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By Kayemba Joel

Statistics and Probability

January 2018

1
...
MA (Econ), BA (Econ)
1
...
It introduces students to basic statistical concepts and methods
used in social-economic analysis
...
2 Course Objectives
This course is aimed at systematically equipping students with analytical
skills in statistical methods and techniques in addressing various socialeconomic problems
...

 Competency in application of statistical theory to data collection,
analysis and interpretation
...

1
...

(ii)Design experiments, collect, organize, analyze and summarize data,
and possibly make reliable predictions or forecasts for future use
...


1
...
In-class exercises, take
home assignments and group exercises where possible will be given to
assess student learning outcomes
...
5 Course Assessment and Grading
There will be two written tests which will constitute 40%, while the final
examination will carry 60% of the overall course grade
...

% Mark

Letter

80-100
75-79
...
9
65-69
...
9
55-59
...
9
45-49
...
P)
5
...
5
4
...
5
3
...
5
2
...
5

Below 45

F

0
...
6 COURSE CONTENT
UNIT 1: Introduction
 Meaning of statistics
 Nature and sources of data in statistical analysis
 Sources of data
 Types of data
 Accuracy of statistical data
 Role of statistics in business management
UNIT 2: Sampling
 Basic Concepts
 Descriptive and Inferential Statistics
 Data Collection and Sampling Techniques
 Descriptive Statistics
UNIT 3: Summarizing and presentation of data (7 hrs)
 Frequency distributions
 Graphing frequency distributions
 Histograms
3

(3 hrs)

UNIT

UNIT

UNIT

UNIT

 Relative frequency distribution
 Cumulative frequency distributions and ogives
4: Measures of Central Tendency (4 hrs)
 Arithmetic mean
 Median
 Mode
 Geometric mean
 Harmonic mean
 Quartiles/ Deciles/ Percentiles
5: Measures of Variability (5 hrs)
 Range
 Semi-inter quartile range
 Percentile range
 Mean deviation
6: CORRELATION THEORY
 Correlation analysis
 Forms of correlation
 Value of correlation co-efficient
 Rank correlation coefficient
 Covariance
 Limitations of the theory of linear correlation
7: Moments of skewness and kurtosis (2 hrs)

UNIT 8: Elementary Probability theory (8 hrs)
 Probability rules
 Joint probability
 Conditional Probability
 Baye’s Theorem
 Permutations and Combinations
UNIT 9: Probability Distributions and Mathematical Expectation
(10 hrs)
 Discrete Random variables and Probability Distributions
 Binomial Distribution
 Continuous Random variables Probability Distributions
 Normal Distributions
UNIT 10: Hypothesis testing (6 hrs)
 Formulating Null and Alternative Hypothesis
 Type I and Type II errors
 Levels of significance
 One and Two tailed Hypothesis tests
 Degrees of freedom and the t-distribution
1
...
Newbold, P
...
Carlson Betty Throne, (2003),
Statistics for Business and Economics
...

2
...
H and Wannacoot, R
...
4th Edition, John Wiley and sons, Inc
...

3
...
R
...
J
...
A
...
Statistics
for Business and Economics
...

4
...
N
...
New Delhi:
Prentice Hall of India Private Limited
...
Murray
...
Spiegel: Theories and Problems of Statistics

5

UNIT 1: INTRODUCTION TO STATISTICS
Statistics is the science of uncertainty; therefore it does not deal with
questions like, what is? But instead what could be? , What might be,
what probability is?
1
...
The
success of any data analysis ultimately depends on the availability of
appropriate data
...
2 TYPES OF DATA
(i) Time Series data
(ii) Cross Sectional data
(iii)
Panel data
(i)
TIME SERIES DATA
This is a set of observations on the values that the variable takes at
different times from period to period
...

(ii)
CROSS-SECTIONAL DATA
This is the data on one or more variables collected at the same point
of time e
...
census of population, consumer expenditure etc
...
They record the behavior of the
same set of individual micro-economic unit
...
g
...
3 SOURCES OF DATA
These include libraries of government agencies, Internet, Bank of
Uganda, ministry of finance, international agencies, World Bank, IMF,
UBOS
...

