Biostatistics ppt.pdf

University of Gondar
College of medicine and health science
Department of Epidemiology and Biostatistics
Basic Biostatistics
Wullo S. (BSc, MPH, Assistant professor)
Tuesday, December 26, 2023 1

Chapter One
1.1 Introduction to Biostatistics
❖ Objectives of the chapter
➢ After completing this chapter, the student will be able to:
– Define Statistics and Biostatistics
– Identify the Branch of biostatistics
– Enumerate the importance and limitations of biostatistics
– Define and Identify the different types of data and
understand why we need to classify variables
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Definition and classification of Biostatistics
 Statistics is the science of
 collecting
 organizing
 Presenting
 analysing and drawing conclusion (inferences) from
data for the purpose of making decision.
❖ Biostatistics: The application of statistical methods
to the fields of biological and health sciences.
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Classification of Biostatistics
Descriptive biostatistics
❖ A statistical method that is concerned with the collection,
organization, summarization, and analysis of data from a
sample of population.
Inferential biostatistics
❖ A statistical method that is concerned with the drawing
conclusions/inference about a particular population by
selecting and measuring a random sample from the population.
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Cont…
collection
organizing
summarizing
presenting of data
Descriptive Statistics
making inferences
hypothesis testing
determining relationship
making the prediction
Inferential Statistics
Biostatistics
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Descriptive Biostatistics
• Some statistical summaries which are especially common in
descriptive analyses are:
✓ Measures of central tendency
✓ Measures of dispersion
✓ Measures of association
✓ Cross-tabulation, contingency table
✓ Histogram
✓ Quantile, Q-Q plot
✓ Scatter plot
✓ Box plot
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Inferential Biostatistics
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1.2 Stages in statistical investigation
There are five stages or steps in any statistical investigation.
1. Collection of data
 The process of obtaining measurements or counts.
2. Organization of data
Includes editing, classifying, and tabulating the data
collected.
3. Presentation of data:
overall view of what the data actually looks like.
facilitate further statistical analysis.
Can be done in the form of tables and graphs or diagrams.
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Cont…
4. Analysis of data
 To dig out useful information for decision making
 It involves extracting relevant information from the data
(like mean, median, mode, range, variance…),
5. Interpretation of data
 Concerned with drawing conclusions from the data
collected and analyzed; and giving meaning to analysis
results.
 A difficult task and requires a high degree of skill and
experience.
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1.3 Definition of Some Basic terms
Population: is the complete set of possible measurements for which
inferences are to be made.
Census: a complete enumeration of the population. But in most real
problems it cannot be realized, hence we take sample.
Sample: A sample from a population is the set of measurements that are
actually collected in the course of an investigation.
Parameter: Characteristic or measure obtained from a population.
Statistic: A statistic (rather than the filed of Statistics) refers to a
numerical quantity computed from sample data (e.g. the mean, the
median, the maximum). 10
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Parameter and statistic
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Cont...
Sampling: The process or method of sample selection from the
population.
Sample size: The number of elements or observation to be
included in the sample.
variable is a characteristic or attribute that can assume different
values in different persons, places, or things.
Some examples of variables include:
▪ Diastolic blood pressure,
▪ heart rate, heights,
▪ The weights
Data: Refers to a collection of facts, values, observations, or
measurements that the variables can assume.
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Uses of statistics:
The main function of statistics is to enlarge our knowledge of
complex phenomena. The following are some uses of statistics:
▪ Estimating unknown population characteristics.
▪ Testing and formulating of hypothesis.
▪ Studying the relationship between two or more variable.
▪ Forecasting future events.
▪ Measuring the magnitude of variations in data.
▪ Furnishes a technique of comparison.
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Limitations of statistics
As a science statistics has its own limitations. The following are
some of the limitations:
▪ Deals with only quantitative information.
▪ Deals with only aggregate of facts and not with individual data
items.
▪ Statistical data are only approximately and not mathematical
correct.
▪ Statistics can be easily misused and therefore should be used
be experts.
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1.5 Types of Variables and Measurement Scales
A variable is a characteristic or attribute that can assume
different values in different persons, places, or things.
Examples :
▪ age,
▪ diastolic blood pressure,
▪ heart rate,
▪ the height of adult males,
▪ the weights of preschool children,
▪ gender of Biostatistics students,
▪ marital status of instructors at University of Gondar,
▪ ethnic group of patients
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A. Depending on the characteristic of the measurement, variable can be:
❖ Qualitative(Categorical) variable
✓ A variable or characteristic which cannot be measured in quantitative
form but can only be identified by name or categories,
✓ for instance place of birth, ethnic group, type of drug, stages of
breast cancer (I, II, III, or IV), degree of pain (minimal, moderate,
sever or unbearable).
✓ The categories should be clear cut, not overlapping, and cover all the
possibilities. For example, sex (male or female), vital status (alive or
dead), disease stage (depends on disease), ever smoked (yes or no).
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Quantitative(Numerical) variable:
➢ is one that can be measured and expressed numerically.
Example: survival time, systolic blood pressure, number of
children in a family, height, age, body mass index.
➢ they can be of two types
Discrete Variables
✓ Have a set of possible values that is either finite or countabl
infinite.
✓ The values of a discrete variable are usually whole
numbers.
✓ Numerical discrete data occur when the observations are
integers that correspond with a count of some sort.
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Some common examples are:
▪ Number of pregnancies,
▪ The number of bacteria colonies on a plate,
▪ The number of cells within a prescribed area upon
microscopic examination,
▪ The number of heart beats within a specified time interval,
▪ A mother’s history of numbers of births ( parity) and
pregnancies
▪ The number of episode of illness a patient experiences
during some time period, etc.
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Continuous Variables
✓ A continuous variable has a set of possible values including
all values in an interval of the real line.
✓ No gaps between possible values.
✓ Each observation theoretically falls somewhere along a
continuum.
Example: body mass index, height, blood pressure, serum
cholesterol level, weight, age etc.
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Con…
✓ Observations are not restricted to take on certain numerical
values: Often measurements (e.g., height, weight, age).
✓ Continuous data are used to report a measurement of the
individual that can take on any value within an acceptable
range.
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Level of measurement which classifies data into mutually exclusive, all
inclusive categories in which no order or ranking can be imposed on
the data.
▪ Assign subjects to groups or categories
▪ No order or distance relationship
▪ No arithmetic origin
▪ Only count numbers in categories
▪ Only present percentages of categories
▪ Chi-square most often used test of statistical significance
Nominal Scale
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Sex Social status
Marital status Days of the week (months)
Geographic location Seasons
Ethnic group Types of restaurants
Brand choice Religion
Job type : executive, technical, clerical
Other Examples
Coded as “0” Coded as “1”
Nominal Scale
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▪Classifies data according to some order or rank
▪With ordinal data, it is fair to say that one
response is greater or less than another.
▪E.g. if people were asked to rate the hotness of 3 chili
peppers, a scale of "hot", "hotter" and "hottest"
could be used. Values of "1" for "hot", "2" for
"hotter" and "3" for "hottest" could be assigned.
Ordinal Scale
Level of measurement which classifies data into categories that can be
ranked. Differences between the ranks do not exist.
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Ordinal Scales
• Arithmetic operations are not applicable but relational
operations are applicable.
• Ordering is the sole property of ordinal scale.
Examples:
Letter grades (A, B, C, D, F).
Rating scales (Excellent, Very good, Good, Fair, poor).
Military status.
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Interval Scales
• Level of measurement which classifies data that can be ranked
and differences are meaningful. However, there is no meaningful
zero, so ratios are meaningless.
• All arithmetic operations except division are applicable.
• Relational operations are also possible.
Examples:
IQ
Temperature in oF.
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▪assumes that the measurements are made in equal units.
▪i.e. gaps between whole numbers on the scale are equal.
▪e.g. Fahrenheit and Celsius temperature scales
▪an interval scale does not have a true zero.
▪e.g. A temperature of "zero" does not mean that
there is no temperature...it is just an arbitrary zero
point.
▪permissible statistics: count/frequencies, mode, median,
mean, standard deviation
Interval Scale
Numerically equal distances on the scale represent equal values in
the characteristic being measured. An interval scale contains all the
information of an ordinal scale, but it also allows you to compare the
differences between objects.
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Ratio Scales
• Level of measurement which classifies data that can be ranked,
differences are meaningful, and there is a true zero. True ratios
exist between the different units of measure.
• All arithmetic and relational operations are applicable.
Examples: Weight
Height
Number of students
Age
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Primary Scales of Measurement
4 81 9
Nominal Numbers
assigned to
runners
Ordinal Rank order of
winners
Third
Place
Second
Place
First
Place
Interval Performance
rating on a 0 to
10 Scale
8.2 9.1 9.6
Ratio Time to finish in
20 seconds 15.2 14.1 13.4
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STATISTICS
SCALE DESCRIPTIVE INFERENTIAL
Nominal Percentages, Mode Chi-square, Binomial test
Ordinal Percentile, Median Rank-order, Correlation,
ANOVA
Interval Range, Mean, SD Correlations, t-tests, ANOVA
Regression, Factor Analysis
Ratio Geometric Mean, Coefficient of Variation (CV)
Harmonic Mean
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Excercise
Categorize the following variables into nominal, ordinal,
interval or ratio
➢ Gender
➢ Grade(A, B, C, D and F )
➢ Rating scale(poor, good, excelent)
➢ Eye colour
➢ Political affilation
➢ Religious affilation
➢ Ranking of tennis players
➢ Majour field
➢ Nationality
30
➢Height
➢Weight
➢Time
➢Age
➢IQ
➢Temprature
➢Salary
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Chapter 2
Organization and Presentation of data
• Having collected and edited the data, the next important step
is to organize it.
• The process of arranging data in to classes or categories
according to similarities is called classification
• Classification is a preliminary and it prepares the ground for
proper presentation of data.
• The presentation of data is broadly classified in to the
following two categories:
• Tabular presentation
• Diagrammatic and Graphic presentation.
Tuesday, December 26, 2023 Wullo S. 31

