribution whose equation is:
for - < x < +
2
2
1
-
e
2
1
f(x)





 
 



x
59

2023
Characteristics of the Normal Distribution
♣ It is a probability distribution of a continuous variable. It extends
from minus infinity( -) to plus infinity (+).
♣ It is unimodal, bell-shaped and symmetrical about x =.
♣ The mean, the median and mode are all equal
♣ The total area under the curve above the x-axis is one square
unit.
♣ The curve never touches the x-axis.
♣ It is determined by two quantities: its mean (  ) and SD (  )
♣ An observation from a normal distribution can be related to a
standard normal distribution (SND) which has a published
table.
60

2023
Properties of Normal Distribution
• The normal Distribution is a family of
Bell-shaped and symmetric distributions as the
allocation is symmetric: one-half (.50 or 50%) lies on
either side of the mean.
Each is characterized by a different pair of mean, ,
and variance,  . That is: [X~N()]. 61

2023
Standard normal distribution
♣ Since the values of  and  will depend on the particular
problem in hand and tables of the normal distribution
cannot be published for all values of  and ,
calculations are made by referring to the standard normal
distribution which has  = 0 and  = 1.
♣ Thus an observation x from a normal distribution with
mean  and standard deviation  can be related to a
Standard normal distribution by calculating :
SND = Z = (x -  )

62

2023
Properties of the Standard Normal Distribution:
Same as a normal distribution, but also...
• Mean is zero
• Variance is one
• Standard Deviation is one
• Areas under the standard normal distribution curve
have been tabulated in various ways.
• The most common ones are the areas between Z=
0 and a positive value of Z
63

The standard normal random variable, Z, is the normal random
variable with mean  = 0 and standard deviation  = 1: Z~N(0,12).
5
4
3
2
1
0
- 1
- 2
- 3
- 4
- 5
0 .4
0 .3
0 .2
0 .1
0 .0
Z
f
(
z
)
Standard Normal Distribution
 = 0
=1
{
The Standard Normal Distribution
2023
64

2023
• Given a normal distributed random variable X
with Mean μ and standard deviation σ
)
(
)
(





 







b
X
a
P
b
X
a
P
)
(
)
(



 






b
Z
a
P
b
X
a
P
65

2023
Examples
1. Find the area under the standard normal
distribution which lies
a. Between
b. Between
c. To the right of
d. Between
96
.
0
0 
 Z
and
Z
0
45
.
1 

 Z
and
Z
35
.
0


Z
75
.
0
67
.
0 

 Z
and
Z
66

2023
• Solutions
• a.
• b.
• c.
• d.
3315
.
0
)
96
.
0
0
( 


 Z
P
Area
4265
.
0
)
45
.
1
0
(
)
0
45
.
1
(








Z
P
Z
P
Area
6368
.
0
50
.
0
1368
.
0
)
0
(
)
35
.
0
0
(
)
0
(
)
0
35
.
0
(
)
35
.
0
(

















Z
P
Z
P
Z
P
Z
P
Z
P
Area
5220
.
0
2734
.
0
2486
.
0
)
75
.
0
0
(
)
67
.
0
0
(
)
75
.
0
0
(
)
0
67
.
0
(
)
75
.
0
67
.
0
(




















Z
P
Z
P
Z
P
Z
P
Z
P
Area
67

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .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
5
4
3
2
1
0
-1
-2
-3
-4
-5
0.4
0.3
0.2
0.1
0.0
Z
f(
z
)
StandardNormal Distribution
1.56
{
Standard Normal Probabilities
Look in row labeled 1.5
and column labeled .06 to
find P(0  z  1.56) =
0.4406
2023
Finding Probabilities of the Standard Normal
Distribution: P(0 < Z < 1.56)
68

Example: A random variable X has a normal distribution
with mean 80 and standard deviation 4.8. What is the
probability that it will take a value
•Less than 87.2
•Greater than 76.4
•Between 81.2 and 86.0
Solution: 8
.
4
,
tan
,
80
, 
 
 deviation
dard
s
mean
with
normal
is
X
9332
.
0
4332
.
0
50
.
0
)
5
.
1
0
(
)
0
(
)
5
.
1
(
)
8
.
4
80
2
.
87
(
)
2
.
87
(
)
2
.
87
(


















