Module 8-S M & T C I, Regular.pptx

Module 8: Statistical Methods and Testing in
curriculum and Instruction
Module Code: TECS: 624
Credit hours: 2
Prof.Omprakash H M
Department of Curriculum and Instructons
College of Education and Behavioral Sciences
Bule Hora University, Adola, Ethiopia

1. Introduction to statistics in curriculum and
Instruction
1.1 The definition of statistics and other related
terms
1.2 Descriptive statistics
1.3 Inferential statistics
1.4 Function and significance of statistics in
education
1.5 Types and levels of measurement scale

1.1 The definition of statistics and
other related terms
Statistics is:
 The scientific study of handling of
quantitative information.
 Concerned with the scientific methods for
collecting, organizing, summarizing,
presenting and analyzing data as well as
deriving valid conclusions and making
reasonable decisions on the basis of this
analysis.
 The systematic collection of numerical data
and its interpretation.

The knowledge of statistics enables us to:
o Present our result in summarized, more meaningful and
convenient form.
o Draw generalization and make prediction.
o In short, it helps us to condense (e.g. tabulation and
classification), compare (e.g. grand total, percentage,
mean, median, range, variance, etc.), forecast (e.g.
regression analysis), estimate the result of the data under
consideration and to test the hypothesis.

I. Variable
o It refers to a property or character of an
object or events that can take on different
values.
Example: Sex, age, grade, height, weight,
intelligence, attitude, depression, etc.
We dichotomize the concept of a variable
in terms of :
a) Independent variable
o The variable which is manipulated by the
researcher
o It is the variable that the experimenter
control it (e.g. teaching material,
motivation, stress, confidence, etc.)

b) Dependent Variable
oVaries as the result of independent variable
oIt is the result of the study (data)
Example: Achievement scores, self-esteem scores,
etc.
II. Data
o The measurements/observations recorded about
the individuals.
o There are two types of data. These are:
a) Numerical/Quantitative/Measurement
Variable/Data

a) Numerical/Quantitative/Measurement Variable/Data
o Is classified according to some characteristics and obtained by
counting or measurements.
Example: Grades on a test, weight, height, self-esteem scores,
number of children, etc.
o We are interested in ‘how much’ of some property a particular
object represents.
There are two types of numerical/ quantitative
data/variable. These are:
1) Discrete data/variables
o The data/variables which can take on integer
values/ whole numbers only (0,1,2,3,---) and are
usually obtained by counting

o It is the variables that have only a countable number of distinct
possible values.
Example. The no. of students in the class, the no. of households,
cows in a village, etc.
2. Continuous Data/Variables
o The data which can take on any value in a set of real
numbers.
Example: length, weight, temperature and age, self-esteem
scores, etc.
b) Categorical/ Qualitative/ Frequency Data/Variables
o Are classified according to some attributes
Example. Gender/sex (male/female), marital status (single,
married, divorced, widowed), hair colors (brown, yellow, red,
gray, etc.)
o They have only discrete data

Exercise.
Identify the types of data collected
on 50 Freshman psychology students
given below.
a) Their age
b) Their height
c) Their region/city administration
d) No. of brothers and sisters they
have
e) No. courses they have enrolled in
f) Whether full time/part time
student
g) Their sex/gender

Score
o It is the raw data collected
o It is an individual value for a variable
Example: Score on a test, self-esteem score
1.2 Descriptive Statistics
Descriptive Statistics
o Refers to procedures for organizing,
summarizing and describing quantitative data
about the sample or about the population
o Does not involve the drawing of conclusions or
drawing inferences from the sample or population

o Data can be summarized by:
1) Tabular representation- Frequency distribution
2. Graphic Representation
o Bar-graph, phi chart, histogram, polygon, etc.
3. Numerical Representation
o A single value present many numbers such as:
a) Measures of central tendency
b) Measures of variability
c) Measures of association/ relationship, etc.
1.3 Inferential Statistics
 It is the method that used to draw conclusions or
inferences about characteristics of populations based
on data from a sample.

o It uses data gathered from a sample to make
decisions/inferences about the larger population from which
the samples was drawn.
 Statistical inference is the process of making an estimate,
prediction, or decision about a population based on a sample
 What can we infer about a population’s parameters based on
a sample’s statistics
 The major inferential statistical methods are:
 The t-test, Analysis of Variance (ANOVA), Analysis of
Covariance (ANCOVA), Regression Analysis, etc.