Tabular presentation of data
• Frequency distribution: is the organization of raw
data in table form using classes and frequencies.
• There are three basic types of frequency
distributions
❖ Categorical frequency distribution
❖Ungrouped frequency distribution
❖Grouped frequency distribution

Categorical frequency Distribution:
Used for data that can be place in specific categories such
as nominal, or ordinal.
Example1: a social worker collected the following data on
marital status for 25 persons.(M=married, S=single,
W=widowed, D=divorced)
M S D W D S S M M M W D S M M W D D S S S W W D D
Class (1) Frequency (3)
Percent (4) M 6
24 S 7
28 D 7
28 W 5
20

Example 2
• Consider for example, the variable birth weight with levels ‘Very
low ’, ‘Low’, ‘Normal’ and ‘Big’. The frequency distribution for
newborns is obtained simply by counting by the number of
newborns in each birth weight category.
Table 2. Distribution of birth weight of newborns between 1976-1996 at TAH.
BWT Freq. Rel.Freq(%) Cum. FreCum.rel.freq.(%)
Very low 43 0.4 43 0.4
Low 793 8.0 836 8.4
Normal 8870 88.9 9706 97.3
Big 268 2.7 9974 100
Total 9974 100

2. Ungrouped frequency Distribution
• Is a table of all the potential raw score values that could possible occur
in the data along with the number of times each actually occurred.
• Example: The following data represent the mark of 20 students. 80
,76, 90 ,85 ,80 ,70 ,60 ,62 ,70 ,85, 65 ,60 ,63 ,74 ,75 ,76 ,70 ,70 ,80, 85
Construct ungrouped frequency distribution
• Mark Frequency
60 2
62 1
63 1
65 1
70 4
74 1
75 2
76 1
80 3
85 3
90 1

3.Grouped frequency distribution
• When the range of the data is large, the data must be grouped
in to classes that are more than one unit in width.
Definitions of same terms:
• Grouped Frequency Distribution: a frequency distribution
when several numbers are grouped in one class.
• Class limits: Separates one class in a grouped frequency
distribution from another. The limits could actually appear in
the data and have gaps between the upper limits of one class
and lower limit of the next.
• Units of measurement (U): the distance between two possible
consecutive measures. It is usually taken as 1, 0.1, 0.01, 0.001,
-----.

Cont…
• Class boundaries: Separates one class in a grouped frequency
distribution from another. The boundaries have one more
decimal places than the row data and therefore do not appear
in the data. There is no gap between the upper boundary of
one class and lower boundary of the next class. The lower
class boundary is found by subtracting U/2 from the
corresponding lower class limit and the upper class boundary
is found by adding U/2 to the corresponding upper class limit.
• Class width: the difference between the upper and lower
class boundaries of any class. It is also the difference between
the lower limits of any two consecutive classes or the
difference between any two consecutive class marks.

Cont…
• Class mark (Mid points): it is the average of the lower and
upper class limits or the average of upper and lower class
boundary.
• Cumulative frequency: is the number of observations less
than/more than or equal to a specific value.
• Cumulative frequency above: it is the total frequency of all
values greater than or equal to the lower class boundary of a
given class.
• Cumulative frequency blow: it is the total frequency of all
values less than or equal to the upper class boundary of a
given class.

Cont…
• Cumulative Frequency Distribution (CFD): it is the
tabular arrangement of class interval together with
their corresponding cumulative frequencies. It can be
more than or less than type, depending on the type
of cumulative frequency used.
• Relative frequency (rf): it is the frequency divided by
the total frequency.
• Relative cumulative frequency (rcf): it is the
cumulative frequency divided by the total frequency.

Steps for constructing Grouped
frequency Distribution
1. Find the largest and smallest values
2. Compute the Range(R) = Maximum - Minimum
3. Select the number of classes desired, usually
between 5 and 20 or use Sturges rule
where k is number of classes desired and n is total
number of observation.
4. Find the class width by dividing the range by the
number of classes and rounding up, not off.

Cont…
5. Pick a suitable starting point less than or equal to the
minimum value. The starting point is called the lower limit of
the first class. Continue to add the class width to this lower
limit to get the rest of the lower limits.
6. To find the upper limit of the first class, subtract U from the
lower limit of the second class. Then continue to add the class
width to this upper limit to find the rest of the upper limits.
7. Find the boundaries by subtracting U/2 units from the lower
limits and adding U/2 units from the upper limits. The
boundaries are also half-way between the upper limit of one
class and the lower limit of the next class. !may not be
necessary to find the boundaries.

Cont…
8. Tally the data.
9. Find the frequencies.
10. Find the cumulative frequencies. Depending on
what you're trying to accomplish, it may not be
necessary to find the cumulative frequencies.
11. If necessary, find the relative frequencies and/or
relative cumulative frequencies

Example*:
• Construct a frequency distribution for the following data.
11, 29 , 6 ,33 , 14 ,31, 22 , 27 , 19 ,20 ,18 ,17 ,22 ,38 ,23 ,21 ,26
,34 ,39 ,27
Solutions:
Step 1: Find the highest and the lowest value H=39, L=6
Step 2: Find the range; R=H-L=39-6=33
Step 3: Select the number of classes desired using Sturges
formula;

• Step 4: Find the class width; w=R/k=33/6=5.5=6
(rounding up)
• Step 5: Select the starting point, let it be the
minimum observation.
6, 12, 18, 24, 30, 36 are the lower class limits.
Step 6: Find the upper class limit; e.g. the first upper
class=12-U=12-1=11
11, 17, 23, 29, 35, 41 are the upper class limits.
So combining step 5 and step 6, one can construct the
following classes.

• Class limits
6 – 11
12 – 17
18 – 23
24 – 29
30 – 35
36 – 41
• Step 7: Find the class boundaries;
• E.g. for class 1 Lower class boundary=6-U/2=5.5 Upper class
boundary =11+U/2=11.5

• Then continue adding w on both boundaries to obtain the rest
boundaries. By doing so one can obtain the following classes.
• Class boundary
5.5 – 11.5
11.5 – 17.5
17.5 – 23.5
23.5 – 29.5
29.5 – 35.5
35.5 – 41.5
Step 8: tally the data.
Step 9: Write the numeric values for the tallies in the frequency
column

Cont…
• . Step 10: Find cumulative frequency.
• Step 11: Find relative frequency or/and relative
cumulative frequency.
• The complete frequency distribution follows:

Diagrammatic and Graphic
presentation of data.
• These are techniques for presenting data in visual
displays using geometric and pictures for a
categorical / qualitative types of data.
Importance:
• They have greater attraction.
• They facilitate comparison.
• They are easily understandable.
Diagrams are appropriate for presenting discrete data.

Cont…
• The three most commonly used diagrammatic presentation
for discrete as well as qualitative data are:
• Pie charts
• Pictogram
• Bar charts
Pie chart
A pie chart is a circle that is divided in to sections or wedges
according to the percentage of frequencies in each category
of the distribution. The angle of the sector is obtained using:
angle of the sector =RF*360

Cont…
• Example: Draw a suitable diagram to represent the following
population in a town.
Men Women Girls Boys
2500 2000 4000 1500
Solutions:
Step 1: Find the percentage.
Step 2: Find the number of degrees for each class.
Step 3: Using a protractor and compass, graph each section and
write its name corresponding percentage.

Cont…
1
2
3
4

Bar Charts
 The frequency distribution of a categorical variable is often
presented graphically as a bar chart or pie chart.
 Bar charts: display the frequency distribution for nominal
or ordinal data.
 In a bar chart the various categories into which the
observation fall are represented along horizontal axis and a
vertical bar is drawn above each category such that the
height of the bar represents either the frequency or the
relative frequency of observation within the class.
 The vertical axis should always start from 0 but the
horizontal can start from any where.
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Tuesday, December 26, 2023

Cont…
• There are different types of bar charts. The most common
being :
• Simple bar chart
• Component or sub divided bar chart.
• Multiple bar charts.
Simple bar chart:
Are used to display data on one variable.
They are thick lines (narrow rectangles) having the same
breadth.
The magnitude of a quantity is represented by the height /length
of the bar.

Cont…
• Example: The following data represent sale
by product, in 1957 a given company for
three products A, B, C.
Product In 1957 Sales($)
A 12
B 24
C 24

Cont…
• Component Bar chart
-When there is a desire to show how a total (or aggregate) is
divided in to its component parts, we use component bar chart.
• Example: Example: The following data represent sale by product,
1957- 1959 of a given company for three products A, B, C.
Product In 1957 Sales($) In 1958 Sales($) In 1959
A 12 14 18
B 24 28 18
C 24 30 36

Cont…
• Multiple Bar charts
- These are used to display data on more than
one variable.
- - They are used for comparing different
variables at the same time. Example: Draw a
component bar chart to represent the sales
by product from 1957 to 1959.