1
...
Errors of omission and commission
2
...
g
...

3
...
The researcher is likely to get 30% response
...
The sampling method used in obtaining data may not be good
...
Statistical data is generally available at high aggregate data
...
Such data may
not tell you much about individual level or micro level
...

1
...
This makes
statistics to be of considerable importance to decision makers in the
following ways:1
...

Statistical information enables the finance manger to find out the cash
flow potentials of alternative investments over a given period of time
so as to make viable investment decisions
...
Statistics enables decision makers to come up with macroeconomic
policies
...

3
...

7

4
...


6
...


8
...
But if these figures are
presented in a table, pie chart, graphs etc, one can draw conclusions
from them
...

It acts as an important tool in quality control
...

Statistics can be used as a basis for comparison amongst various
phenomenons under study
...

This is because it enables marketing managers to collect information
about customer’s tastes, competitor’s action, disposable incomes, and
population structures etc all of which are important when designing
marketing strategies
...


9
...

10
...
It can present an
active picture of the phenomenon under investigation and it enables
the interpretation of the condition by developing possible causes of
the described results
...
e determining what is likely to happen-the likely
outcomes
...

Each of the above (i) and (ii) involves sampling
...
g
...
e
...
She would therefore opt for a subset (a sample) of the
population
...

8

1
...


3
...


2
...

I
...
it is an observed subset of the population
...
e
...

Parameter & statistic
...

 Statistic:
This is a descriptive measure computed from sample data
...
g
...

The objective of statistics is to make inference (i
...
predictions and
decisions) about the population based on the sample collected
...

(ii)
The design of the experiment
(iii) Collection and analysis of data
(iv) The procedure for making predictions and decisions about the
population based on the sample
...

All these operations must make predictions and decisions about the
population based on the sample collected
...
The experiment
may select a sample from a wrong population or collect data wrongly,
and therefore the data may become difficult or impossible to analyze or
it may comply little or no pertinent information and the sample may not
represent the population of interest
...

2
...
Therefore
descriptive statistics consists of data collection, organization, summation
and presentation of data
...
Inferential statistics uses probability i
...
the chance of an
event occurring
...
5 DATA COLLECTION AND SAMPLING TECHINQUES
Data is organized information
...
Data can be collected in
a variety of ways
...
The surveys can be done by using a variety of methods
namely: - Telephone surveys,
- Mailed questionnaire surveys,
- Personal interview surveys
...
e
...

This method of sampling is also known as unrestricted random
sampling or chance sampling
...

(2) Stratified sampling method
...
For example, suppose the
headmaster of a given school wants to learn how students feel about
their teachers, the headmaster would select the student from each
group to use in the sample
...

This is a sampling method that involves selection of sampling units at
equal intervals after all these sampling units have been arranged in
10

some order
...
g
...

In this case, cluster samples are selected by using intact groups called
clusters
...
E
...
suppose a
researcher wishes to survey slums in Kampala city, if there are 10
slums in the city, the researcher can select at random 3 slums from
10 slums and interviews all the residents of these slums
...
6 DESCRIPTIVE STATISTICS
This is a branch of mathematics dealing with collection, interpretation,
presentation and analysis of data
...

Statistical data
(a) Discrete and continuous data
Information collected by counting is discrete and usually takes
integral values e
...
number of students in a class, school etc
...
g
...

(b) Crude and classified data
Crude data are individual values of a variable in no particular order of
magnitude, written down as they occurred or as they were measured
...

(c)Populations and samples
A population is the total set of items under consideration and is
defined by some characteristics of those items
...

UNIT 3: SUMMARIZING AND PRESENTATION OF DATA
The various ways of representing data include:(i)
Bar graphs
(ii)
Histogram
(iii) Frequency Polygon
(iv) The ogive(cumulative frequency curve)
3
...

11

3
...

Example 1
The table below shows the weights of some freshers in 2015/2016
academic year who underwent medical examination at the sick bay
...