Population
o It is the set of all the individuals of interest in a particular study
o Anything we might be interested in studying
Example: College students, elder people, single parent families,
people with depression, etc.
Sample
o A set of data or individuals selected from a population
oIt is the subset of the population
oUsually intended to represent the population in a research study

Parameter
oIt is a descriptive measure of a population
oIt is a value that describe a population
oUsually derived from measurements of the individuals in the
population
oIt is the real entities of interest Example. Mean, variance,
standard deviation, etc. of the population
Statistic
o A descriptive measure of a sample
o A value that describes of a sample
o Usually derived from measurements of individuals in the
sample.
o Are guesses at reality
Example: Mean, variance, standard deviation, etc. of the

1.4 Function and significance of statistics in education
Function of Statistics:
1. It helps in collecting and presenting the data in a systematic
manner.
2. It helps to understand unwisely and complex data by
simplifying it.
3. It helps to classify the data.
4. It provides basis and techniques for making comparison.
5. It helps to study the relationship between
different phenomena.
6. It helps to indicate the trend of behaviour.
7. It helps to formulate the hypothesis and test it.
8. It helps to draw rational conclusions.

Significance of Statistics in Education
1. It helps the teacher to provide the most exact
type of description.
2. It makes the teacher definite and exact in
procedures and thinking.
3. It enables the teacher to summarize the results in
a meaningful and convenient form.
4. It enables the teacher to draw general conclusions.
5. It helps the teacher to predict the future perfor-
mance of the pupils.
6. Statistics enables the teacher to analyse some of the
causal factors underlying complex and otherwise be-
wildering events.

1.5 Types and levels of measurement scale
1. What is measurement?
 Measurement is the process of assigning
numbers to objects and events according to
logically acceptable rules.
2. What are levels/scales of measurement?
 There are four levels/scales of me
asurement, these are:
 Nominal Scale
 Ordinal Scale
 Interval Scale and
 Ratio Scale

 Before discussing these one by one, it is better to look at
some of the attributes possessed by these scales.
a) Magnitude: The quantitative value that exist to an
attribute, will determine whether the value of a given attribute
is >, < or = than other
b) Equal Interval: The same magnitude (value interval) with in
the same interval of any other two points along the same scale
c) Absolute zero point: Refers to the absence or presenc of true
zero (0) point. Whether the attribute has no value at all.

 Nominal Scale
A Nominal Scale is a measurement scale, in which numbers serve
as “tags” or “labels” only, to identify or classify an object.
A nominal scale measurement normally deals only with non-
numeric (quantitative) variables or where numbers have no
value. Below is an example of Nominal level of measurement.
 Also known as classificatory scale because it has
the property of classification and labeling.
 The objects are classified or labeled without
ranking or order associated with such classified data.
 It reflects qualitative differences rather than
quantitative ones

 The least precise method of quantification
and does not have any of the three
attributes (magnitude, equal interval and
absolute zero point)
 Uses categorical/qualitative variables and
Found by counting
Example: Yes/No, Pass/Fail, Male/Female,
political party affiliation (democratic,
republican 0r independent), nationality,
race, occupation, religious affiliation,
football players number,

 Ordinal Scale
Ordinal Scale is defined as a variable
measurement scale used to simply depict the
order of variables and not the difference
between each of the variables. These scales are
generally used to depict non-mathematical
ideas such as frequency, satisfaction,
happiness, a degree of pain, etc. It is quite
straight forward to remember the
implementation of this scale as ‘Ordinal’
sounds similar to ‘Order’, which is exactly the
purpose of this scale.

 It is also termed as ranking scale and
possesses only the attributes of magnitude.
 Incorporates the property of nominal scale
and in addition it introduces the meaning of
ordering (ranking).
 Has no absolute values and the real
difference between adjacent ranks may not
be equal.
 Numbers are used as labels and do not
indicate amount or quantity. It is used
simply as description.
 Uses categorical/qualitative variables.

Example: Students in rank, professional career
structures, likert scale of pain (No pain, low pain,
moderate pain, high pain), marathon runners medals
award (Gold, silver, bronze), position at the end of
race-1st, 2nd, 3rd, etc.), state of buildings (Excellent
condition, moderate condition, poor condition).

 Interval Scale
Interval Scale is defined as a numerical scale where the order
of the variables is known as well as the difference between
these variables. Variables that have familiar, constant, and
computable differences are classified using the Interval scale.
It is easy to remember the primary role of this scale too,
‘Interval’ indicates ‘distance between two entities’, which is
what Interval scale helps in achieving.
 Is an arbitrary scale based on equal units of measurement
that indicate how much of a given characteristics is present.
 Possesses two of the attributes, that are magnitude and
equal intervals. Zero point is arbitrary in this scale.
 Uses quantitative variables.

 The difference amount of characteristics possessed by
persons with scales of 90 and 91 is assumed to be
equivalent to that between persons with scores of 60 and
61 or the difference in temperature between 10OF and
20OF is the same as the difference between 80OF and
90OF
 Its primary limitation is the lack of absolute zero point.
 It does not have the capacity to measure the complete
absence of the trait, and rations between numbers on the
scale are not meaningful (arbitrary).
 we cannot say, for example, that 40OF is half as hot as
80OF or twice as hot as 20OF

oTherefore, numbers cannot be multiplied and divided
directly.
oPsychological tests and inventories are interval scales and
have this limitation, although they can be added, subtracted,
multiplied and divided.
Example: IQ test, Temperature, Motivation Score, Attitude
 Ratio Scale
Ratio Scale is defined as a variable measurement scale that
not only produces the order of variables but also makes the
difference between variables known along with information on
the value of true zero. It is calculated by assuming that the
variables have an option for zero, the difference between the
two variables is the same and there is a specific order between
the options.