Graphical Presentation of data
The histogram, frequency polygon and cumulative frequency
graph or ogive are most commonly applied graphical
representation for continuous data.
Procedures for constructing statistical graphs:
• Draw and label the X and Y axes.
• Choose a suitable scale for the frequencies or cumulative
frequencies and label it on the Y axes.
• Represent the class boundaries for the histogram or ogive or
the mid points for the frequency polygon on the X axes.
• Plot the points.
• Draw the bars or lines to connect the points.

Stem-and-Leaf
Represents data by separating each value into
two parts: the stem (the left most digit) and
the leaf (such as the rightmost digit).
• Are most effective with relatively small data
sets
• Are not suitable for reports and other
communications,
• Help researchers to understand the nature of
their data
Example: 43, 28, 34, 61, 77, 82, 22, 47, 49, 51, 29,
36, 66, 72, 41

Histogram
• A graph which displays the data by using vertical bars of various heights
to represent frequencies.
• Class boundaries are placed along the horizontal axes.
• Class marks and class limits are some times used as quantity on the X
axes.
Table 3: Distribution of the age of women at the time of marriage
Age group No. of women
15-19 11
20-24 36
25-29 28
30-34 13
35-39 7
40-44 3
45-49 2

Cont…
• A histogram of the age of women at the time of marriage
Age of women at the time of marriage
0
5
10
15
20
25
30
35
40
14.5-19.5 19.5-24.5 24.5-29.5 29.5-34.5 34.5-39.5 39.5-44.5 44.5-49.5
Age group
No
of
women

Frequency Polygon
• Frequency Polygon : - A line graph. The
frequency is placed along the vertical axis
and classes mid points are placed along the
horizontal axis.
• It is customer to the next higher and lower
class interval with corresponding frequency
of zero, this is to make it a complete
polygon. Example: Draw a frequency
polygon for the above data

Ogive (cumulative frequency polygon)
- A graph showing the cumulative frequency (less than
or more than type) plotted against upper or lower
class boundaries respectively.
- That is class boundaries are plotted along the
horizontal axis and the corresponding cumulative
frequencies are plotted along the vertical axis.
- The points are joined by a free hand curve.
- Example: Draw an ogive curve(less than type) for the
above data.

Chapter 3
Numerical summary measures
A. Measures of location
• It is often useful to summarize, in a single number or
statistic, the general location of the data or the point at
which the data tend to cluster.
• Such statistics are called measures of location or
measures of central tendency.
• We describe them mean, median and mode.
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Cont…
Arithmetic mean
• The arithmetic mean, usually abbreviated to
‘mean’ is the sum of the observations divided
by the number of observations.
• We use the following data set of 10 numbers
to illustrate the computations:

Arithmetic Mean
19 21 20 20 34 22 24 27
27 27
• Then, mean = (19 + 21 + … +27) = 24.1
10
• General formula
a) Ungrouped mean
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Estimation of the mean from a grouped frequency distribution
In calculating the mean from grouped data, we assume that all
values falling into a particular class interval are located at the
mid-point of the interval.
It is calculated as follow:

Cont…
where,
k = the number of class intervals
xi = the mid-point of the ith class interval
fi = the frequency of the ith class interval

Properties of the arithmetic mean
• The mean can be used as a summary measure for both
discrete and continuous data, in general however, it is not
appropriate for either nominal or ordinal data.
• For given set of data there is one and only one arithmetic
mean.
• The arithmetic mean is easily understood and easy to
compute.
• Algebraic sum of the deviations of the given values from their
arithmetic mean is always zero.
• The arithmetic mean is greatly affected by the extreme values.
• In grouped data if any class interval is open, arithmetic mean
can not be calculated.
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Reading Assignment
A. Weighted mean
B. Correct and wrong mean
C. Combined mean
D. Geometric mean
E. Harmonic

Median
• With the observations arranged in increasing or decreasing
order, the median is defined as the middle observation.
• If the number of observations is odd, the median is defined as
the [(n+1)/2]th observation.
• If the number of observations is even the median is the
average of the two middle
{(n/2)th +[(n/2)+1]th }/2 values i.e
• If the number of observations is even, so that there is no
middle observation, the median is defined as the average of
the two middle observations.
• Example : 19 20 20 21 22 24 27
27 27 34
• Then, the median = (22 + 24)/2 = 23
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Median for Grouped data
In calculating the median from grouped data, we assume that the
values within a class-interval are evenly distributed through the
interval.
– The first step is to locate the class interval in which it is located.
– Find n/2 and see a class interval with a minimum cumulative
frequency which contains n/2.
• Whereas:
• LCB= lower class boundary of the median class
• Fc= cumulative frequency just before the median class
• fc=frequency of the median class
• W =class width and n=number of observations.
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Properties of median
• The median can be used as a summary measure for discrete
and continuous data, in general however, it is not
appropriate for nominal data.
• There is only one median for a given set of data
• The median is easy to calculate
• Median is a positional average and hence it is not drastically
affected by extreme values
• Median can be calculated even in the case of open end
intervals
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Mode
• Any observation of a variable at which the distribution reaches
a peak is called a mode.
• Most distributions encountered in practice have one peak and
are described as uni-modal.
• E.g. Consider the example of ten numbers
19 21 20 20 34 22 24 27
27 27
• In the above data set, the mode is 27, because the value 27
occurs three times (the most frequent).
• The mode of grouped data, usually refers to the modal class,
where the modal class is the class interval with the highest
frequency.
• If a single value for the mode of grouped data must be
specified, it is taken as the mid point of the modal class
interval
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Properties of mode
• The mode can be used as a summary measure for nominal,
ordinal, discrete and continuous data, in general however, it is
more appropriate for nominal and ordinal data.
• It is not affected by extreme values
• It can be calculated for distributions with open end classes
• Often its value is not unique
• The main drawback of mode is that often it does not exist
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proportion
• As we have seen from the previous section, a variable can be
either categorical or numerical
• Proportion is one of summery measures for categorical variable
and also numerical variable if we are counting the number of
cases under the specific category.
• If we denote “x” as the number of success in an experiment and
“n” is the number of trials then the proportion of success from n
number of trials is given by x/n.
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Percentiles, Quartiles and Inter-quartile Range
• The quartiles are sets of values which divide the distribution into
four parts such that there are an equal number of observations
in each part.
– Q1 = [(n+1)/4]th
– Q2 = [2(n+1)/4]th
– Q3 = [3(n+1)/4]th
• The inter-quartile range is the difference between the third and
the first quartiles.
– Q3 - Q1
Example1: We use the data set of 11 numbers:
19 21 20 20 34 22 24 27 27 27
28
– The first quartile is 20 and the third quartile is 27
– The inter quartile range = 27 – 20 = 7.
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Percentiles, Quartiles and Inter-quartile Range
• Percentiles divide the data into 100 parts of observations in each
part.
• It follows that the 25th percentile is the first quartile, the 50th
percentile is the median and the 75th percentile is the third
quartile.
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B. Measures of Dispersion (Variation)
• The scatter or spread of items of a distribution is known as
dispersion or variation.
• In other words the degree to which numerical data tend to
spread about an average value is called dispersion or
variation of the data.
• The most commonly used measures of dispersions are:
1) Range and relative range
2) Quartile deviation and coefficient of Quartile deviation
3) Mean deviation and coefficient of Mean deviation
4) variance
5) Standard deviation and coefficient of variation.

Range
• The range is the largest score minus the smallest
score.
• It is a quick and dirty measure of variability
• Because the range is greatly affected by extreme
scores and its only depends on two observations
Relative Range (RR)
• It is also some times called coefficient of range and
given:

Cont..
• Example:
1. If the range and relative range of a series are 4 and
0.25 respectively. Then what is the value of Smallest
observation and Largest observation
The Quartile Deviation (Semi-inter quartile range)
The inter quartile range is the difference between the
third and the first quartiles of a set of items and
semi-inter quartile range is half of the inter quartile
range.

Variance
• variance is the "average squared deviation from the mean".
• A good measure of dispersion should make use of all the data.
• For the case of frequency distribution it is expressed as:
the variance is limited as a descriptive statistic because it is not
in the same units as in the observations.

Cont….
For the case of frequency distribution it is
expressed as:

Coefficient of Variation (C.V)
Is defined as the ratio of standard deviation to
the mean usually expressed as percents.
The distribution having less C.V is said to be less
variable or more consistent.