Weights(Kgs) No
...
5-44
...
5-49
...
5-54
...
5-59
...
5-64
...
5-69
...
5-74
...
5-79
...
5-84
...

NB: Additional classes are added to either ends in order to end with zero
frequency to make the polygon a closed figure
...

Heights
in Number
of
cm
prisoners
150-154
3
155-159
7
160-164
10
165-169
15
170-174
25
175-179
12
180-184
6
185-189
2
Question;
Construct a frequency polygon for the above data
...

This is known as super imposing a frequency polygon
...

Class limits Class
Class Frequency
Class
boundaries
marks
145-149
144
...
5
150-154
149
...
5
155-159
154
...
5
160-164
159
...
5
165-169
164
...
5
Σf=80
170-174
175-179
180-184
185-189
190-194

169
...
5
174
...
5
179
...
5
184
...
5
189
...
5

25

172

12

177

6

182

2

187

0

192

Superimposing a frequency polygon on a histogram
15

3
...

Example
...

Weight
Number of children
10-14

5

15-19

9

20-24

12

25-29

18

30-34

25

35-39

15

40-44

10

45-49

6

Question
Draw a cumulative frequency curve (Ogive) for the data given above
...
5-9
...
5-14
...
5-19
...
5-24
...
5-29
...
5-34
...
5-39
...
5-44
...
5-49
...
4 ADVANTAGES AND DISADVANTAGES OF HISTOGRAMS
Histograms are frequently used to display grouped frequency
distributions graphically
...
They display clearly the comparative frequency of occurrence of data
items within classes
...

2
...
The wider the histogram, the larger the range of values it’s
presenting and the narrower the histogram, the smaller is the range
of the corresponding values
...
They indicate the concentration of the values in the distribution (i
...

their skeweness) e
...

17

(i) Skewed to the left

(ii)Skewed to the right
...

1
...
This is because the
sample data may not be an exact representation of the population as a
whole
...
Histograms in which the class widths vary may not be readily understood
by the lay man
...
5 MORE PRESENTATION OF DATA
Frequency Distributions
Frequency distribution is a table that lists a set of scores and their
frequency (how many times each score occurs)
...

18

Example
Consider the following set of data which represents high temperatures
recorded for 30 consecutive days
...

STEPS
- First identify the highest and lowest values(i
...
51 & 43)
- Then create the various columns (i
...
tempreture,tally and frequency)
Solution
Frequency distribution for high temperatures
Temperature
Tally
Frequency
43
3
44
3
45
4
46
3
47
3
48
0
49
6
50
4
51
4
Σf=30
Cumulative Frequency distribution
It can be created from a frequency distribution by adding an additional
column called “cumulative frequency”
...
5 GROUPED FREQUENCY DISTRIBUTION
In some cases, it is necessary to group the value of the data to
summarize the data properly
...

Example
Consider the following data of high temperatures for 50 days
...

NB: In this case, the highest temperature is 59 and the lowest
temperature is 39
...
e
...

This is greater than 20 values, so we should create a grouped frequency
distribution
...

Each interval must be the same size and they must not overlap
...
e
...

20

Solution;
Grouped frequency distribution for high temperatures
Interval
Class
Tally
mid point Frequency
interval
39-41
40
4
42-44

43

6

45-47

46

7

48-50

49

9

51-53

52

11

54-56

55

7

57-59

58

6
N=50

Cumulative grouped frequency distribution
In this case, we add a cumulative frequency column to the grouped
frequency distribution
...

Measures of central tendency are also referred to as measures of central
location
...
1 MEAN/AVERAGE/ARITHMETIC MEAN
This is also known as average and it is given by:
Sum of a set of numeric observations
Number of observations
e
...

Given a population of N- observations
i
...
x1,x2,…
...

1
...
5
Also let x1, x2,…
...
+ xn

n

Example
22

Given the following percentage earnings per share ,13
...
5, 43
...
8, -13
...
3, 36
...
3
...