oThe numerals of the ratio scale have the qualities of real
numbers and can be added, subtracted, multiplied and divided.
 We can say that 5 grams is one half of 10 grams, 15 grams is
three times 5 grams, a weight of 0kg means no weight (mass) at
all.
All statistical measures can be used for a variable measured at
the ratio level, since all the necessary mathematical operations
are defined.
Almost exclusively confined to use in physical sciences. In
other words, it is almost non existence in psychology and other
social sciences.
oUses quantitative variables
Example: Mass, length, time,
energy, number of correct answers
on a test.

2. Introduction to SPSS Software

A frequency distribution is an overview of all distinct
values in some variable and the number of times they
occur. That is, a frequency distribution tells
how frequencies are distributed over
values. Frequency distributions are mostly used for
summarizing categorical variables.
3. Frequency Distribution:

Frequency Distribution:
 Is the easiest method of organizing data, which converts
raw data into a meaningful pattern for statistical analysis.
Lists all the classes and the number of values (frequency)
that belong to each class
Exhibits how the frequencies are distributed over various
categories
Besides it makes the data easy and brief to be understood.
However, individual score loss its identity.

Frequency Distributions and Graphs for
Ungrouped Data
a) Frequency Distributions for Ungrouped Data
First we arrange numbers/data in ascending or descending
order.
 Then, we tally the numbers/data
Finally, we use a frequency table
E.g. Age of 20 students
25,20,18,23,21,23,22,24,19,20,18,26,18,25,18,24,18,21,24,18

Table 1: Frequency Distribution of Age of 20
students in a class
Age Tally Frequency (f)
18 ////// 6
19 / 1
20 // 2
21 // 2
22 / 1
23 // 2
24 /// 3
25 // 2
26 / 1
Total 20

b) Graphs for ungrouped/
Discrete Data
What is graph in statistics?
 It is the geometrical
image of frequency
distribution
It is when frequency
distributions are converted
into visual models to
facilitate understanding
(e.g. pie chart/circle graph,
bar graph, line graph,
histogram, frequency
polygon, etc.)

For ungrouped data we use bar graph, pie chart/circle graph, line
graph, etc.
a) Bar Graph/Chart
It is a detached graph made up of bars whose heights represent
the frequencies of the respective categories.
It is used to summarize categorical data.

b) Pie chart/Circle Graph
It is divided into portions that
represent the frequency, relative
frequency, or percentage of a
population or a sample belonging to
different categories.
Relative frequency of a category =
Frequency of a category/Sum of all
frequencies
Percentages = Relative frequency
x 100%

c) Line Graph
It is a graph consists of independent and dependent
variables, where the former (IV) is written on x-axis and
the latter (DV) on y-axis and points are joined by line
segments.

Frequency Distribution and Graphs for
Grouped Data
 Some basic technical terms when continuous frequency
distribution is formed or data are classified (grouped) according
to class interval.
a) Class limits
 It is the lowest and the highest values that can be included in
the class. E.g. 30-40; 30 is the lower (lowest) limit (L) and 40
is the upper (highest) limit (U).
 Sometimes, a class limit is missing either at the end of the
first class interval or at the end of the last class interval or
both are not specified. In such case the classes are known as
open-end classes.
Example: below 18, 18-20, 21-23, 24-26, 27 and above years of
age

b) Class boundaries: Are the midpoints of the upper class
limit of one class and the lower class limit of the
subsequent class (LCB = LCL-0.5, UCB = UCL+.0.5). Example:
class limits class boundaries
30-40 29.5-40.5
41-50 40.5-50.5
c) Class interval
It is the size of each grouping of data 50-75, 76-101, 102-
127, are class intervals.
d) Width or size of the class interval (c) c = Range/no. of
classes
The difference between the lower or upper class limit of
one class and the lower or upper class limit of the previous
class.

e) Number of Class Intervals
Should not be too many (5-20).
To decide the number of class intervals, we
choose the lowest and the highest values.
The difference between them will enable us
to determine class intervals.
f) Range (R)
 The difference between the largest and the
smallest value of the observation.
g) Mid value or mid point
 It is the central point of a class boundary/interval
 It is found by adding the upper and lower limits of a
class and dividing the sum by 2
Mid point = L+U
2

h) Frequency (f)
 Number of observations falling within a
particular class interval
Weight (in Kg) No. of persons (f)
30-40 25
41-50 15
51-60 45
Total 85

Rules to Construct Frequency Distribution of
Grouped Data
1) There should be between 5-20 classes
2) The class width should be an odd number.
This ensures that the midpoint of each class
has the same place value as the data. Always
we have to round up.
3) The classes must be mutually exclusive.
Mutually exclusive classes have non over
lapping class limits so that data cannot be
placed into two classes E.g. 10-20,
21-31, 32-42, 43-53
4) The classes must be continuous. Even if
there are no values in a class, the class must
be included in the frequency distribution