Which measures to use?
• When the distribution of the data is symmetric and unimodal (i.e.
the data are approximately normally distributed), it is usual to
summarize the data using means and standard deviations.
• However when the data are skewed, it is preferable to use the
median and quartiles as summary statistics.
• Median and quartiles are not easily influenced by extreme values
in a skewed distribution unlike means and standard deviations.
• Remark:
– The mean and median of symmetric distribution coincide.
– When the distribution is skewed to the right, its mean is larger than its
median.
– When the distribution is skewed to the left, its mean is smaller than its
median. [See Figures 7(a-c)].
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Median Mode Mean
Fig. 2(a). Symmetric Distribution
Mean = Median = Mode
Mode Median Mean
Fig. 2(b). Distribution skewed to the right
Mean > Median > Mode
Mean Median Mode
Fig. 2(c). Distribution skewed to the left
Mean < Median < Mode
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Chapter 4
Introduction to probability
Objectives
• After completing this chapter, you should be able to
– Determine sample spaces and ﬁnd the probability of an events
– Understand the different properties of probability
– Explain the various types of probability distributions – emphasis will be
given to the two widely used probability distributions

Introduction
• Many medical decisions are made on a statistical
basis since individuals differ in their reactions to
medications or surgery in an unpredictable way.
• In that case the treatment applied is based on
getting the best outcome for as many patients as
possible
– The life experienced consists of a series of events
– “Probability” is a very useful concept and are used in
everyday communication.
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Introduction con'td…
 An understanding of probability is fundamental for
 quantifying the uncertainty in the decision-making
process
 drawing conclusions about a population of patients based on known
information about a sample of patients drawn from that population.
Probability can be deﬁned as the chance of an event
occurring.
Many people are familiar with probability from
observing or playing games of chance, such as card
games, slot machines, or lotteries.
Probability theory is used in the various ﬁelds of area
like insurance, investments, and weather forecasting
and other areas.
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Basic concepts
• The following definitions and terms are used in
studying the theory of probability.
– Random experiment: a chance process that leads to
well-deﬁned results called outcomes, that is the result
cannot be predicted. E.g. Tossing of coins, throwing of
dice are some examples of random experiments.
– Trial: Performing a random experiment is called a
trial.
– Outcomes: The results of a random experiment are
called its outcomes. When two coins are tossed the
possible outcomes are HH, HT, TH, TT.
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Basic concepts con'td….
– Sample space: Each conceivable outcome of an
experiment is called a sample point. The totality of all
sample points is called a sample space and i s denoted
by S.
– Event: An outcome or a combination of outcomes of a
random experiment is called an event. It is a subset of
the sample space of a random experiment.
– Equally-likely Approach: If an experiment must result in
n equally likely outcomes, then each possible outcome
must have probability 1/n of occurring.
– Mutually exclusive events: when the occurrence of any
one event excludes the occurrence of the other event.
Mutually exclusive events cannot occur simultaneously.
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Basic concepts cont’d…..
• Some sample spaces for various probability experiments are.
• Probability attempts to quantify an uncertain situation and relative
tries to make it more concrete the occurrence of events.
• Probability is used to quantify the likelihood, or chance, that an
outcome of a random experiment will occur.
• Probability is a number between 0 and 1 that expresses how likely the
event is occur.
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Basic concepts cont’d…..
• Example: Find the sample space for the gender of the children if
a family has three children. Use B for boy and G for girl.
– Solution: There are two genders, male and female, and each
child could be either gender. Hence, there are eight
possibilities, as shown here.
S= {BBB, BBG, BGB, GBB, GGG, GGB, GBG, BGG}
• Note: the way to ﬁnd all possible outcomes of a probability
experiment (the sample spaces)
– by observation and reasoning;
– use a tree diagram (a device consisting of line segments
emanating from a starting point and also from the outcome
point.)
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Tree diagram of the above example
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Types of probability
1. Classical Probability
If the number of outcomes belonging to an event E is NE, and the total
number of outcomes is N, then the probability of event E is defined
as P(E)=NE/N
Example: A couple wants to have exactly 3 children. Assume that each
child is either a Boy or a Girl and that there are no duplicate births.
Find the probability that two of them will be boys? List all possible
orderings for the three children.
Solution: S {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}
Then the event E={BBG, BGB, GBB}
P(E)=3/8
2. Relative Frequency Probabilities
This approach to probability is well-suited to a wide range of scientific
disciplines. It is based on the idea that the underlying probability of
an event can be measured by repeated trials

• In this view, probability is treated as a quantifiable level of belief
ranging from 0 (complete disbelief) to 1 (complete belief)
• For instance, an experienced physician may say “this patient has a
50% chance of recovery.”
• An appreciation of the various types of probability are not mutually
exclusive. And fortunately, all obey the same
• mathematical laws, and their methods of calculation are similar.
• All probabilities are a type of relative frequency—the number of
times something can occur divided by the total number of
possibilities or occurrences.
3. Subjective probability

Rules of Probability
• Any probability assigned must be a nonnegative number.
• The probability of the sample space (the collection of all possible
outcomes) is equal to 1.
• The probability of A or B involves addition.
• P(A or B) = P(A) + P(B) if the two are mutually exclusive.
• The probability of A and B involves multiplication
• P(A and B) = P(A) P(B) if the two are independent
• P( Not A) = 1- P(A)
• P(At least one) = 1- P(none)
• P(none) = P(each event not happening)^number of events

Addition Rule
• General rule: if two events, A and B, are not mutually exclusive,
then the probability that event A or event B occurs is:
– P(A or B) = P(A) + P(B) – P(A and B)
– P(AuB) = P(A) + P(B) – P(AnB)
• Special case: when two events, A and B, are mutually exclusive,
then the probability that event A or event B occurs is:
– P(A or B) = P(A) + P(B)
– P(AuB) = P(A) + P(B)
• since P(A and B) = 0 for mutually exclusive events
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Conditional Probabilities
• Sometimes the chance a particular event happens depends on the
outcome of some other event. This applies obviously with many
events that are spread out in time.
• The probability that an event occurs subject to the condition that
another event has already occurred is called conditional probability
• If A and B are events with Pr(A) > 0, the conditional probability of
B given A is
• Example: Drug test
• Given A is independent from B, what is the relationship between
Pr(A|B) and Pr(A)?
Pr( )
Pr( | )
Pr( )
AB
B A
A
=
97
Women Men
Success 200 1800
Failure 1800 200
A = {Patient is a Women}
B = {Drug fails}
Pr(B|A) = 1800/2000
Pr(A|B) =
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Tuesday, December 26,
2023

Conditional probability:
• The conditional probability of an event A given an
event B is present is:
– P(A | B) =P(AnB)/P(B)
• where P(B)≠0
• Joint probability
• The joint probability of an event A and an event B is
– P(AnB) = P(A and B)
• When events A and B are mutually exclusive, then
– P(A and B) = 0
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Multiplication rule
• General rule: the multiplication rule specifies the
joint probability as:
• P(AnB)=P(B)P(A/B)
• Special case: When events A and B are independent,
then:
– P(A|B) = P(A)
– P(AnB)=P(A)P(B)
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Example
1. Find the Probability of at least one male birth in ten consecutive
births?
Solution
• P(At least one male) = 1- P(all females)
• P(all females) = P(each single birth is a female)10 = (0.5)10
= 9.77 x 10-4
• So P(At least one male) = 1 – 9.77 x 10-4 = 0.999023.

Exercises
1. 50% of the students in a school weigh more than 140 pounds, but
70% weigh less than 170. If a student is randomly selected, what
is the probability the student will weigh between 140 and 170?
• Solution
• P(w<140)=0.5, P(w>170)=0.3, and P(w<140)+P(140<w<170)+
P(w>170)= 1
• Then
• 0.5+P(140<w<170)+ 0.3= 1 P(140<w<170)=1-0.8=0.2

Probability Distribution
• A probability distribution is a table or an equation
that links each outcome of a statistical experiment
with its probability of occurrence.
• Probability distribution for discrete variable
• Probability distribution for continuous variable
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Probability Distribution for discrete Variable
1. Binomial distribution
• A binomial experiment (also known as a Bernoulli trial) is a statistical
experiment that has the following properties:
• The experiment consists of n repeated fixed number of trials.
• Each trial can result in just two possible outcomes. We call one of these
outcomes a success and the other, a failure.
• The probability of success, denoted by P, is the same on every trial.
• The trials are independent; that is, the outcome on one trial does not
affect the outcome on other trials.
• The probability distribution of this experiment is known as binomial
probability distribution.
• The binomial distribution describes the distribution of "success" in a series
of trials, that is, out of N tries, what is the probability that X of them
succeed.
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Binomial formula
• If the probability of success on an individual trial is P, then the binomial
probability is defined by:
• Where
• K=the number of success
• P=probability of success
• n=the number of experiments
• 1-p=probability of failure
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2. Poisson Distribution
• The Poisson Distribution is a discrete distribution which takes on the values
X = 0, 1, 2, 3, and so on.
• It is often used as a model for the number of events in a specific time
period.
• It is determined by one parameter, lambda. The Poisson random variable
satisfies the following conditions:
• The number of successes in two disjoint time intervals is independent.
• The probability of a success during a small time interval is proportional to
the entire length of the time interval.
• Some of the examples in which Poisson distribution is appropriate are:
➢ birth defects and genetic mutations
➢ rare diseases (like Leukemia, but not AIDS because it is infectious and so
not independent)
➢ car accidents
➢ traffic flow and ideal gap distance, and so on

Poisson formula
• The probability distribution of a Poisson random variable X representing the
number of successes occurring in a given time interval or a specified region of
space is given by the formula:
Where
• X=Number of successes per unit time
• e=The base of the natural log
• λ= The expected number of successes per unit time
• If λ is the average number of successes occurring in a given time interval or
region in the Poisson distribution, then the mean and the variance of the
Poisson distribution are both equal to λ.

