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BUSINESS STATISTICS STT O6O
1
...
1 Meaning of Statistics
1
...
Statistics are numerical statement of facts in any department of enquiry placed
interrelation to each other- Bouly
3
...
By Statistics we mean quantitive data affected to a marked extent by a multiplicity of
causes- Youle and Kendal
5
...
A
...
2

The Scope and Importance of Statistics
1
...
Almost all over the world the governments are going
back to planning for economic development
2
...
Such as wages, price, time series
analysis and demand analysis
3
...
Statistical
techniques are being used more in studying the desires of the customers
4
...
This is in Production
engineering to find out whether the product is confirming to the specifications or not
...

5
...
Statistics and Modern Science: The statistical tools for collection, presentation and
analysis of observed facts relating causes and incidence of diseases and results of various
drugs and medicine applications are of importance
7
...
t
...

8
...


STATISTICS IN BUSINESS AND MANAGEMENT


Marketing: Statistical analysis provides information for decision making
...
An analysis of production, purchasing power, man power, habits of
competitors, consumer habits, transportation cost are a must consideration for new
markets



Production: The decision what to produce? How to produce? When to produce? For
whom to produce is based largely on statistical analysis



Finance: For financial organizations to have an effective finance function they will
depend heavily on statistical analysis of peat and tigers



Banking: Research departments are established to gather and analyze information
regarding their business, general economic situation and other external business segments
of interest



Investment: Investors are greatly assisted in making clear and valued judgment in
selecting safe securities with good income yielding prospects



Purchase: Statistical data is utilized to frame suitable purchase policies such as what to
buy? What quantity to buy? What time to buy? Where to buy? Whom to do the buying?



Accounting: Auditing in accounts utilizes the technique of sampling and destination



Control: This combines statistics and accounting in making the overall budget for the
coming years including sales, materials, labor, other costs, net profits and capital
requirements
...
3 CATEGORIES OF STATISTICS
Definitions:
Data: These are the values (measurements or observations) that the variables can assume
Data set: This is a collection of data values
...
These areas are:1
...
Inferential Statistics
Descriptive Statistics consists of the collection, organization, summarization and presentation of
data
...
Data is collected, organized and summarized
...

Inferential Statistics consists of generalizing from samples to populations, performing estimations
and hypothesis tests, determining relationships among variables and making predictions
...
Inferential Statistics utilizes
probability (the chance of an event occurring)
...

A Population consists of all subjects (human or otherwise) that are being studied
...

PARAMETERS VERSUS STATISTICS
The population under investigation can be summarized by numerical parameters
...

So sample statistics are used to make inference about these parameters
...

A Parameter is an unknown numerical summary of the population
...
4 STATISTICAL ENQUIRIES
The goal of Statistics is to gain understanding from data
...
This includes assessment of the extent of
uncertainty involved in these inferences
...
It involves
the systematic gathering of data for a particular purpose from various sources, that has been
systematically observed, recorded and organized
...

The Purpose of Data Collection
 To obtain information
 To keep on record
 To make decisions about important issues
 To pass information on to others
2
...
g your own questionnaire
SECONDARY DATA
Data gathered and recorded by someone else prior to and for a purpose other than the
current project
Secondary data is data that has been collected for another purpose
It involves less cost, time and effort
This is data that is being re used, usually in a different context e
...

2
...
Observation

Advantages:




Collects data where and when an event is occurring
Does not rely on people‟s willingness to provide information
Directly see what people do rather than relying on what they say they do
5

Disadvantages:




Susceptible to observer bias
Hawthorne effect: People usually perform better when they know they are being observed
Does not increase understanding of why people behave the way they do
...
Document Review

Advantages:






Relatively inexpensive
Good source of background information
Unobtrusive
Provides a “behind the scenes” look at a program that may not be directly observable
May bring up issues not noted by other means

Disadvantages:





Information may be in applicable, disorganized, unavailable or out of date
Could be biased because of selective survival of information
Information may be incomplete or inaccurate
Can be time consuming to collect, review and analyze many documents

3
...
Focus Groups
Advantages:



Quick and relatively easy to set up
Group dynamics can provide useful information that individual data collection does not
provide
6



Useful in gaining insight into topics difficult to gather info through the other methods

Disadvantages
 Susceptible to facilitator bias
 Discussions can be dominated by few individuals
 Data analysis is time consuming
 Doesn‟t provide valid information at the individual level
 The information is not representative of other groups
5
...
Town hall meetings and other large group events
7
...
Illustrated Presentations- Photo Voice, Power Voice
...
3 COMPLETE ENUMERATION (CENSUS) AND SAMPLING
Survey: A set of statistical activities designed to obtain data
Types of survey: Census surveys, Sample surveys
Census survey: A complete enumeration of the population of interest
...
4 SAMPLING METHODS
SAMPLING
Once you have decided to carry out a sample survey, there are decisions which have to be made
before you start collecting information
...