5. The classes must be exhaustive. There
should be enough classes to accommodate
all the data
6. The classes must be equal in width. This
avoids a distorted view of data. Exception
is in open-ended

E.g. The number of hours 40 employees spends on their job
for the last 30 working days is given below. Construct a
group frequency distribution for these data using 8 classes
62 50 35 36 31 43 43 43
41 31 65 30 41 58 49 41
37 62 27 47 65 50 45 48
27 53 40 29 63 34 44 32
58 61 38 41 26 30 47 37

Solution:
Step1. Max= 65, Mim= 26 that Range= 65- 26= 39
Step2. It is given 8 classes
Step3. Class width C= 39/8 = 4.875 ~ 5
Step4. Starting point 26= lower limit of the first class.
Hence the lower class limits become
26 31 36 41 46 51 56 61
Step 5. The upper limit of the first class = 31- 1= 30 and
hence the upper class limit become
30 35 40 45 50 55 60 65

The lower and the upper class limits can
be written as follows:

Exercise:
Construct a group frequency distribution for the
weight in kg of 35 students data using 8 classes
52 60 56 81 72 72 55
73 60 58 55 65 75 56
55 81 64 60 50 70 58
70 55 52 56 55 50 64
65 60 50 52 60 53 64

 Some basic types of graph we use for
grouped/ continuous data are the
following:
a) Histogram
 A histogram is a graph in which class
limits/boundaries are marked on the horizontal
axis (x) and the frequencies, relative frequencies or
percentages are marked on the vertical axis (y).
 The frequencies, relative frequencies, percentages
are represented by the heights of bars.
 The bars are drawn adjacent to each other.
 A histogram is called a frequency, a relative
frequency or a percentage histogram.
 Used to summarize measurement data.

 Relative frequency = Frequency of aclass
Sum of all frequencies
 Percentage = Relative Frequency x 100%
Weight (in Kg) No. of persons
(f)
Relative
frequency
Percentage
(%)
29.5-40.5 25 0.25 25
40.5-50.5 15 0.15 15
50.5-60.5 45 0.45 45
60.5-70.5 10 0.10 10
70.5-80.5 5 0.05 5
Total 100 1 100

B) Frequency Polygon
A graph formed by joining the midpoints of the tops of
successive bars in a histogram with a straight lines.
A polygon with a relative frequency marked on the vertical axis
is known as a relative frequency polygon.
Similarly a polygon with percentages marked on the vertical axis
is called a percentage polygon.
Class
boundaries
Frequency Mid point
5.5-10.5 1 8
10.5-15.5 2 13
15.5-20.5 6 18
20.5-25.5 8 23
25.5-30.5 3 28
Total 30

C) The Cumulative Frequency Graph/
Ogive
 It is a curve drawn for the
cumulative frequency, cumulative
relative frequency or cumulative
percentage by joining with straight
lines the dots marked above the
upper boundaries of classes at
height equal to cumulative
frequencies, cumulative relative
frequencies or cumulative
percentages of the respective
classes.

Data Example:
Class
Boundary
Frequency Cumulative
Frequency
Cumulative
Relative Frequency
Cumulative
Percentage
5.5-10.5 1 1 0.05 5
10.5-15.5 2 3 0.15 15
15.5-20.5 6 9 0.45 45
20.5-25.5 8 17 0.85 85
25.5-30.5 3 20 1 100
Total 20

Less than and More than Cumulative Frequency
Example:
Class limits Frequency Less than
Cf
More than Cf
5-9 1 1 35
10-14 2 3 34
15-19 6 9 32
20-24 8 17 26
25-29 15 32 18
30-34 3 35 3
Total

4. Normal Curve and Standard Score
Definition: The normal distribution curve can be used to study
many variable that are approximately normal.
 Normal distribution curve depend on the mean and standard
deviation.
 when the majority of the data fall to the right of
the mean, the distribution said to be negatively
skewed. The tail of the curve is to the left and vise a
versa.
Skewness: It refers to a distortion or asymmetry that deviates from
the symmetrical bell curve, or normal distribution, in a set of data. If
the curve is shifted to the left or to the right, it is said to be skewed.
Skewness can be quantified as a representation of the extent to which
a given distribution varies from a normal distribution. A normal
distribution has a skew of zero, while a lognormal distribution, for
example, would exhibit some degree of right-skew.

Key Points:
Skewness, in statistics, is the degree of asymmetry observed in a
probability distribution.
Distributions can exhibit right (positive) skewness or left
(negative) skewness to varying degrees. A normal distribution
(bell curve) exhibits zero skewness.
Investors note right-skewness when judging a return distribution
because it, like excess kurtosis, better represents the extremes
of the data set rather than focusing solely on the average.