Probability Distribution for continuous Variables
• If a random variable is a continuous variable, its probability
distribution is called a continuous probability distribution.
• A continuous probability distribution differs from a discrete
probability distribution in several ways by:
• Under different circumstances, the outcome of a random
variable may not be limited to categories or counts.
• Because a continuous random variable X can take on an
uncountable infinite number of values, the probability
associated with any particular one value is almost equal to zero.
• As a result, a continuous probability distribution cannot be
expressed in tabular form.
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Normal distribution
• The normal distribution refers to a family of continuous
probability distributions described by the normal equation and
described as follows:
where
• X is a normal random variable,
• μ is the mean
• σ is the standard deviation
• pi is approximately 3.14159, and e is approximately 2.71828.
• The random variable X in the normal equation is called the
normal random variable. 108
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Characteristics of Normal Distribution
• It links frequency distribution to probability distribution
• Has a Bell Shape Curve and is Symmetric
• It is Symmetric around the mean: Two halves of the curve are the
same (mirror images)
• Hence Mean = Median
• The total area under the curve is 1 (or 100%)
• Normal Distribution has the same shape as Standard Normal
Distribution.
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Normal Curve
• The graph of the normal distribution depends on two factors:
✓the mean and the standard deviation.
• The mean of the distribution determines the location of the center of the
graph, and the standard deviation determines the height and width of the
graph.
• When the standard deviation is large, the curve is short and wide; when the
standard deviation is small, the curve is tall and narrow.
• All normal distributions look like a symmetric, bell-shaped curve.
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Standard Normal Distribution
• It makes life a lot easier for us if we standardize our normal curve, with a mean
of zero and a standard deviation of 1 unit.
• We can transform all the observations of any normal random variable X with
mean μ and variance σ to a new set of observations of another normal random
variable Z with mean 0 and variance 1 using the following transformation:
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• About 95% of the area under the curve falls within 2 standard deviations
of the mean.
• About 99.7% of the area under the curve falls within 3 standard deviations
of the mean.
• A graph of this standardized (mean 0 and variance 1) normal curve is given
in Graph:
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Table of normal
• Example 1: Suppose we want to compute the area under the
normal curve to the right of 1.45
• This area can be computed by finding the probability under
the normal curve. The probability can be read at the normal
curve by combining the value of 1.4 under the first column
and 0.05 under the first row.
• The green shaded area in the diagram represents the area
that is within 1.45 standard deviations from the mean. The
area of this shaded portion is 0.4265 (or 42.65% of the total
area under the
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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 0.3304 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 114
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Example 2
• Assuming the normal heart rate (H.R) in normal healthy individuals is
normally distributed with Mean = 70 and Standard Deviation =10 beats/min
Then:
1) What area under the curve is above 80 beats/min?
Now we know, Z =X-M/SD,
Z=? X=80, M= 70, SD=10 .
So we have to find the value of Z.
For this we need to draw the figure…..and find the area which corresponds to Z.
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• Since M=70, then the area under the curve which is above 80 beats
per minute corresponds to above + 1 standard deviation.
•
• The total shaded area corresponding to above 1+ standard
deviation in percentage is 15.9% or Z= 15.9/100 =0.159.
• Or we can find the value of z by substituting the values in the
formula Z= X-M/ standard deviation.
• Therefore, Z= 70-80/10 -10/10= -1.00 is the same as +1.00. The
value of z from the table for 1.00 is 0.159. How do we interpret this?
•
• This means that 15.9% of normal healthy individuals have a heart
rate above one standard deviation (greater than 80 beats per
minute).
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13.6%
2.2%
0.15
-3 -2 -1 μ 1 2 3
Diagram of Exercise
0.159
33.35%
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Example …
2) What area of the curve is above 90
beats/min?
3) What area of the curve is between
50-90 beats/min?
4)What area of the curve is above 100
beats/min?
5) What area of the curve is below 40 beats per
min or above 100 beats per min?
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solution
2) 2.3% or 0.023
3) 95.4% or 0.954
4) 0.15 % or 0.015
5) 0.3 % or 0.015 (for each tail)
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Example 3
• Suppose scores on an IQ test are normally distributed. If the test has a
mean of 100 and a standard deviation of 10, what is the probability that a
person who takes the test will score between 90 and 110?
• Solution:
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Application/Uses of Normal Distribution
• It’s application goes beyond describing distributions
• It is used by researchers and modelers.
• The major use of normal distribution is the role it plays
in statistical inference.
• The z score along with the t –score, chi-square and F-
statistics is important in hypothesis testing.
• It helps managers/management make decisions.
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Exercises
• Find the probability under the normal curve of the
following:
• The area greater than 1.25
• The area lower than 0.87
• The area greater than -2.36
• The area lower than -1.96
• The area between 0.87 and 1.25
• The area between -1.96 and 1.25
• The value of z which cuts the lower 20%
• The value of z which cuts the upper 10%
• The values of z which cut the middle 80%
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Chapter s
Sampling methods and sample size
determination
• Sampling is a procedure by which some members of
the given population are selected as representative
of the entire population
➢ Theoretical population
➢ Target population
➢ Study population
➢ sampling frame
➢ Sampling unit
➢ Study unit

Hierarchy of sampling
Wullo S. 124

Error in sampling
• No sample is the exact mirror image of the population
• Potential Source of Error in research are two.
1. Sampling error is any type of bias that is attributable to
mistakes in either drawing a sample or determining the
sample size
➢ Sampling error (chance ) Can not be avoided or totally
eliminated
➢ The causes are
– One is chance: That is the error that occurs just because of bad
luck
– Design error
– Un representativeness of the sample
Wullo S. 125

Error in sampling con’d..
2. Non-sampling error is any error which will be
committed during data collection, coding, entry, and
so on
✓ Observational error
✓ Respondent error
✓ Lack of preciseness of definition
✓ Error in editing and tabulation of the data
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Wullo S. 127

Advantage of sampling
➢Feasibility it may be the only feasible method
of collecting data
➢Reduced cost sampling reduces demands on
resource such as finance, personal and
material
➢Greater accuracy sampling may lead to better
accuracy of collecting data/ detailed
information
➢Greater speed data can be collected and
summarized more quickly
Wullo S. 128

Disadvantage of sampling
• There is always sampling error
• Sampling may create a feeling of discrimination within
the population
• It may be inadvisable where every unit in the
population is legally required to have a record
• Demands more rigid control in undertaking sample
operation
• The presence of bias creates difference between the
parameter and the statistic
• Minority and smallness in number of sub-groups often
render study to be suspected.
• Sample results are good approximations at best.
Wullo S. 129

Types of sampling
I. Probability sampling
• Is any method of sampling that utilizes some form of random
selection.
• probability sampling is a procedure for sampling from a
population in which
– The selection of a sample unit is based on chance
– Every element of the population has a known and non-zero
probability of being selected
– Random sampling helps produce representative samples by
eliminating voluntary response bias and guarding against under
coverage bias
❖ Every individual of the target population has equal chance to be
included in the sample.
Wullo S. 130

1. Simple random sample (SRS)
• Objective: To select n units out of N
• If the population is homogenous
• If frame is available
• If the study area is not very wide
– Note: Homogeneity refers to the similarity of the population with regard
to the outcome variable .
• Procedure:
✓Use a table of random numbers: takes on values
0,1,2,…….,9 with equal probability
✓ a computer random number generator
✓ mechanical device to select the sample.
✓RAND() function from Excel sheet if frame is available
✓Lottery method
Wullo S. 131

Wullo S. 132

2. Stratified random sampling
• Stratified Random Sampling involves dividing your population
into homogeneous subgroups and then taking a simple random
sample in each subgroup.
• Objective: Divide the population into non-overlapping groups
(i.e., strata) N1, N2, N3, ... Ni, such that N1 + N2 + N3 + ... + Ni = N.
Then do a simple random sample depending on the type of
allocation
➢ Proportional allocation:
➢ Equal allocation
Example:
➢ An agency has clients from three ethnic groups and the agency
wants to asses clients view of quality of service for the last year.
i
i N
N
n
n *
=
Wullo S. 133

Stratified random sampling
Wullo S. 134

3. Systematic random sampling
▪ Here are the steps you need to follow in order to achieve a
systematic random sample:
❖number the units in the population from 1 to N
❖decide on the n (sample size) that you want or need
❖k = N/n = the interval size
❖randomly select an integer between 1 to k
❖then take every kth unit
• Assumptions
– Homogeneous population
– Frame is not available
– If the study area is not very wide
Wullo S. 135

Systematic random sampling
• Example
Wullo S. 136

Wullo S. 137

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4. Cluster (area) random sampling
• The problem with random sampling methods when we have to
sample a population that's disbursed across a wide geographic
region is that you will have to cover a lot of ground geographically
in order to get to each of the units you sampled
• In cluster sampling, we follow these steps:
✓ divide population into clusters (usually along geographic
boundaries)
✓ randomly sample clusters
✓ measure all units within sampled clusters
Wullo S. 139

Example
• In the figure we see a map of the counties in New York State.
Let's say that we have to do a survey of town governments that
will require us going to the towns personally. If we do a simple
random sample state-wide we'll have to cover the entire state
geographically
Wullo S. 140

5. Multi-stage sampling
• The four methods we've covered so far -- simple, stratified,
systematic and cluster -- are the simplest random sampling
strategies
• When we combine sampling methods, we call this multi-stage
sampling
• Consider the problem of sampling students in grade schools.
We might begin with a national sample of school districts
stratified by educational level. Within selected districts, we
might do a simple random sample of schools. Within schools,
we might do a simple random sample of classes or grades.
And, within classes, we might even do a simple random
sample of students. In this case, we have three or four stages
in the sampling process and we use both stratified and simple
random sampling.
Wullo S. 141

Wullo S. 142

II. Non probability sampling
• Non probability sampling does not involve random selection
• Does that mean that non probability samples aren‘t representative
of the population?
• It does mean that non probability samples cannot depend upon the
rationale of probability theory
• Most sampling methods are purposive in nature because we usually
approach the sampling problem with a specific plan in mind.
Wullo S. 143