Solution:
N

X=  X i
i 1

n

=

13
...
5+ 43
...
8+ -13
...
3+ 36
...
3
8

= 112
8
X =14

Properties of Arithmetic Mean
...
It
has several important properties
...
Every set of interval-level and ratio level data has a mean
...
All the values are included in computing the mean
...
A set of data has only one mean, implying that the mean is unique
...
The mean is a useful measure for comparing two or more populations
...
g
...

5
...
e
...
2 MEDIAN
This is the middle value when the data is arranged in a specified order
(ascending or descending)
...
I
...
if there are N-observations arranged in increasing order, the
median is: -

(

)

th

observation, when N is odd
...


23

Where,
N= the total number of observations in a given distribution
...

Find the median of the following data
...
10, 11, 12, 13, 14, 15, 16
b
...
(a)
From

(

)

th

observation
...


th

2
2
th
th
4 + 5 observations
2
13+14
2
= 13
...
5
Question
...
86, 78, 75, 72,
65, 62, 70, 69, 74 and 68 find the medium weight
...
3 MODE
The mode of a set of observations is the value that occurs most
frequently
...

The radios were checked and the number of defects was recorded as
follows
...

Modal number of defects is 2
...

1
...
The points awarded to 10 contestants were recorded as follows
...
4 GEOMETRIC MEAN
This is usually used to measure growth rates
...
For the grouped data, x1,x2,…
...
M = antilog

N

  log xi
i 1

n
OR
N

G
...

Given the following observations, compute the geometric mean
...

N

From G
...
699
0
...
845
1
...
301
0

...
878
∑ log xi= 3
...
878 ( antilog)
6
=0
...
43
Question
...


UNIT 5: MEASURE OF DISPERSION/VARIABILITY
The measures of central tendency do not provide an adequate summary
or characteristics of a set of data
...
The measures of dispersion include
...
1 VARIANCE AND STANDARD DEVIATION
Let x1, x2,…
...
e
...
D=√variance
S
...
2

MEAN ABSOLUTE DEVIATION (MAD)
26

Let X1, X2---XN denote members of the population whose mean is m ,then
the MAD is given by:
MAD =  | Xi - m |
N
Absolute deviations are those differences without regard to the signs
i
...
All the differences are made positive
...
3 THE RANGE
The range of a set of data is the difference between the largest and the
smallest observations e
...

Given the following observations;
24500, 20700, 22900, 26000, 24100, 23800, 22500
Find the following;(i)The variance
(ii)The standard deviation
(iii)The MAD
(iv)The range
Solution:
From, m 2 = ∑ (Xi- m )2
N

Xi

m

24500
20700
22900
26000
24100
23800
22500
∑Xi
164,500

23500
23500
23500
23500
23500
23500
23500

(i)

Where m = ∑Xi
N
(Xi- m )
1000
-2800
-600
2500
600
300
-1000

(Xi- m )2

|xi- m |

1000,000
7840000
360,000
6,250,000
360,000
90,000
1,000,000
∑(xi- m )2

1000
2800
600
2500
600
300
1000
∑|xi- m |

=16,900,000 =8800
Variance
From m 2 = ∑(Xi- m )2
27

N
m 2 =16,900,000
7
Variance = 2,414,286
(i)

Standard Deviation
S
...
D= 2, 414, 286

(ii)

S
...
14

(iii)

Range
26,000-20,700
Range = 5,300

Question: 1
Given the following observations of assessment rates of a population in
percentages
21, 22, 27, 36, 22, 29, 22, 23, 22, 28, 36, 33, 30, 33
Find the following:(a)The mean assessment rate
(b)The median
(c)The mode
(d)The variance
(e)The standard deviation
(f)The mean absolute deviation
(g)The range
Question: 2
The weights in Kgs of students selected from a class were recorded
as follows:58, 54, 65, 84, 82, 69, 57, 82, 52 and 60
Find the figure:
(a) The mean weight
(b) The median
(c) The mode
28

(d)
(e)
(f)
(g)

The
The
The
The

variance
standard deviation
MAD
range

UNIT 6: CORRELATION THEORY
There are various methods of measuring relationships existing between
economic variables
...

6
...
The degree of relationship between two variables is
called Simple correlation
...