 Skewedness: when the majority of the data fall to the left or
right of the mean, the distribution said to be skewed.
The mean of positively skewed data will be greater than the median.
In a distribution that is negatively skewed, the exact opposite is the
case: the mean of negatively skewed data will be less than the
median. If the data graphs symmetrically, the distribution has zero
skewness, regardless of how long or fat the tails are.

Kurtosis: it is the degree of peakedness of a distribution
It is a measure of whether the data are flat/peaked relative to a
normal distribution.
The data sets with high kurtosis tend to have a distinct sharp peak
near the mean (+ve kurtosis), declining rather rapidly and have
heavy tail.
The data sets with low kurtosis tend to have a flat top (-ve
kurtosis) near the mean rather than a sharp peak.
Kurtosis is a statistical measure that defines
how heavily the tails of a distribution differ
from the tails of a normal distribution. In
other words, kurtosis identifies whether the
tails of a given distribution contain extreme
values.

1. Mesokurtic:
Data that follows a mesokurtic distribution
shows an excess kurtosis of zero or close to
zero. This means that if the data follows a
normal distribution, it follows a mesokurtic
distribution.
Types of Kurtosis:
The types of kurtosis are determined by the excess kurtosis of a
particular distribution. The excess kurtosis can take positive or
negative values, as well as values close to zero.

2. Leptokurtic:
Leptokurtic indicates a positive excess
kurtosis. The leptokurtic distribution shows
heavy tails on either side, indicating
large outliers. In finance, a leptokurtic
distribution shows that the investment
returns may be prone to extreme values on
either side. Therefore, an investment whose
returns follow a leptokurtic distribution is
considered to be risky.

3. Platykurtic:
A platykurtic distribution shows a negative
excess kurtosis. The kurtosis reveals a
distribution with flat tails. The flat tails
indicate the small outliers in a
distribution. In the finance context, the
platykurtic distribution of the investment
returns is desirable for investors because
there is a small probability that the
investment would experience extreme
returns.

Characteristics of Normal Curve:
 The curve is a bell-shaped distribution of a variables.
 The mean, median and mode are equal and located at the
center of the distribution.
 It is a unimodal distribution /only one mode/
 The curve is symmetrical about the mean.
 The curve is continuous, i. e there are no gap.
 The curve never touches the x-axis. But it gets increasingly
closer to the axis.
 The total area under the normal distribution curve is equal to
1 or 100%. This fact may seem unusual, since the curve never
touches the x-axis.
 The area under the normal curve that lies b/n
-1 and +1 s is approximately 0.68or 68%
 -2 and +2 s is approximately 0.95 or 95%
 -3 and +3 s is approximately 0.997 or 99.7%

5. Confidence Interval for the Mean, Proportions,
and Variances
In statistics, a confidence interval (CI) is a type
of estimate computed from the statistics of the observed
data. This gives a range of values for an
unknown parameter (for example, a population mean). The
interval has an associated confidence level that gives the
probability with which the estimated interval will contain the
true value of the parameter. The confidence level is chosen
by the investigator. For a given estimation in a given sample,
using a higher confidence level generates a wider (i.e., less
precise) confidence interval. In general terms, a confidence
interval for an unknown parameter is based on sampling
the distribution of a corresponding estimator.

 Confidence Interval for the Mean:
A confidence interval gives an estimated range of values which
is likely to include an unknown population parameter, the
estimated range being calculated from a given set of sample
data.
The common notation for the parameter in question is . Often,
this parameter is the population mean , which is estimated
through the sample mean .
The level C of a confidence interval gives the probability that the
interval produced by the method employed includes the true
value of the parameter .

Example:
Suppose a student measuring the boiling temperature of a
certain liquid observes the readings (in degrees Celsius)
102.5, 101.7, 103.1, 100.9, 100.5, and 102.2 on 6 different
samples of the liquid. He calculates the sample mean to be
101.82. If he knows that the standard deviation for this
procedure is 1.2 degrees, what is the confidence interval for
the population mean at a 95% confidence level?
In other words, the student wishes to estimate the true
mean boiling temperature of the liquid using the results of
his measurements. If the measurements follow a normal
distribution, then the sample mean will have the
distribution N ( , ). Since the sample size is 6, the
standard deviation of the sample mean is equal to 1.2/sqrt
(6) = 0.49.