Convenience sampling
Example
➢ Man in the street (attitude of foreigners about Ethiopia)
➢ College students for psychological study
Wullo S. 144

Purposive sampling
• In purposive sampling, we sample with a purpose in mind
1. Modal Instance Sampling
✓ In statistics, the mode is the most frequently occurring value
in a distribution.
✓ we are sampling the most frequent case, or the "typical"
case
✓ We could say that the modal voter is a person who is of
average age, educational level, and income in the
population
Wullo S. 145

Purposive sampling…
2. Expert Sampling
✓ Expert sampling involves the assembling of a sample of persons
with known or demonstrable experience and expertise in some
area.
3. Quota Sampling
✓ In quota sampling, you select people non randomly according to
some fixed quota
✓ There are two types of quota sampling: proportional and non
proportional
4. Heterogeneity Sampling
✓ We sample for heterogeneity when we want to include all
opinions or views, and we aren't concerned about representing
these views proportionately.
✓ Another term for this is sampling for diversity.
Wullo S. 146

Purposive sampling…
5. Snowball sampling
✓ In snowball sampling, you begin by identifying someone who
meets the criteria for inclusion in your study
✓ You then ask them to recommend others who they may know
who also meet the criteria
✓ Snowball sampling is especially useful when you are trying to
reach populations that are inaccessible or hard to find
✓ For instance, if you are studying the homeless, you are not
likely to be able to find good lists of homeless people within a
specific geographical area. However, if you go to that area and
identify one or two, you may find that they know very well
who the other homeless people in their vicinity are and how
you can find them
Wullo S. 147

Summary
▪ Selecting a sampling method depends on
• Population to be studied
✓ Size and geographic distributions
✓ Heterogeneity with respect to the variable studied
• Resource available
• Level of precision required
• Importance of having a precise estimate of sampling error
Wullo S. 148

Sample size determination
• Determining the sample size for a study is a crucial
component of study design.
• The goal is to include sufficient numbers of subjects so that
statistically significant results can be detected.
• Among the questions that a researcher should ask when
planning a survey or study is that "How large a sample do I
need?"
• The answer will depend on the aims, nature and scope of the
study and on the expected result.
• All of which should be carefully considered at the planning
stage
Wullo S. 149

Sample size determination ….
• In general, sample size depends on
– Objective of the study
– Design of the study
– Plan for statistical analysis
– Accuracy of the measurement to be made
– Degree of precision required for generalization
– Degree of confidence
• We can use three approaches to determine sample size
– Rules of thumb for determining the sample size
– Statistical formula
• Confidence interval approach
• Hypothesis testing approach
Wullo S. 150

Rules of thumb
• For smaller samples(N < 100), there is little point in sampling.
Survey the entire population.
• If the population size is around 500 (give or take 100), 50%
should be sampled.
• If the population size is around 1500, 20% should be sampled.
• Beyond a certain point (N = 5000), the population size is
almost irrelevant and a sample size of 400 may be adequate.
• Statistician maximalist: at least 500
• To make generalizations about entire population, need a total
sample size of 200-400 (depending on total population and
confidence level desired)
Wullo S. 151

confidence interval
• Hence the absolute precision denoted by d is given as
• Where s.e is the standard error of the estimator of the
parameter of interest.
• The margin of error (d) measures the precision of the estimate
– Small value of w indicates high precision
– It lies in the interval (0%; 5%]
– For p close to 50%, w is assumed to be close to 5%
– For smaller value of p, w is assumed to be lower than 5%
e
s
z
proportion
mean .
)
(
2


e
s
z
d .
2

=
Wullo S. 152

Estimating a single population mean
Where the standard deviation δ can be estimated by;
✓From previous study, if there is
✓From pilot study
✓From educate guess
Wullo S. 153

Single population proportion
• Let p denotes proportion of success, then
Where the standard deviation p can be estimated by;
✓From previous study, if there is
✓From pilot study
✓P=50%
Wullo S. 154

Point to be considered
Wullo S. 155

Wullo S. 156

Example
Wullo S. 157

Chapter six
Inferential statistics
• After complete this session you will be able to do
– Parameter estimations
– Point estimate
– Confidence interval
– Hypothesis testing
– Z-test
– T-test
– Testing associations
– Chi-Square test

Introduction
159
Wullo S.

Introduction con'td……..
• Before beginning statistical analyses
– it is essential to examine the distribution of the variable for
skewness (tails),
– kurtosis (peaked or ﬂat distribution), spread (range of the
values) and
– outliers (data values separated from the rest of the data).
• Information about each of these characteristics
determines to choose the statistical analyses and can
be accurately explained and interpreted.
160
Wullo S.

Introduction con’td …….
• Statistical tests can be either parametric or non-
parametric
• The path way for the analysis of continuous variables
is shown below
161
Wullo S.

Sampling Distribution
• The frequency distribution of all these samples forms the sampling
distribution of the sample statistic
162
Wullo S.

Sampling distribution .......
 Three characteristics about sampling distribution of a statistic
its mean
its variance
its shape
 If we repeatedly take sample of the same size n from a
population the means of the samples form a sampling
distribution of means of size n is equal to population mean.
 In practice we do not take repeated samples from a population
i.e. we do not encounter sampling distribution empirically, but it
is necessary to know their properties in order to draw statistical
inferences.
163
Wullo S.

The Central Limit Theorem
• Regardless of the shape of the frequency distribution of
a characteristic in the parent population,
• the means of a large number of samples (independent
observations) from the population will follow a normal
distribution (with the mean of means approaches the
population mean μ, and standard deviation of σ/√n ).
• Inferentialstatisticaltechniqueshavevariousassumptionst
hatmustbemetbeforevalid conclusions can be obtained
• Samples must be randomly selected.
• sample size must be greater (n>=30)
• the population must be normally or approximately normally distributed if the
sample size is less than 30.
164
Wullo S.

Sampling Distribution......
• E.g. Sampling Distribution of the mean
• Suppose we choose a random sample of size n, the
sampling distribution of the sample mean x posses the
following properties.
– The sample mean x will be an estimate of the population
mean μ.
– The standard deviation of x is σ/√n (called the standard
error of the mean).
– Provided n is large enough the shape of the sampling
distribution of x is normal.
165
Wullo S.

Sampling Distribution ..........
• Proportion
 Suppose we choose a random sample of size n, the sampling
distribution of the sample means p posses the following
properties.
 The sample proportion p will be an estimate of the
population mean p.
________
 The standard deviation of p is = √p(1-p) /n called the
standard error of the proportion).
 Provided n is large enough the shape of the sampling
distribution of p is normal.
166
Wullo S.

Parameter Estimations
• In parameter estimation, we generally assume that the
underlying (unknown) distribution of the variable of interest is
adequately described by one or more (unknown) parameters,
referred as population parameters.
• As it is usually not possible to make measurements on every
individual in a population, parameters cannot usually be
determined exactly.
• Instead we estimate parameters by calculating the
corresponding characteristics from a random sample
estimates .
• the process of estimating the value of a parameter from
information obtained from a sample.
• Point estimation
• Interval estimation
167
Wullo S.

Point estimation
• A point estimate is a speciﬁc numerical value estimate of a
parameter.
• Sample measures (i.e., statistics) are used to estimate population
measures (i.e., parameters). These statistics are called estimators.
• Point estimate for population mean µ is
• Point estimate for population proportion is given by
• Where x is the total number of success (events) 168
n
x
=
x
n
1
=
i
i

n
=
p
x

Wullo S.

Three Properties of a Good Estimator
• The estimator should be an unbiased estimator. That is, the
expected value or the mean of the estimates obtained from
samples of a given size is equal to the parameter being
estimated.
• The estimator should be consistent. For a consistent
estimator, as sample size increases, the value of the estimator
approaches the value of the parameter estimated.
• The estimator should be a relatively efficient estimator. That
is, of all the statistics that can be used to estimate a parameter,
the relatively efficient estimator has the smallest variance.
169
Wullo S.

Some BLUE estimators
170
Wullo S.

Confidence Interval estimate
• However the value of the sample statistic will vary from
sample to sample therefore, to simply obtain an
estimate of the single value of the parameter is not
generally acceptable.
– We need also a measure of how precise our estimate is likely
to be
– We need to take into account the sample to sample variation
of the statistic
• A confidence interval defines an interval within which
the true population parameter is like to fall (interval
estimate).
171
Wullo S.

Confidence intervals…
• Confidence interval therefore takes into account the sample to
sample variation of the statistic and gives the measure of
precision.
• The general formula used to calculate a Confidence interval is
Estimate ± K × Standard Error, k is called reliability coefficient
• Confidence intervals express the inherent uncertainty in any
medical study by expressing upper and lower bounds for
anticipated true underlying population parameter.
• The conﬁdence level is the probability that the interval estimate
will contain the parameter, assuming that a large number of
samples are selected and that the estimation process on the
same parameter is repeated
• Most commonly the 95% confidence intervals are calculated,
however 90% and 99% confidence intervals are sometimes used.
172
Wullo S.

Confidence interval ……
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173
A (1-α) 100% confidence interval for unknown population mean
and population proportion is given as follows;
Wullo S.

Interval estimation
174
Wullo S.