6
...
Linear correlation:
This is basically when points on a scattered diagram are clustered
along the line
...

a)
Positive linear correlation

b)

Negative linear correlation

29

2
...
Zero-correlation

Here, there is no relationship between X and Y i
...
The two variables are
un-correlated when they tend to change with no correlation to each
other
...
If all points
lie on the line (curve), the correlation is said to be perfect
...

For instance how poverty rates behave as growth rates change how
growth rates change as inflation rates change etc
Correlation theory involves looking at covariance and correlation analysis
...
This yields a linear
correlation coefficient (Pearson r)
...

Use the data in the previous example to calculate the linear correlation
coefficient
...

From
31

-90
-30
0
-25
-70
Σxi,yi=215

rxy

= Σxi,yi
√Σxi2 Σyi2
=-125
√250X200
= -0
...
If r is
positive, x and y increase or decrease together
...
5 VALUE OF CORRELATION CO-EFFICIENT
The correlation coefficient is the measure of the degree of covariability of
the variables X and Y
...
If r = +1, it implies perfect positive correlation between X and
Y
...
When r = 1, it implies perfect negative correlation between X and Y
...
i
...
0
< r <1
...

6
...
g
...

We assign ranks to the data and measure the relationship between their
ranks instead of their actual numerical values
...
e
...

Thus,
r1= 1 - 6ΣD2
n(n2-1)
Where:
D – is the difference between the ranks of the corresponding pairs of the
observation
...
e
...

NB:
It does not matter whether observations are arranged in ascending or
descending order so long as you use the same rules of ranking for both
observations
...
We would like to find
out whether there is a relationship between the accomplishment of the
students during the whole semester and their performance in the exams
...
24
10(102-1)
= 1- 144
990
= 0
...


Question
...
Basing on this information, compute the
spearman’s correlation co-efficient
...
7 COVARIANCE
This is the measure of linear relationship between two paired variables
...
The positive
value indicates an increasing direction/relationship, while a negative
value indicates a decreasing linear relationship
...

The population covariance is give by:

Sxy = ∑xi yi
N
The sample covariance is given by:

Sxy = ∑xi yi
n-1
Where xi = X-X
Where X =∑xi
N
And yi = Yi-Y
Where Y = ∑yi
N

Example:
The table below provides information of smoking and savings from 5
computer science students
...

Solution:
Number
of Saving($)
cigarettes(X)
month (Y)
per day
0
45

per xi=Xi-X

yi=Yi-Y

xiyi

-10

9

-90

5

42

-5

6

-30

10

33

0

-3

0

15

31

5

-5

-25

20

29

10

-7

-70

ΣXi=50

ΣYi=180

Σxi=0

Σyi=0

Σxi,yi=-215

From
Y = ∑yi
n
= 180
5
Y = 36

X = ∑xi
n
= 50
5
X = 10
The population covariance
∑xy = ∑xi yi
N
=-215
5
= - 43
Sample covariance
From; Sxy = ∑xi yi
n-1
=-215

35

5-1
= -215
4
= -53
...
e
...
Generally
variables can have positive, negative and zero co-variance
...
8 LIMITATIONS OF THE THEORY OF LINEAR CORRELATION
Correlation analysis has serious limitations as a technique of the study of
economic relationships
...

The formular “r” applies only when the relationship is linear
...

ii
...

iii
...
e
...
It only assumes one direction i
...
X influences Y
...


UNIT 7: PROBABILITY THEORY
Although it is not possible from the basis of a sample to derive certain
knowledge about the population, it may be possible to make precise
decision about the nature of uncertainty
...


36

Probability is the measure of occurrence of an event or outcome
...

Examples include: Tossing a coin (two outcomes)
Throwing a dice (six outcomes)
That a consumer is asked which of the two products he
or she prefers (two outcomes), etc
...
Therefore, a random
experiment is a process leading to atleast two possible outcomes with
uncertainty as to which will occur
...
g
...
If a dice is rolled, the result will be
one of the numbers (1, 2, 3, 4, 5, 6)
...
g
...