NB: Bias can rarely be eliminated completely



Decide on the Sampling method

8

METHODS OF SAMPLING
Probability and Non-probability Sampling
The choice of a sampling method will depend on the:

Aim of the survey



Type of Population involved



Time and funds at your disposal

In probability sampling, every item in the Population has a known chance of being selected as a sample
member
...

Probability Sampling:
1
...
The most convenient method for drawing a sample for a survey is to use a table of random
numbers
...

2
...
It involves the selection of a certain proportion of the
population
...
This method reduces the amount of time the sample takes to
draw
...
The major
disadvantage is that the sampling frames can easily cause over or under representation of certain
characteristics in the sample
...
Stratified sampling
The population is divided into groups or strata, according to different sections of the population, and a
simple random sample is taken from each stratum
...
The
advantage is that the results are never distorted by undue emphasis on extreme observations while the
disadvantage is that of defining the strata, its time consuming, expensive and complicated to analyze
...
Multistage sampling
5
...
Quota Sampling
The interviewers are allocated with an area, the number and type of sampling units needed
...
Th interviews then go out, select units
that make up their quota
...
The
disadvantage is it‟s difficult in assessing the degree of confidence in the conclusions
...

2
...
For
example a sample of experts in a field who are thought can provide useful information on a research topic
...
Snow ball Sampling
A small sample is first selected and then each sample member is asked to give acquaintances, in this
bigger samples are selected
...
Convenience sampling

10

CHAPTER THREE
ORGANIZATION AND PRESETATION OF DATA
3
...

It shows the frequency of occurrence of different values of a single phenomenon
...

EXAMPLE:
In a survey of 40 families in a village, the number of children per family was recorded and the
following data obtained
...

Solution:
Number of Children

Tally marks

Frequency

0

///

3

1

//// //

7

2

//// ////

10

3

//// ///

8

4

//// /

6

5

////

4

6

//

2

TOTAL

40

11

In the continuous frequency distribution refers to groups of values
...

E
...
Wage distribution of 100 employees
Weekly Wages (Kshs)
50-100
100-150
150-200
200-250
250-300
300-350
350-400
Total

Number of Employees
4
12
22
33
16
8
5
100

3
...
1 NATURE OF CLASS
When data is classified according to class intervals or some continuous frequency distribution the
following are some associated technical terms:
(a) Class Limits
These are the lowest and highest values that can be included in the class e
...
These can also be termed as the
upper and lower limits of the class
...

(b) Class Boundaries
These are used to separate the classes so that there exists no gaps in the frequency
distribution
...
g
CLASS LIMITS
CLASS BOUNDARIES
24-30
23
...
5
31-37
30
...
5
38-44
45-51
52-58
59-65
*Find the boundaries by subtracting 0
...
5 to 37 (the upper class limit)
...
E
...

(e) Width or size of the class interval
The difference between the lower and upper class limits is called width or size of class
interval and is denoted by “C”
(f) Range
The difference between the largest and smallest value of the observation is called the
range and is denoted by “R”
(g) Mid value or Mid point
The central point of a class interval is called the mid value or midpoint
...

(h) Frequency
The number of observations falling within a particular class interval is called frequency of
that class
...
According
to Sturges’ rule the number of classes can be determined by:K = 1 + 3
...

Therefore with 10 as the number of observations, then the number of class intervals is
K = 1 + 3
...
322 4
(j) The size of class interval

= Range
Number of Class Interval
= Range
1 + 3
...