175
Wullo S.

176
Wullo S.

177
Wullo S.

• The 95% confidence interval is calculated in such a way that,
under the conditions assumed for underlying distribution, the
interval will contain true population parameter 95% of the time.
• Loosely speaking, you might interpret a 95% confidence interval
as one which you are 95% confident contains the true parameter.
• 90% CI is narrower than 95% CI since we are only 90% certain that
the interval includes the population parameter.
• On the other hand 99% CI will be wider than 95% CI; the extra
width meaning that we can be more certain that the interval will
contain the population parameter. But to obtain a higher
confidence from the same sample, we must be willing to accept a
larger margin of error (a wider interval).
178
Wullo S.

• For a given confidence level (i.e. 90%, 95%, 99%) the
width of the confidence interval depends on the
standard error of the estimate which in turn depends
on the
– 1. Sample size:-The larger the sample size, the narrower the
confidence interval (this is to mean the sample statistic will
approach the population parameter) and the more precise
our estimate. Lack of precision means that in repeated
sampling the values of the sample statistic are spread out or
scattered. The result of sampling is not repeatable.
179
Wullo S.

- To increase precision (of an SRS), use a larger sample.
You can make the precision as high as you want by
taking a large enough sample. The margin of error
decreases as√n increases.
• 2. Standard deviation:-The more the variation among
the individual values, the wider the confidence
interval and the less precise the estimate. As sample
size increases SD decreases.
• Z is the value from SND
• 90% CI, z=1.64
• 95% CI, z=1.96
• 99% CI, z=2.58 180
Wullo S.

Confidence interval ……
• If the population standard deviation is unknown and
the sample size is small (<30), the formula for the
confidence interval for sample mean is: x ± t (s/√n)
– x is the sample mean
– s is the sample standard deviation
– n is the sample size
– t is the value from the t-distribution with (n-1) degrees of freedom
181
Wullo S.

Mean Example
 A SRS of 16 apparently healthy subjects yielded the following values of
urine excreted (milligram per day);
0.007, 0.03, 0.025, 0.008, 0.03, 0.038, 0.007, 0.005, 0.032, 0.04, 0.009,
0.014, 0.011, 0.022, 0.009, 0.008
Compute point estimate of the population mean
Construct 90%, 95%, 98% confidence interval for the mean
(0.01844-1.65x0.0123/4, 0.01844+1.65x0.0123/4)=(0.0134, 0.0235)
(0.01844-1.96x0.0123/4, 0.01844+1.96x0.0123/4)=(0.0124, 0.0245)
(0.01844-2.33x0.0123/4, 0.01844+2.33x0.0123/4)=(0.0113, 0.0256)
182
01844
.
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Wullo S.

Proportion example
• In a survey of 300 automobile drivers in one city, 123 reported
that they wear seat belts regularly. Estimate the seat belt rate of
the city and 95% confidence interval for true population
proportion.
• Answer : p= 123/300 =0.41=41%
n=300,
Estimate of the seat belt of the city at 95%
CI = p ± z ×(√p(1-p) /n) =(0.35,0.47)
183
Wullo S.

Summary
• Students sometimes have difficulty deciding whether to use
Za/2 or t a/2 values when ﬁnding conﬁdence intervals
184
Wullo S.

HYPOTHESIS TESTING
Introduction
– Researchers are interested in answering many types of questions. For example, A
physician might want to know whether a new medication will lower a person’s
blood pressure.
– These types of questions can be addressed through statistical hypothesis testing,
which is a decision-making process for evaluating claims about a population.
185
Wullo S.

Hypothesis Testing
• The formal process of hypothesis testing provides us with a
means of answering research questions.
• Hypothesis is a testable statement that describes the nature of
the proposed relationship between two or more variables of
interest.
• In hypothesis testing, the researcher must deﬁned the
population under study, state the particular hypotheses that will
be investigated, give the signiﬁcance level, select a sample from
the population, collect the data, perform the calculations
required for the statistical test, and reach a conclusion.
186
Wullo S.

Idea of hypothesis testing
187
Wullo S.

type of Hypotheses
• Null hypothesis (represented by HO) is the statement about the value of the
population parameter. That is the null hypothesis postulates that ‘there is no
difference between factor and outcome’ or ‘there is no an intervention effect’.
• Alternative hypothesis (represented by HA) states the ‘opposing’ view that
‘there is a difference between factor and outcome’ or ‘there is an intervention
effect’.
188
Wullo S.

Methods of hypothesis testing
• Hypotheses concerning about parameters
which may or may not be true
• The three methods used to test hypotheses
are
• The traditional method
• The P-value method (New approach)
• The conﬁdence interval method
189
Wullo S.

Steps in hypothesis testing
1
Identify the null hypothesis H0 and
the alternate hypothesis HA.
190
3
Select the test statistic and
determine its value from the sample
data. This value is called the
observed value of the test statistic.
Remember that t statistic is usually
appropriate for a small number of
samples; for larger number of
samples, a z statistic can work well if
data are normally distributed.
4
Compare the observed value of the statistic to
the critical value obtained for the chosen a.
5
Make a decision.
6
Conclusion
2
Choose a. The value should be small, usually less
than 10%. It is important to consider the
consequences of both types of errors.
Wullo S.

Test Statistics
 Because of random variation, even an unbiased sample may not
accurately represent the population as a whole.
 As a result, it is possible that any observed differences or
associations may have occurred by chance.
• A test statistics is a value we can compare with known
distribution of what we expect when the null hypothesis is true.
• The general formula of the test statistics is:
Observed _ Hypothesized
Test statistics = value value .
Standard error
• The known distributions are Normal distribution, student’s distribution , Chi-
square distribution ….
191
Wullo S.

Critical value
• The critical value separates the critical region from the noncritical region
for a given level of significance
192
Wullo S.

Decision making
• Accept or Reject the null hypothesis
• There are 2 types of errors
• Type I error is more serious error and it is the level of significant
• power is the probability of rejecting false null hypothesis and it is
given by 1-β
193
Type of decision H0 true H0 false
Reject H0 Type I error (a)
Correct decision (1-
β)
Accept H0
Correct decision (1-
a)
Type II error (β)
Wullo S.

194
Wullo S.

195
Wullo S.

196
Wullo S.

H0: m = m0
H1: m < m0
0
0
0
H0: m = m0
H1: m > m0
H0: m = m0
H1: m  m0


/2
Critical
Value(s)
Rejection Regions
One tailed test
Two tailed test
Types of testes
Wullo S. 197

Two tailed test
198
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

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
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

m
m


m
m
Wullo S.

Steps in hypothesis testing…..
199
If the test statistic falls in the critical
region:
Reject H0 in favour of HA.
If the test statistic does not fall in the
critical region:
Conclude that there is not enough
evidence to reject H0.
Wullo S.

One tailed tests
200







=
=
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Decision
H
H
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not
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m
m
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m


m
m


m
m

Wullo S.

The P- Value
• In most applications, the outcome of performing a hypothesis
test is to produce a p-value.
• P-value is the probability that the observed difference is due to
chance.
• A large p-value implies that the probability of the value observed,
occurring just by chance is low, when the null hypothesis is true.
• That is, a small p-value suggests that there might be sufficient
evidence for rejecting the null hypothesis.
• The p value is defined as the probability of observing the
computed significance test value or a larger one, if the H0
hypothesis is true. For example, P[ Z >=Zcal/H0 true].
201
Wullo S.

P-value……
• A p-value is the probability of getting the
observed difference, or one more extreme, in
the sample purely by chance from a
population where the true difference is zero.
• If the p-value is greater than 0.05 then, by
convention, we conclude that the observed
difference could have occurred by chance and
there is no statistically significant evidence (at
the 5% level) for a difference between the
202
Wullo S.

How to calculate P-value
o Use statistical software like SPSS, SAS……..
o Hand calculations
—obtained the test statistics (Z Calculated or t-
calculated)
—find the probability of test statistics from
standard normal table
—subtract the probability from 0.5
—the result is P-value
Note if the test two tailed multiply 2 the result.
Wullo S. 203

P-value and confidence interval
• Confidence intervals and p-values are based upon the same
theory and mathematics and will lead to the same conclusion
about whether a population difference exists.
• Confidence intervals are referable because they give
information about the size of any difference in the population,
and they also (very usefully) indicate the amount of
uncertainty remaining about the size of the difference.
• When the null hypothesis is rejected in a hypothesis-testing
situation, the conﬁdence interval for the mean using the same
level of signiﬁcance will not contain the hypothesized mean.
204
Wullo S.

The P- Value …..
• But for what values of p-value should we reject the null
hypothesis?
– By convention, a p-value of 0.05 or smaller is considered
sufficient evidence for rejecting the null hypothesis.
– By using p-value of 0.05, we are allowing a 5% chance of
wrongly rejecting the null hypothesis when it is in fact
true.
• When the p-value is less than to 0.05, we often say that the
result is statistically significant.
205
Wullo S.

SUMMARY
206
Wullo S.

Hypothesis testing for single population mean
• EXAMPLE 1: A researcher claims that the mean of the IQ for 16
students is 110 and the expected value for all population is 100
with standard deviation of 10. Test the hypothesis .
• Solution
1. Ho:µ=100 VS HA:µ≠100
2. Assume α=0.05
3. Test statistics: z=(110-100)4/10=4
4. z-critical at 0,025 is equal to 1.96.
5. Decision: reject the null hypothesis since 4 ≥ 1.96
6. Conclusion: the mean of the IQ for all population is different
from 100 at 5% level of significance.
207
Wullo S.