In the example above, the student calculated the sample mean of the
boiling temperatures to be 101.82, with standard deviation 0.49. The
critical value for a 95% confidence interval is 1.96, where (1-0.95)/2
= 0.025. A 95% confidence interval for the unknown mean is
((101.82 - (1.96*0.49)), (101.82 + (1.96*0.49))) = (101.82 - 0.96,
101.82 + 0.96) = (100.86, 102.78).
As the level of confidence decreases, the
size of the corresponding interval will
decrease. Suppose the student was
interested in a 90% confidence interval for
the boiling temperature. In this case, C =
0.90, and (1-C)/2 = 0.05. The critical
value z* for this level is equal to 1.645, so
the 90% confidence interval is ((101.82 -
(1.645*0.49)), (101.82 + (1.645*0.49))) =
(101.82 - 0.81, 101.82 + 0.81) = (101.01,
102.63)

 Confidence Interval for the Proportions:
Suppose we wish to estimate the proportion of people with
diabetes in a population or the proportion of people with
hypertension or obesity. These diagnoses are defined by specific
levels of laboratory tests and measurements of blood pressure
and body mass index, respectively. Subjects are defined as
having these diagnoses or not, based on the definitions. When
the outcome of interest is dichotomous like this, the record for
each member of the sample indicates having the condition or
characteristic of interest or not. Recall that for dichotomous
outcomes the investigator defines one of the outcomes a
"success" and the other a failure. The sample size is denoted by
n, and we let x denote the number of "successes" in the sample.

For example, if we wish to estimate the proportion of
people with diabetes in a population, we consider a
diagnosis of diabetes as a "success" (i.e., and individual
who has the outcome of interest), and we consider lack of
diagnosis of diabetes as a "failure." In this example, X
represents the number of people with a diagnosis of
diabetes in the sample. The sample proportion is p̂ (called
"p-hat"), and it is computed by taking the ratio of the
number of successes in the sample to the sample size, that
is:

The point estimate for the population proportion is the sample
proportion, and the margin of error is the product of the Z value
for the desired confidence level (e.g., Z=1.96 for 95% confidence)
and the standard error of the point estimate. In other words, the
standard error of the point estimate is:
This formula is appropriate for samples with at least 5 successes
and at least 5 failures in the sample. This was a condition for the
Central Limit Theorem for binomial outcomes. If there are fewer
than 5 successes or failures then alternative procedures,
called exact methods , must be used to estimate the population
proportion.

Example: During the 7th examination of the Offspring cohort
in the Framingham Heart Study there were 1219 participants
being treated for hypertension and 2,313 who were not on
treatment. If we call treatment a "success", then x=1219 and
n=3532. The sample proportion is:
This is the point estimate, i.e., our best estimate of the
proportion of the population on treatment for hypertension is
34.5%. The sample is large, so the confidence interval can be
computed using the formula:

Substituting our values we get
which is
So, the 95% confidence interval is (0.329, 0.361).
Thus we are 95% confident that the true proportion of
persons on antihypertensive medication is between 32.9%
and 36.1%.

 Confidence Interval for the Variances
When using a sample to calculate a statistic we are estimating a
population parameter. It is just an estimate and the sample due to
the nature of drawing a sample may not create a value (statistic)
that is close to the actual value (parameter).
We can calculate confidence interval about the statistic to
determine where the true and often unknown parameter may
exist. This includes the calculation of a variance statistic.
If you were to draw many different samples all the same size from
a population and plot the variance statistic the resulting
distribution is likely to fit a χ2 distribution. Plotting the means
creates a normal distribution which is symmetrical and produced
symmetrical confidence intervals. The χ2 distribution is not
symmetrical and will produce asymmetrical intervals.

Confidence Intervals for Variances and Standard Deviations:
Estimates of population means can be made from sample means,
and confidence intervals can be constructed to better describe
those estimates. Similarly, we can estimate a population standard
deviation from a sample standard deviation, and when the original
population is normally distributed, we can construct confidence
intervals of the standard deviation as well.
The Theory:
Variances and standard deviations are a very different type of
measure than an average, so we can expect some major differences
in the way estimates are made.
We know that the population variance formula, when used on a
sample, does not give an unbiased estimate of the population
variance. In fact, it tends to underestimate the actual population
variance. For that reason, there are two formulas for variance, one
for a population and one for a sample.

The sample variance formula is an unbiased estimator of the
population variance. (Unfortunately, the sample standard deviation is
still a biased estimator.)
Also, both variance and standard deviation are nonnegative
numbers. Since neither can take on a negative value, the domain of
the probability distribution for either one is not (−∞,∞)(−∞,∞), thus
the normal distribution cannot be the distribution of a variance or a
standard deviation. The correct PDF must have a domain
of [0,∞)[0,∞). It can be shown that if the original population of data
is normally distributed, then the
expression (n−1)s2σ2(n−1)s2σ2 has a chi-square distribution
with n−1n−1 degrees of freedom.

The chi-square distribution of the quantity (n−1)s2σ2(n−1)s2σ2 allows us to
construct confidence intervals for the variance and the standard
deviation (when the original population of data is normally
distributed). For a confidence level 1−α1−α, we will have the
inequality χ21−α/2≤(n−1)s2σ2≤χ2α/2χ1−α/22≤(n−1)s2σ2≤χα/22.
Solving this inequality for the population variance σ2σ2, and then
the population standard deviation σσ, leads us to the following pair
of confidence intervals.
It is worth noting that since the chi-
square distribution is not symmetric,
we will be obtaining confidence
intervals that are not symmetric
about the point estimate.