(k) Class Width
Find the class width by dividing the range by the number of classes
Width=

R
Number of classes

EXAMPLE:
The following are the weights of 50 College Students
42
47
54
49
56

62
50
39
61
38

46
58
51
41
45

54
49
58
40
52

41
51
47
58
46

37
42
64
49
40
13

54
46
43
59
63

44
37
48
57
41

32
42
49
57
51

45
39
48
34
41

(1) Find the class interval
(2) The number of class interval
(3) Draw a table with the class interval, tally marks and frequency
...
2
...
2
...
It is constructed by adding the frequency of the
first class interval to the frequency of the second class interval and so on
...
The data set is
A
B B AB O
O
O B AB B
B
B O
A O
A
O O
O AB
AB A O B
A
Construct a frequency distribution for the data, and inform the blood type with more people
...
2 BAR CHARTS AND LINE GRAPHS
BAR CHARTS/GRAPHS
A bar chart represents the data by using vertical or horizontal bars whose heights or lengths
represent the frequencies of the data
...

Draw a bar graph and line graph for the data
...

800
700
600
500
400
300
200
100
0
Electronics

Room Decor

Clothing

Figure 2: ANU FIRST YEAR STUDENTS SPENDING

15

Shoes

3
...
3
...
If it is not
continuous, then first make it continuous:CLASS BOUNDARIES
FREQUENCY
99
...
5
2
104
...
5
8
109
...
5
18
114
...
5
13
119
...
5
7
124
...
5
1
129
...
5
1
20
18
16
14
12
10
8
6
4
2
0
99
...
5 104
...
5 109
...
5 114
...
5 `119
...
5 124
...
5 129
...
5

Figure 3: Record High Temperatures

16

3
...
2 FREQUENCY POLYGON
A frequency polygon is a graph that displays the data by using lines that connect points plotted
for the frequencies at the midpoints of the classes
...

EXAMPLE:
Using the previous example: Find the midpoints for each class
CLASS BOUNDARIES
MIDPOINTS
99
...
5
102
104
...
5
107
109
...
5
112
114
...
5
117
119
...
5
122
124
...
5
127
129
...
5
132

FREQUENCY
2
8
18
13
7
1
1

Record High Tmperatures:
20
18
16
14
12
10
8
6
4
2
0
102

107

112

117

Figure 4: Record High Temperatures

17

122

127

132

3
...
4
...
It is used in the study of wealth and income
...
t
...

It is specially used in the study of the degree of inequality in the distribution of income and
wealth between countries or between different periods
...
The curve starts from the origin (0,0) and ends at
(100, 100)
...
4
...
Negative Z scores means the value is to the left of the mean while a
Positive z-score means that a value is to the right of the mean
...
It is the distribution
that occurs when a normal random variable has a mean of zero and a standard deviation of one
...
Every normal random variable X can be transformed into a z-score via the following
equation:
𝑧 = (𝑋 − 𝜇) / 𝜎

Where X is a normal random variable, μ is the mean of X, and σ is the standard deviation of X
...
An antipollution advocate does not believe the
government claim
...

(i)
Does the sample data support the government claim at 5% level of significance?
Interprete your answer
...
80 and H1 P> 0
...
64 standard normal deviation units
for H0
= 5% for the sample,
P= 56/64=0
...
80)(0
...
05
n
64

since
z= P- P = 0
...
80 = 1
...
05
It falls within the acceptance region for H0
...
8 at the 5% level of significance
...
88 as before
δ P‟ = (0
...
2) = 0
...
88- 0
...
04

and would fall in the rejection region for H0 (so that these would be no evidence
against the government claim that P> 0
...
Note that increasing (and holding
everything else the same) increases the probability of accepting the government
claim
...
0 MEASURES OF CENTRAL TENDENCY (LOCATION)
In a given population study, you may get a large number of observations
...
Such a representative
number is the central value for all the observations
...
Among these averages are the
Mean, Median and Mode
...
1 THE MEAN:
The mean is the sum of the observations divided by the number of observations
...
Xn then the mean, X, is given by:X = X1 + X2

+ X3

+…………………
...
1
...
88
60

Example 2
The following is the distribution of persons according to different income groups
...

0-10
10-20
20-30
30-40
40-50
50-60
60-70
Income
(Kshs)
6
8
10
12
7
4
3
Number of
Persons
Solution:
Income

Number of
Persons (f)
6
8
10
12
7
4
3
50

0-10
10-20
20-30
30-40
40-50
50-60
60-70

Mean= X = A +

𝑓𝑑

Mid (X)
5
15
25
35 (A)
45
55
65

𝑁

= 35- 20 X 10
50
= 35-4
= 31

21

d= X -A
c
-3
-2
-1
0
1
2
3

Fd
-18
-16
-10
0
7
8
9
-20

4
...