Sample Point
This is an element of a sample space
...
g
...
e
...
E
...
from tossing a dice, if A = even number appears, then A =
{2, 4, 6}; if B = at least 4 appears, then B = {4, 5, 6}
...
i
...
If A and B are two events, then their
intersection is a set of all basic outcomes in the sample space that
belong to both A and B
...


37

It’s possible that events A and B have no common basic outcomes
...
i
...


If sets A and B have no common basic outcomes, they are called
mutually exclusive events
...
e
...
Then is follows that A∩B cannot occur
...
For example, in a dice throw experiment, the outcomes {2, 4, 5,
6}, all belong to at least one of the events “Even number results” or
“number of at least four results”
...
A =Even number results i
...
A= {2,
4, 6} B number of at least four results
...
e
...
It shows all elements belonging A or B or both
...
IF E1∪E2∪Ek=S,
these events are said to be collectively exhaustive
...
Therefore A and A1 are
mutually exclusive
...
e
...
e
...

Find the following
...
A1 and B1
b
...
A∪B
d
...
A∪A1
Solution;
From:
S= {1, 2, 3, 4, 5, 6}
A= {2, 4, 6}
B= {4, 5, 6}
a
...
A∩B ={4,6}
c
...
A∩A1 = {}
e
...

E
...
whether it rains in Luweero or not, it does not affect the flights at
Entebbe airport
Conditional event
Events are said to be conditional if for one of them to occur, another one
must have occurred already
...
g
...
Conditional events are written as E1/E2, which refers to the
occurrence of event E2 given that event E1 has already occurred
...
e
...

(2) (a) P(S) =1
...
e
...

(b) P( ) =0
...
e
...

(3) ) P (A) + P (A1) =1 i
...
If A is an element in the sample space,
then A and A1 are complementary events
...

NB;
The third law/rule holds, if A and A1 are mutually exclusive
The table below shows the probabilities on the number of computer
system failure in a week
...
31
0
...
20
0
...
02

1) What is the probability that there will be less than two failures in a
particular week?
2) What is the probability that there will be more than two failures is
a particular week?
3) What is the probability that there will be atleast one failure in a
particular week?
40

4) The five probabilities of a table sum up to one (1), why must this
be so?
Solution:
1
...
31 +0
...
69
2
...
09+0
...
11
3
...
38+0
...
09+0
...
69
4
...
e
...

Example 2
A corporation takes delivery to some new machinery that must be
installed and checked before it becomes operational
...

Number
3
of days
Probability 0
...
24

0
...
20

0
...

Find the following
...
41+0
...
07
P(A) =0
...
08+0
...
41
=0
...
68
=0
...
41

(e)

P(A∪B)
A∪B={3,4,5,6,7}
P(A∪B)=P(3)+P(4)+P(5)+P(6)+P(7)
=0
...
24+0
...
20+0
...

(f) Since A∪B ={3,4,5,6,7} i
...
the union of events form a whole sample
space, then events A and B are collectively exhaustive
...
The table below shows the
42

probability of delays
...
”
Delay
in None
days
Probability 0
...
19

0
...
12

0
...
09

Find
...
A1,A2_ _ _ _
_An, i
...
(Ai) and B1, B2,_ _ _ _ _Bn, i
...
(Bi)
...
e
...
Then these two events are said to be BIVARIATE and the
probabilities are referred to as Bivariate probabilities
...
The probabilities for individual events P (Ai) and P (Bi)
are called joint marginal probabilities
...
04
Occasionally 0
...
12
Totals
0
...
13
0
...
17
0
...
04
0
...
22
0
...
21
0
...
51
1

The table shows that 10% of the families have high incomes and
occasionally watch TV series
...
e
...
10
...
17

P (Occasionally watch) =0
...
11+0
...
27
P (Regularly watch) =0
...
13+0
...
21
P (high Y) =0
...
41
P (low Y) =0
...
11
0
...
27
P

(middle Y) =
Occasionally watch

=

P(occasionally watch ∩ middle Y)
P(middle Y)

P (occasionally ∩ middle Y)
P(occasionally watch)

= 0
...
27
= 0
...