Hypothesis testing for single proportions
• Example: In the study of childhood abuse in psychiatry patients, brown found
that 166 in a sample of 947 patients reported histories of physical or sexual
abuse.
a) constructs 95% confidence interval
b) test the hypothesis that the true population proportion is 30%?
• Solution (a)
• The 95% CI for P is given by
208
]
2
.
0
;
151
.
0
[
0124
.
0
96
.
1
175
.
0
947
825
.
0
175
.
0
96
.
1
175
.
0
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1
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−


n
p
p
z
p 
Wullo S.

Example……
• To the hypothesis we need to follow the steps
Step 1: State the hypothesis
Ho: P=Po=0.3
Ha: P≠Po ≠0.3
Step 2: Fix the level of significant (α=0.05)
Step 3: Compute the calculated and tabulated value of the test statistic
209
96
.
1
39
.
8
0149
.
0
125
.
0
947
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7
.
0
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3
.
0
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.
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=
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−
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−
−
=
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tab
cal
z
n
p
p
Po
p
z
Wullo S.

Example……
• Step 4: Comparison of the calculated and tabulated values of the
test statistic
• Since the tabulated value is smaller than the calculated value of
the test the we reject the null hypothesis.
• Step 6: Conclusion
• Hence we concluded that the proportion of childhood abuse in
psychiatry patients is different from 0.3
• If the sample size is small (if np<5 and n(1-p)<5) then use student’s
t- statistic for the tabulated value of the test statistic.
210
Wullo S.

Two sample mean and proportion
• Still now we have seen estimate for only single mean and single
proportion. However it is possible to compute point and interval
estimation for the difference of two sample means.
• let x1, x2, …, xn1 are samples from the first population and y1, y2,
…, yn2 be sample from the second population.
• Sample mean for the first population be
• Sample mean for the second population
• Then the point estimate for the difference of means (µ1-µ2) is
given by
211
)
( Y
X −
Y
X
Wullo S.

Two sample estimation
• Confidence interval estimation
• A (1-α)100% confidence interval for the
difference of means is given by
• If are unknown, then can be estimated
by
212
2
2
2
1
2
1
2
)
(
n
n
z
y
x


 +

−
2
1, 
 and
2
1, s
and
s
Wullo S.

Hypothesis testing for two sample means
• The steps to test the hypothesis for difference of means is the
same with the single mean
Step 1: state the hypothesis
Ho: µ1-µ2 =0
VS
HA: µ1-µ2 ≠0, HA: µ1-µ2 <0, HA: µ1-µ2 >0
Step 2: Significance level (α)
Step 3: Test statistic
213
2
2
2
1
2
1
2
1 )
(
)
(
n
n
y
x
zcal


m
m
+
−
−
−
=
Wullo S.

Hypothesis …






−




−


−






−
=
=
o
cal
cal
o
cal
cal
A
o
cal
cal
o
tab
cal
A
o
tab
cal
o
tab
cal
A
tabulated
tabulated
H
reject
not
do
z
z
if
H
reject
z
z
if
H
For
H
reject
not
do
z
z
if
H
reject
z
z
if
H
For
H
reject
not
do
z
z
if
H
reject
z
z
if
H
For
test
tailed
one
for
z
z
test
tailed
two
for
z
z
0
:
0
:
|
|
|
|
0
:
2
1
2
1
2
1
2
m
m
m
m
m
m


214
Wullo S.

Small sample size and population variance is not given
• The test statistic will be student’s t-statistic with degree of
freedom equals to n1+n2 -2
• Hence the tabulated value of t is read from the table.
• The decision remains the same
215
Wullo S.

Example
• A researchers wish to know if the data they have collected provide
sufficient evidence to indicate a difference in mean serum uric
acid levels between normal individual and individual with down’s
syndrome. The data consists of serum uric acid readings on 12
individuals with down’s syndrome and 15 normal individuals. The
means are 4.5mg/100ml and 3.4 mg/100ml with standard
deviation of 2.9 and 3.5 mg/100ml respectively.
216
0
:
0
:
2
1
2
1

−
=
−
m
m
m
m
A
O
H
H
Wullo S.

SOLUTION
217
96
.
1
33
.
5
23
.
1
6
.
1
5178
.
1
6
.
1
15
5
.
3
12
9
.
2
0
)
4
.
3
3
.
4
(
)
(
)
(
025
.
0
2
2
2
2
2
2
1
2
1
2
1
=
=
=
=
=
+
−
−
=
+
−
−
−
=
z
z
n
n
y
x
zcal



m
m
Wullo S.

Estimation and hypothesis testing for two population proportion
• Let n1 and n2 be the sample size from the two population. If x and
y are the out come of interest then the point estimate for each
population is given by p1=x/n1 and p2=y/n2 respectively.
• The point estimates π1-π2 =p1-p2
• The interval estimate for the difference of proportions is given by
• If the sample size is large and n1p1>5, n1 (1-p1)>5, n2p2>5, then
218







 −
+
−

−
2
2
2
1
1
1
2
2
1
)
1
(
)
1
(
n
p
p
n
p
p
z
p
p 
Wullo S.

Hypothesis testing for two proportions
• To test the hypothesis
Ho: π1-π2 =0
VS
HA: π1-π2 ≠0
The test statistic is given by
219
2
2
2
1
1
1
2
1
2
1
)
1
(
)
1
(
)
(
)
(
n
p
p
n
p
p
p
p
zcal
−
+
−
−
−
−
=


Wullo S.

Small sample size
• If the sample size is small and n1p1>5, n2p2<5,
then use student’s t-statistic at n1+ n2-2
degrees of freedom with the given level of
significant.
220
Wullo S.

Chi-square test
• In recent years, the use of specialized statistical methods for
categorical data has increased dramatically, particularly for
applications in the biomedical and social sciences.
• Categorical scales occur frequently in the health sciences, for
measuring responses.
• E.g.
• patient survives an operation (yes, no),
• severity of an injury (none, mild, moderate, severe),
and
• stage of a disease (initial, advanced).
• Studies often collect data on categorical variables that can be
summarized as a series of counts and commonly arranged in
a tabular format known as a contingency table
221
Wullo S.

Chi-square Test Statistic cont’d…
• As with the z and t distributions, there is a different chi-square
distribution for each possible value of degrees of freedom.
Chi-square distributions with a small number of degrees of freedom
are highly skewed; however, this skewness is attenuated as the
number of degrees of freedom increases.
The chi-squared distribution is concentrated over nonnegative
values. It has mean equal to its degrees of freedom (df), and its
standard deviation equals √(2df ). As df increases, the distribution
concentrates around larger values and is more spread out.
The distribution is skewed to the right, but it becomes more bell-
shaped (normal) as df increases.
222
Wullo S.

The degrees of freedom for tests of hypothesis that involve an rxc
contingency table is equal to (r-1)x(c-1);
223
Wullo S.

Test of Association
• The chi-squared (2) test statistics is widely used in the
analysis of contingency tables.
• It compares the actual observed frequency in each group
with the expected frequency (the later is based on theory,
experience or comparison groups).
• The chi-squared test (Pearson’s χ2) allows us to test for
association between categorical (nominal!) variables.
• The null hypothesis for this test is there is no association
between the variables. Consequently a significant p-value
implies association.
224
Wullo S.

Test Statistic: 2-test with d.f. = (r-1)x(c-1)
225
( )

−
=
j
i ij
ij
ij
E
E
O
,
2
2

n
C
R
i
E
j
i
th
ij

=

=
total
grand
al
column tot
j
total
raw th
Oij=observed frequency, Eij=expected frequency of the cell at the
juncture of I th raw & j th column
Wullo S.

Procedures of Hypothesis Testing
1. State the hypothesis
2. Fix level of significance
3. Find the critical value (x2 (df, α))
4. Compute the test statistics
5. Decision rules; reject null hypothesis if test statistics > table
value.
226
Wullo S.

Test of associations for 2x2 tables
• If we call the frequencies in the four cells of 2x2 table a, b, c and d then
the table is given by
227
Disease status Row total
D ND
Exposur
e Status
E a b a+b
NE c d c+d
Column total a+c b+d N
Wullo S.

Test of association
• If the contingency table is 2x2
• Is the table is rxc then
228
( )
)
)(
)(
)(
(
2
2
d
c
b
a
d
b
c
a
bc
ad
n
+
+
+
+
−
=

( )

−
=
j
i ij
ij
ij
E
E
O
,
2
2

Wullo S.

Assumptions of the 2 - test
The chi-squared test assumes that
• Data must be categorical
• The data be a frequency data
– the numbers in each cell are ‘not too small’. No expected frequency should be
less than 1, and
– no more than 20% of the expected frequencies should be less than 5.
• If this does not hold row or column variables categories can
sometimes be combined (re-categorized) to make the expected
frequencies larger or use Yates continuity correction.
229
Wullo S.

Yates correction
• It is a requirement that a chi-squared test be applied to discrete data. Counting
numbers are appropriate, continuous measurements are not. Assuming
continuity in the underlying distribution distorts the p value and may make false
positives more likely.
• Frank Yates proposed a correction to the chi-squared formula. Adding a small
negative term to the argument. This tends to increase the p-value, and makes
the test more conservative, making false positives less likely. However, the test
may now be *too* conservative.
• Additionally, chi squared test should not be used when the observed values in a
cell are <5. It is, at times not inappropriate to pad an empty cell with a small
value, though, as one can only assume the result would be more significant with
no value there.
230
Wullo S.

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