EXAMPLE:
Find the Median for the following data
25,18,27,10,8,30,42,20,53
Solution:
Arranging the data in the increasing order 8, 10, 18, 20, 25, 27, 30, 42, 53
When odd number values are given, The middle value is the 5th item
...
e
...
Find the median for the following data
5,8,12,30,18,10,2,22
Arranging the data in the increasing order 2,5,8,10,12,18,22,30
Here the median is the middle two items i
...
e
=

10 +12
2
= 11

= Using Formula =

9 th
2
item= 4
...
2
...
F)
1
21
63
118
180
225
255
280
295
313
323
325

Where L= Lower Limit of the median class
m= cumulative frequency preceding the median
c= width of the median class
f= frequency in the median class
N= Total frequency

N = 325 = 162
...
5 -118 X 50
62

= 250 + 35
...
89
4
...
It is an actual value,
which has the highest concentration of items in and around it
...

4
...
1 THE MODE FOR GROUPED DATA
See the highest frequency and the corresponding value of X is mode
...
3
...
Then apply the formula
...
4THE RELATIONSHIP BETWEEN THE MEAN, MEDIAN AND THE
MODE:
If in a distribution mean= median= mode, then that distribution is known as symmetrical
distribution
...

(a) Symmetrical distribution

MEAN = MEDIAN= MODE

The spread of the frequencies is the same on both sides of the centre point of the
curve
...
The distribution of frequencies are spread on the right hand than on the left
...
The median lies in between
...

4
...
1 KARL-PEARSON COEFFICIENT OF SKEWNESS
According to Karl Pearson, the absolute measure of Skewness= Mean – mode
...
To avoid this, the use of relative
measure of skewness called Karl- Pearson‟s Coefficient of skewness is given by
= Mean – Mode
Standard Deviation
In case the Mode is indeterminate, the coefficient of skewness is

= 3(Mean – Median)
Standard Deviation

26

CHAPTER FIVE:
MEASURES OF DISPERSION
5
...

In symbols Range = L- S
Where L = Largest value
S= Smallest value

COEFFICIENT OF RANGE

𝐿−𝑆

Coefficient of Range=

𝐿+𝑆

Example:
Find the value of range and it‟s co-efficient for the following data
7, 9, 6, 8, 11, 10, 4
Solution L= 11, S= 4
Range= L – S = 11- 4 = 7
Coefficient of Range =

𝐿−𝑆
𝐿+𝑆
11−4

= 11+5
7

=15 = 0
...
2 THE VARIANCE
The variance of a set of values, which we denote by σ2, is defined as
σ2 = (𝑥 2 – 𝑋 )2
𝑛
Where x is the mean, n is the number of data values and x stands for each data
value in turn
...
Similarly
(X2 – X )2 means you subtract the mean from each data value, square and finally add up
the resulting values
...
11)

5
...
The standard deviation, unlike the
variance, will be measured in the same units as the original data
...
5820
Standard Deviation, S = √S2 = √7
...
75
Thus, the standard deviation of the number of orders received at the office of this
mail-order company during the past 50 days is 2
...

5
...
45

C
...
45
20

𝑥 100

= 12
...
6= 3
...
V
...
69
15

𝑥 100

= 24
...


31

CHAPTER SIX
REGRESSION AND CORRELATION:
6
...

Types of regression
Regression analysis can be classified into
 Simple and Multiple
Considers only two variables e
...

 Total and Partial
Total relationship considers all the important variables, while in partial not
all but a few
...
(2)
a, b are constants
...
The two regression lines show the average relationship
between the two variables
...
2 SCATTER DIAGRAMS
In regression, a line is drawn between these points either by free hand or scale rule
in such a way that the squares of the vertical or the horizontal distances between the
points and the line of regression is the least
...
The line
has the following rules
 The algebraic sum of deviation in the individual observations with reference
to the regression line may be equal to zero
...

6
...
Regression equations through regression coefficient
2
...
4
...