Z
P
Z
P
Z
P
Z
P
X
P
X
P




a
2023
69

2023
Exercise: Diskin et al. studied common breath metabolites such
as ammonia, acetone, isoprene, ethanol and acetaldehyde
in five subjects over a period of 30 days. Each day, breath
samples were taken and analyzed in the early morning on
arrival at the laboratory. For subject A, a 27-year-old female,
the ammonia concentration in parts per billion (ppb)
followed a normal distribution over 30 days with mean 491
and standard deviation 119. What is the probability that on
a random day, the subject‘s ammonia concentration is
between 292 and 649 ppb?
70

Chapter-7
Sampling methods
and
Sample size estimation

Sampling
• Sampling: The process or method of sample selection from the
population.
• Reference population (or target population): the population of
interest to whom the researchers would like to make generalizations.
• Study population: the actual group in which the study is conducted
• Sampling unit: the ultimate unit to be sampled or the units on which
information will be collected: persons, housing units, etc.
• Sampling frame:-List of all the sampling units from which sample is
drawn.
. Thursday, November 16,
2023
72

Eg. The performance of freshman students in institution of health
science in JU,
Target population: All JU freshman students
Study population: All freshman students in institution of health
Sampling unit: selected freshman students in in institution of
health
Sampling frame: List of students in the registrar office.
2023
73

e.g. Researchers are interested to know about factors
associated with ART use among HIV/AIDS patients
attending JU referal hospital
Target population
= All HIV/AIDS
patients in the JU ref.H.
Study population
= All ART HIV/AIDS patients in
JU referral hospital
Sample units
=Selected HIV/AIDS HIV patients
2023
74

Exercise
Assessment of obstetric Fistula repair success
rate and associated factors in Jimma Town.
Prevalence of emergency contraceptive
utilization and associated factors among students
in Jimma University, 2015
2023
75

Sampling (Cont.)
• Errors in sample survey:
• There are two types of errors
– Sampling error: Is the discrepancy between the
population value and sample value.
• May arise due to in appropriate sampling
techniques applied
• Non sampling errors: are errors due to procedure bias
such as:
– Due to incorrect responses
– Measurement
– Errors at different stages in processing the data.
2023
76

Sampling (Cont.)
• Advantages of sampling approach over that of census are:-
• -Reduced cost
• Greater speed
• Greater accuracy
• More detailed information can be obtained
2023
77

Sampling cont …
To draw a sample from the total population, we
must consider the following questions:
1. What is the group of people (study
population) we are interested in from which
we want to draw a sample?
2. How many people do we need in our sample?
3. How will these people be selected?
2023
78

Types: probability and non-probability
– Probability
– Non-probability
• Probability sampling technique:
– Involves using random selection procedures to ensure that
each unit of the sample is chosen on the basis of chance.
– All units of the study population should have an equal, or at
least a known non-zero chance of being included in the
sample.
– Sample drawn in such a way that it is representative of the
population
2023
79

Probability sampling methods include:
– Simple random sampling
– Systematic sampling
– Stratified sampling
– Cluster sampling
2023
80

1. Simple random sampling
• Selecting required number of sampling units randomly
from list of all units
• Every item has equal chance of being selected
• Possible samples
If selection is without replacement
If selections is with replacement
• Random selection – Lottery method, table of random
numbers or using computer programs
 )
n
N C
n
N
2023
81

2. Systematic sampling
• A complete list of all elements within the population is
required.
• The procedure starts in determining the first element to be
included in the sample.
• Important if the reference population is
arranged in some order:
– Order of registration of patients
– Numerical number of house numbers
– Student’s registration books
.
int
,
, erval
sampling
n
N
k
size
sample
n
size
population
N 