44

Visits

Purchase of Plastic products

Frequent
Often
Infrequency 0
...
06

Sometimes Never

Total

0
...
12

0
...
08

0
...
21

a) What is the probability that the customer is both a frequent
shopper and often purchases plastic products?
b) What is the probability that a customer who never buys plastic
products visits the store frequently?
c) What is the probability that the customer who frequently visits the
store often buys the plastic products?
d) What is the probability that a customer frequently visits the store?
e) What is the probability that a customer never buys plastic products?
CONDITIONAL PROBABILITY OF JOINT RANDOM VARIABLE
Let X and Y be a pair of jointly distributed random variables, then;
= Pxy (X,Y)
PX(X)
Similarly the conditional probability function of X/Y is given by:= Pxy(X,Y)
Py(Y)
Example;
A super market manager is interested in the relationship between the
number of times a sales item is advertised in local newspaper during the
week and the volume of demand for the item
...
09
0
...
04
0
...
15
0
...
10
0
...
06
0
...
10
0
...
3
0
...
24

i
...


X=0 given Y=3
X=1 given Y=3

b) Find the conditional probability of:i
...

Y=3 given X=0

Solution:
a
...
e
...
P

x/y=

(ii)
...
17

x/y

(1/3) = P(X =1 ∩ Y=3)
P(Y=3)

=0
...
24

=0
...
24

=0
...
Conditional probability of Y=2 given x=0
(i)
...
P

y/x

y/x

(2/0) = P(X =0 ∩ Y=2)
P(X=0)
=0
...
04
0
...
04
0
...
21

Revision question:
From the previous table (table above),
Find the conditional probability of: a) X=1 given Y=2
b) X=2 given Y=2
c) X=0 given Y=1
d) Y=3 given X=1
e) Y=1 given X=2
f) Y=2 given X=2

Some definition
46

Sampling Distribution
This is the probability distribution of statistic
...

Statistical Inference
This refers to conclusions reached after analyzing data and therefore
making complete information of the data
...
And in this context, data refer to
facts and figures collected for a purpose
...
Estimation
b
...

ESTIMATION
We use sample statistics to obtain information about population
parameters
...
As the name suggest, we just try to
estimate these parameters but we do not come with exact values
...
Point estimation
2
...
POINT ESTIMATION

This is a process where by we use a single value of the sample statistic
to represent the population parameter of interest
...

E
...
if X=1,2,3,4
then X =Σx
n
= 1+2+3+4
4
=10
4
=2
...
5 is the estimate
...

Properties of a good estimator
 It should be unbiased
...
e
...

 It should be consistent
...

Interval Estimation
In interval estimation, we choose two limits A and B within which the two
parameters must lie
...
g
...

Confidence coefficient
This is the probability that the true population parameter will lie within
the stated interval denoted by (1-∝)
...
The
hypothesis is statistical information made concerning the population
...
This is the hypothesis made by
manufacturer about the life of his car
...

Alternative hypothesis (H1)
This is the opposite of null hypothesis and it is categorized as one-tailed
and two-tailed alternative
...

If the observed results from the sample differ significantly from the
expected results according to H0, we say that the observed difference is
significant and then we would reject H0, otherwise we would accept Ho
...
The procedures to enable us accept or reject Ho
48

are referred to as tests of hypothesis, tests of significance or rules of
decision
...
The probability of committing type I error
is the level of significance (∝)
...
Unless we have a specific alternative hypothesis,
type II error will be impossible to compute
...
If 5% level of
significance is chosen in designing the test of hypothesis, then there are
5 chances in every 100 that we will reject Ho when it is true
...
e
...

One tailed and two tailed tests
One-Tailed test
...
In this case, the critical region lies to the side of only one
tail of the bell shaped curve
...
e
...

A statistical hypothesis test where alternative is two sided is called a two
tailed test
...
i
...


49

Steps
involved
in
setting
1
...
Set the alternative hypothesis (H1)
3
...
State the critical region (Zc)

attest

of

hypothesis

i
...
Perform the necessary computation and hence obtain the value of
the statistic
2
...


50



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