The two regression equations
33

for X on Y; X = a + bY
And for Y on X; Y = a + bX
Where X, Y are variables, and a, b are constants whose
Values are to be determined
For the equation, X = a + bY
The normal equations are
X = na + b Y and
XY = aY + bY2
For the equation, Y= a + bX, the normal equations are
Y = na + b X and
XY = aX + bX2
From these normal equations the values of a and b can be
determined
...
(2)
Multiplying (1) by 6
240 = 30a + 180b……
...
65

Now, substituting the value of „b‟ in equation (1)
40 = 5a – 19
...
5
a = 59
...
9
Hence, required regression line Y on X is Y = 11
...
65 X
...
(3)
214 = 40a + 340b …
...
(5)
(4) – (5) gives
-26 = 20b
b = - 26
20
= - 1
...
3 in equation (3) gives
30 = 5a – 52
5a = 82
a = 82
5
= 16
...
4 – 1
...
4 CORRELATION AND CORRELATION COEFFICIENT
CORRELATION
Correlation refers to the relationship of two variables or more
...

Uses of Correlation
 It is used in physical and social sciences
 It is useful for economists in the study of relationship between variables like
price, quantity e
...
c
 Useful in measuring the degree of relationship between variables like
demand and supply e
...
c
 Sampling error can be calculated
 It is the basis for the concept of regression
36

Scatter Diagrams
It is the simplest method of studying the relationship between two variables
diagrammatically
...
The direction of the dots shows the scatter
or concentration of various points
...

Types of Correlation:
Correlation is classified into various types
...

ii) Linear and non-linear
...

iv) Simple and Multiple
...
This measure is called the measure of correlation (or)
correlation coefficient and it is denoted by „ r‟
...

KARL PEARSON’ S COEFFICIENT OF CORRELATION:
Karl Pearson, a great biometrician and statistician, suggested a mathematical
method for measuring the magnitude of linear relationship between the two

37

variables
...
It is denoted by „ r‟
...
D of x and y
(ii) r=  xy
n σx σ y
(iii) r =  xy

√ x2
...
Simple formula is the third one
...

Steps:
1
...

2
...


X = x- x , Y = y- y
3
...

4
...
This is covariance
...
Substitute the values in the formula
...
√(y- y)2

(i)

n

n

The above formula is simplified as follows
r=  xy

√ x
...
Correlation coefficient lies between –1 and +1
2
...

3
...

4
...

5
...

6
...


rxy = ryx
EXAMPLE
Find Karl Pearson‟s coefficient of correlation from the following data between
height of father (x) and son (y)
...


39

6
...

This method is based on ranks
...
The
individuals in the group can be arranged in order and there on, obtaining for each
individual a number showing his/her rank in the group
...
It is defined as;-

r = 1 - 6D2
n3 – n

r = rank correlation coefficient

NB: some authors use the symbol  for rank correlation
...

n= number of pairs of observations
...
If r = +1 there is complete agreement in
order of ranks and the direction of ranks is also same
...

EXAMPLE
In an evaluation of answer script the following marks are awarded by the
examiners
...
357 = 0
...
643 shows fair in awarding marks in the sense that uniformity has arisen in
evaluating the answer scripts between the two examiners
...
0 INTRODUCTION TO PROBABILITY THEORY AND
DISTRIBUTIONS:
7
...
Probability is used in playing
games of chance such as card games, slot machines or Lotteries
...
Probability is the basis of inferential statistics for example, predictions are
based on probability and hypotheses are tested by using probability
...
It is a number lying between 0 and 1
...

TERMS USED IN PROBABILITY
Random experiment- This is an experiment that has two or more outcomes which vary in
unpredictable manner from trial to trial when conducted under uniform conditions
Sample Point- This is every indecomposable outcome of a random experiment
...
It is the set of all possible outcomes of an experiment
...
g when a coin is tossed, the sample space is (Head, Tail)
Event- An event is the result of a random experiment
...
E
...
2 DIFFERENT TYPES OF STATISTICAL EVENTS
An elemen tary even t
An el ement ar y event is ever y one of the elem ents which form s the sample
space
...

Compound Event
A compound event i s an y subset of the sample space
...

Sure Even t
The sure event, S, i s form ed b y all possi ble results of the sample space
...

I mpossible Event
The impossible event,

, does not have an elem ent
...