2023
82

• Then the technique is to take the kth item from the
sampling frame.
Example: To select a sample of 100 students
from 2500, first calculate sampling
interval=2500/100=25. Then randomly select
select the first student and finally pick every
25th student
2023
83

3. Stratified sampling
• Used when the population structure consists distinct
subgroups/strata
• The population will be divided in to non-overlapping but
exhaustive groups called strata
• Simple random samples will be chosen from each stratum.
• It is applied if the population is between a group
heterogeneous with in a group homogeneous.
• Some of the criteria for dividing a population into strata are:
Sex (male, female); Age (under 18, 18 to 28, 29 to 39, etc);
Occupation and other…..
2023
84

Stratified Random Sample
2023
85

4. Cluster sampling
• The population is divided in to non-overlapping groups called
clusters.
• A simple random sample of groups or cluster of elements is
chosen and all the sampling units in the selected clusters will be
surveyed.
• Clusters are formed in a way that elements within a cluster are
heterogeneous between cluster are homogeneous
• All the study units in the selected clusters are included in the
study
• Used in geographically scattered areas where visiting dispersed
study units is time consuming and costly
• Example: a simple random sample of 5 villages from 30 villages
2023
86

Non Random Sampling or non-probability sampling.
• It is a sampling technique in which the choice of
individuals for a sample depends on the basis of
convenience, personal choice or interest.
• Examples:
–Judgment sampling.
–Convenience sampling
–Quota Sampling.
2023
87

non-probability sampling
• Judgment Sampling: The person taking the
sample has direct or indirect control over which
items are selected for the sample.
• Convenience Sampling: decision maker selects a
sample from the population in a manner that is
relatively easy and convenient
• Quota Sampling: The decision maker requires
the sample to contain a certain number of items
with a given characteristic
2023
88

2023
89

Sample size
Quantitative Qualitative
 
2
2
2
1
1
d
P
P
Z
no









S=standard deviation
d= margin of error
p=population proportion
d= margin of error
 
2
2
2
2
1
d
S
Z
no








2023
90

Sample size con..
If greater than 5%,the sample size must be
adjusted as follow:
N
n
n
n
/
1 0
0


N
n /
0
2023
91

End of CH.7
2023
92

Chapter Eight
Estimation and Hypothesis testing

Estimation
• Estimation: is the process of estimating the value of the parameter from
information obtained by using a sample..
Two types of estimations
• Point estimation: is a specific numerical value estimate of a parameter.
• The best point estimate of the population mean and proportion is the
sample mean and the sample proportion respectively
• Interval estimation: is an interval or range of values used to estimate
the population parameter.
• This estimate may contain the value of the parameter being estimated.
2023
94

Confidence interval
 Confidence interval for the population mean is
2023
95
.
Estimate Critical Value S E
 
2
( )
X SE
z

=

2023
96
Interval estimation of a population means

n
s
Z
X 
n
Z
X 

n
s
t
X 

Example
1. A physical therapist wished to estimate, with 99 percent confidence,
the mean maximal strength of a particular muscle in a certain group
of individuals. He is willing to assume that strength scores are
approximately normally distributed with a variance of 144. A sample
of 15 subjects who participated in the experiment yielded a mean of
84.3.
Ans: 84.3  2.58 (3.0984)= 76.3, 92.3
1. The mean reading speed of a random sample of 81 adults with long
sight problem is 325 words per minute. Find a 90% C.I. For the mean
reading speed of all adults (μ) if it is known that the standard
deviation for the sampled adults is 45 words per minute.
Ans: 325 ± (1.64 x 5 )=(316.8, 333.2)
2023
97

Example(Cont.)
2023
98

Example (cont.)
• The sponsor of a TV programmer targeted at the
children’s market wants to find out the average
amount of time children spend watching TV. A
random sample of 25 children indicated the
average time spent by these children watching
television per week to be 27.2 hours. From
previous experience, the population standard
deviation of the weekly extent of television
watched is known to be 8 hours. Construct a
95% confidence interval.
2023
99