Example:
A statistician chooses 27 randomly selected dates, and when examining
the occupancy records of a particular motel for those dates, finds a
standard deviation of 5.86 rooms rented. If the number of rooms rented
is normally distributed, find the 95% confidence interval for the
population standard deviation of the number of rooms rented.
For a sample size of n=27n=27, we will
have df=n−1=26df=n−1=26 degrees of freedom. For a 95% confidence
interval, we have α=0.05α=0.05, which gives 2.5% of the area at each
end of the chi-square distribution. We find values
of χ20.975=13.844χ0.9752=13.844 and χ20.025=41.923χ0.0252=41.
923. Evaluating (n−1)s2χ2(n−1)s2χ2, we obtain 21.297 and 64.492. This
leads to the inequalities 21.297≤σ2≤64.49221.297≤σ2≤64.492 for the
variance, and taking square roots, 4.615≤σ≤8.0314.615≤σ≤8.031 for
the standard deviation.

Calculate the Variance and Standard Deviation of the
Data Given Below. 35, 45, 30, 35, 40, 25
Variance ad standard deviation for
grouped data


6. Hypothesis Testing with One and two Sample
Hypothesis is a suggested answer to the problem under
investigation. – John T. Townsend
A hypothesis is a tentative generalization, the validity of
which remains to be tested. – J. W. Best
A hypothesis is a proposition which can be put to test to
determine its validity. It may be proved correct or incorrect.
– Good & Hatt
A hypothesis is a conjectural/hypothetical statement of the
relation between two or more variables. – F. N. Kerlinger
Meaning of Hypothesis:
 On the basis of the definitions, we can say that hypothesis is an
assumption that is still not proved but shows the probable solution
of the problem or predicts the relationship between two or more
variables.
 The assumption is proved true or false by testing it.

 We will not have the solution to the problem until the
assumption is tested.
 Three points, regarding such assumptions, are very
important.
 The assumptions are made on the basis of previous
experiences or primary evidences or by thinking logically.
 Whether the assumptions are true or false is decided by
testing them.
 Testing of assumptions lead to the solution of the problem.
 Suppose we are watching some television programme and
suddenly the TV gets off.
 What will be our reactions to this problem? We start thinking
of the reasons of the problem like.

 Perhaps there is an interruption in the flow of electricity or
 There may be a problem in particular channel or
 There may be a loose connection of the cable with TV or
 There may be a problem in the system of cable operator.
 We will make such assumptions on the basis of our previous
experiences. Now we will check all the possible reasons of the
problem. For that, first, we will check if there is any problem in the
flow of electricity.
 If electric supply is found okay, we will check if other channels
are working or not.
 If other channels are found okay, we will check whether the
cable connection is proper or not.
 If everything is found okay, then we will call cable operator to
solve the problem.

 In this way, we will collect the evidence and
analyse it logically. By testing all the evidences, we
come to the conclusion about the solution of the
problem. We make many assumptions in our routine
life to find the solution to our daily problems.
Characteristics of a good hypothesis:
 A good hypothesis never opposes the universal truth and
natural law and rules.
 It is written in simple and easy language.
 Only one assumption is made in one hypothesis.
 The hypothesis is written in such a language that, after
testing, it can be clearly rejected or not rejected.

 Hypothesis is written in present tense because it is
not a prediction or opinion but it is an assumption
that is based on present factual information.
 A good hypothesis assures that the tool required for
testing it (hypo) is available or can be prepared
(developed) easily.
 Before formulating the hypothesis, it is assured
that the data will be available for testing it.
Types of Hypothesis:
It is very difficult to give such a classification of hypotheses as
can be accepted universally because different scholars have
classified the hypotheses in different ways.

 On the basis of different classifications, the types of hypotheses
can be described as follows.
 A hypothesis is of two types:
 Null hypothesis: Null hypothesis is a type of hypothesis in which
we assume that the sample observations are purely by chance. It is
denoted by H0.
 Alternate hypothesis: The alternate hypothesis is a hypothesis
in which we assume that sample observations are not by chance.
They are affected by some non-random situation. An
alternate/Declarative hypothesis are denoted by H1 ,H2, H3 Hn or
Ha.
 Some Special features of alternate /declarative hypotheses are as
follows:
 Researcher formulates the declarative hypotheses on the basis of
pre-experience, study of research material or on the basis of the
findings of previous researches.

 Such hypotheses are formulated on the basis of expected
findings of the research.
 Such hypothesis is accepted when null hypothesis is
rejected.
Hypothesis in Question Form:
 In this type of hypothesis, instead of expecting a certain
result, a questions is formed asking whether certain type of
result will be there or not.
E.g. in the context of the research title ‘STUDY OF EXAM
ANXIETY OF HIGHER SECONDARY SCHOOLS’ students in the
context of their stream’ the question type hypothesis will be as
follows:
 Is there difference between exam anxiety of arts, commerce
and science students of higher secondary schools? OR

 Is the exam anxiety of arts students of higher secondary
schools more than that of science students?
OR
 Is the exam anxiety of commerce students of higher
secondary schools more than that of science students? OR
 Is the exam anxiety of commerce students of higher
secondary schools more than that of arts students? OR
Null Hypothesis:
If, in the context of dependent variable, the hypothesis
indicates ‘no difference’ between two or more levels of
independent variable, it is called null hypothesis. Null
hypothesis indicates no relationship between two variables, if
correlational study is there.