Disjoint or Mutuall y Exclusive Events
Two events, A and B, are disjoi nted or m utuall y exclusive when the y don´t
have an el ement i n common
...

Independent Events
Two events, A and B are independent if t he probabilit y of the succeeding event
is not affect ed b y the out come of the preceeding event
...

43

Dependen t Even ts
Two events, A and B are dependent if the probabilit y of the succeeding event is
affected b y the out come of the preceedi ng event
...

Complemen tary Event
The compl ement ar y event of A is another event that is realized when A is not
realized
...


For exampl e, the complement ar y event of obtai ning an even nu mber when
rolling a die is obt ai ning an odd number
...
3 PERMUTATIONS AND COMBINATIONS
Permutation is an arrangement where order is important
...
A seating arrangement is an example of a permutation because
the arrangement of the “n” objects is in a specific order
...

When the order does not matter, it is a combination, because you are only interested in the group
...
An arrangement where order is important is called a permutation
...

In how many ways can these offices be filled? 4 x 3 x 2 = 24
...

2
...

Example:
1
...
How many different
amounts can he get?

4C3 =

4P3

3!

=

4X3X2

= 24

3X2X1

=4

6

44

2
...
In how many ways 4 mangoes from the basket can be
selected?
Hint: n C r = n!
( n-r)! r!
EXERCISE

n= 10

r= 4

(A) Determine if the situation represents a permutation or a combination:
I
...
In how many ways can three student-council members be elected from five candidates?
III
...

IV
...

(B) Suppose a business owner has a choice of 5 locations in which to establish her business
...
How many different ways can she rank the 5 locations?

FACTORIAL RULE
A collection of n different items can be arranged in order n! different ways
...
)

PERMUTATIONS RULE (WHEN ITEMS ARE ALL DIFFERENT)
Requirements:
1
...
(This rule does not apply if some of the items are
identical to others
...
We select r of the n items (without replacement)
...
(The permutation of
ABC is different from CBA and is counted separately
...
There are n items available, and some items are identical to others
...
We select all of the n items (without replacement)
...
We consider rearrangements of distinct items to be different sequences
...
nk alike, the
number of permutations (or sequences) of all items selected without replacement is

n!
n1!
...
nk!
COMBINATIONS RULE
Requirements:
1
...

2
...

3
...
(The combination of ABC
is the same as CBA
...


46

7
...
The Normal distribution describes a special class of such distributions that are symmetric
and can be described by two parameters
(i)

µ = The mean of the distribution

(ii)

σ = The standard deviation of the distribution

Changing the values of µ and σ alter the positions and shapes of the distributions
...

The probability density of the Normal distribution is given by:-

F (x) = 1 exp −(x−µ)2/2σ2
σ √2π
CALCULATING PROBABILITIES FROM THE NORMAL DISTRIBUTION
For a discrete probability distribution we calculate the probability of being less than some value
x, i
...
P(X < x), by simply summing up the probabilities of the values less than x
...
e
...

Suppose Z ∼ N (0, 1), WHAT IS p (Z < 0)?

P ( Z < 0)

47

Symmetry ----- P (Z < 0) = 0
...
Probability tables are tables of
probabilities that have been calculated on a computer
...


EXAMPLE:
(a) An aptitude test was conducted for selecting officers to work in 4 banks from 1000 students
...
Assume normal distribution for
scores and find:(i)The number of candidates whose score exceed 58
Given N= 1000
μ = 42
σ = 24
X= 58
Z=?

58−42
Z value =

24

=

16
24

= 0
...
667 (Area between 0 and 0
...
2486
: Area above 0
...
5- 0
...
2514
; Number of students whose score exceed 58 = 0
...
4
= 251 students

0
...
5 BUSINESS APPLICATIONS TO PROBABILITY
 Probability theory is used in the calculation of long term gains and losses
...
It is also used to determine the possibility of financial
success of a new product considering competition, demand, market value
and manufacturing costs
...

The worst case scenario would contain some value from the lower end of
probability distribution, the likely scenario towards the middle and the best
case in the upper end of the scenario
 Risk Evaluation
For example a company considering entering a new business line
...
500,000 to break even, and their
probability distribution tells them that there is a 10% chance for the revenues
to be less
...
This will have
the business base its plans on the likely scenario but still be aware of the
alternative possibilities

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