Example
• A researcher found that in a sample of 591 patients admitted to
a psychiatric hospital, 204 admitted to using cannabis at least
once in their lifetime. We wish to construct a 95 percent C.I
for the proportion of lifetime cannabis users in the sampled
population of psychiatric hospital admissions.
Ans: 0.3452  0.0383  0.3069, 0.3835
2023
100

Hypothesis Testing
There are two types of hypothesis:
1. Null hypothesis:
• It is the hypothesis of equality or the hypothesis of no
difference.
• Usually denoted by H0.
2. Alternative hypothesis:
• It is the hypothesis available when the null hypothesis has to
be rejected.
• It is the hypothesis of difference.
2023
101

Types of errors:
2023
102
Decision
Reject H0 Don't reject H0
Truth
H0 Type I Error Right Decision
H1 Right Decision Type II Error

Steps for hypothesis testing
• State the null and alternative hypotheses.
• Choose a fixed significance level α.
• Choose an appropriate test statistic and establish the critical
region based on critical values.
• From the computed test statistic, reject Ho if the test statistic is
in the critical region. Otherwise, do not reject.
• Decision
• conclusions
2023
103

Example
• Assume that in a certain district the mean
systolic blood pressure of persons aged 20 to 40
is 130 mm Hg with a standard deviation of 10
mm Hg. A random sample of 64 persons aged 20
to 40 from village x of the same district has a
mean systolic blood pressure of 132 mm Hg.
Does the mean systolic blood pressure of the
dwellers of the village (aged 20 to 40) differ
from that of the inhabitants of the district (aged
20 to 40) in general, at a 5% Level of
significance?
2023
104

Example
• A physician claims that jogger’s maximal
volume oxygen uptake is lower than the
average of all adults , a sample of 15 joggers
has a mean of 38.6 ml/kg and a standard
deviation of 6 ml/kg. If the Average of all adults is
36.7 ml/kg, test the physician’s claim at α=0.01
• Test the hypotheses that the average height content of
containers of certain lubricant is 10 liters if the contents of
a random sample of 10 containers are 10.2, 9.7, 10.1, 10.3,
10.1, 9.8, 9.9, 10.4, 10.3, and 9.8 liters. Use the 0.01 level
of significance.
2023
105

Example
• Suppose a historical data show that the mean
prone resting systolic blood pressure (BP) for male
army aged 18-24 is 126. A sample of 87 recent
male army recruits in this age group shows a
mean prone resting systolic blood pressure of
128.46 with a standard deviation of 7. At α = 0.05,
does the recruits' mean BP exceed the historical
average of 126?
2023
106

Test of Association
• The chi-square procedure test is used to test the
hypothesis of independency of two attributes .For
instance we may be interested
• Whether the presence or absence of
hypertension is independent of smoking habit or
not.
• Whether the size of the family is independent of
the level of education attained by the mothers.
• Whether there is association between father and
son regarding boldness.
2023
107

Test of Association(cont.)
2023
108
 
  






 

r
i
c
j ij
ij
ij
cal
e
e
O
1 1
2
2 )
(

.
.
B
of
j
and
A
of
i
category
to
belong
that
frequency
Expected
e
B
of
j
and
A
of
i
category
to
belong
that
units
of
number
the
O
Where
ij
ij


n
C
R
e
j
i
ij
*


Example
• Random samples of 200 men was followed for the study
of weather there is association between Alcohol usage
and allergic therapy, the follow-up result is as follows;
Test the hypothesis that Allergic therapy is independent of
Alcohol usage of men (Use 5% level of significance)
2023
109
Allergic test Use Alcohol
Always Sometimes Never
Positive 14 37 32
Negative 31 59 27

Example
• Incidence of three types of malaria in three tropical
regions
Test whether there is no relationship between location
and type of malaria.
2023
110

Solution
2023
111

2023
112
End of
Ch.8

Elementary Probability Concepts

Recommended

Recommended

More Related Content

Similar to Elementary Probability Concepts

Similar to Elementary Probability Concepts (20)

Recently uploaded

Recently uploaded (20)

Elementary Probability Concepts