 Null hypothesis is indicated by the symbol Ho. Such hypothesis is
also called ‘no difference’ type of hypothesis or ‘no relation’ type of
hypothesis.
 Let’s take examples to understand this type of hypothesis.
 For the study of the impact of instructional method on the
achievement of the students of grade nine in English, the null
hypotheses will be as follows:
Special features of Null Hypothesis:
 Researchers prefer to formulate null hypotheses due to their
some special features. These features are as follows:
 It is formulated objectively and not affected by the subjectivity of
the researcher.
 It believes in ‘no difference’ or ‘no relationship’. So the
researcher does not tend to be biased for certain type of the result
and works freely.

 It helps in making the entire research process objective
(unbiased).
 It is tested at certain level of significance.
Hypothesis Testing:
Hypothesis testing is a part of statistics in which we make
assumptions about the population parameter. So, hypothesis
testing mentions a proper procedure by analysing a random
sample of the population to accept or reject the assumption.
Steps of Hypothesis Testing:
 The process to determine whether to reject a null hypothesis
or to fail to reject the null hypothesis, based on sample data is
called hypothesis testing. It consists of four steps:
 Define the null and alternate hypothesis

 Define an analysis plan to find how to use sample data to
estimate the null hypothesis
 Some analysis on the sample data to create a single
number called ‘test statistic’.
 Understand the result by applying the decision rule to
check whether the Null hypothesis is true or not.
 If the value of t-test is less than the significance level we
will reject the null hypothesis, otherwise, we will fail to
reject the null hypothesis.
 Technically, we never accept the null hypothesis, we say
that either we fail to reject or we reject the null hypothesis.

Errors in hypothesis testing:
We have explained what is hypothesis testing and the steps to
do the testing. Now during performing the hypothesis testing,
there might be some errors.
 We classify these errors in two categories.
 Type-1 error: Type 1 error is the case when we reject the
null hypothesis but in actual it was true. The probability of
having a Type-1 error is called significance level alpha(α).
 Type-2 error:
A type II error is a statistical term used within the context
of hypothesis testing that describes the error that occurs
when one accepts a null hypothesis that is actually false. or

 Type 2 error is the case when we fail to reject the null
hypothesis but actually it is false. A type II error produces a
false negative, also known as an error of omission. For
example, a test for a disease may report a negative result,
when the patient is, in fact, infected.
 This is a type II error because we accept the conclusion of
the test as negative, even though it is incorrect.
 where as a type II error describes the error that occurs when
one fails to reject a null hypothesis that is actually false. The
error rejects the alternative hypothesis, even though it does
not occur due to chance.
 The probability of having a type-2 error is called beta(β).
Therefore,
 α= (Null hypothesis rejected | Null hypothesis is true)
 β= (Null hypothesis accepted | Null hypothesis is false)

Two-Tailed test:
All that we are interested in is a difference. Consequently, when
an experimenter wishes to test the null hypothesis, Ho: M1 - M2,
against its possible rejection and finds that it is rejected, then he
may conclude that a difference really exists between the two
means. But he does not make any assertion about the direction of
the difference.
Such a test is a non-directional test. It is also named as two-
tailed test, because it employs both sides, positive and negative,
of the distribution (normal or t distribution) in the estimation of
probabilities. Let us consider the probability at 5% significance
level in a two-tailed test with the help of figure 1.1

Mean of the distribution of scores concerning the difference
between sample means) Figure 1.1

 Therefore, while using both the tails of the distribution we may
say that 2.5% of the area of the normal curve falls to the right of
1.96 standard deviation units above the mean and 2.5% falls to
the left of 1.96 standard deviation units below the mean.
 The area outside these limits is 5% of the total area under the
curve. In this way, for testing the significance at the 5% level, we
may reject a null hypothesis if the computed error of the
difference between means reaches or exceeds the yardstick 1.96.
 Similarly, we may find that a value of 2.58 is required to test
the significance at the 1% level in the case of a two-tailed test.

One-tailed test:
 Here, we have reason to believe that the experimental group
will score higher on the mathematical computation ability test
which is given at the end of the session.
 In our experiment we are interested in finding out the gain in
the acquisition of mathematical computation skills (we are not
interested in the loss, as it seldom happens that coaching will
decrease the level of computation skill).
 In cases ,we make use of the one-tailed or directional test,
rather than the two-tailed or non-directional test to test the
significance of difference between means.
 Consequently, the meaning of null hypothesis, restricted to an
hypothesis of on difference with two-tailed test, will be somewhat
extended in a one-tailed test to include the direction--positive or
negative-of the difference between means.

One-tailed test at the 5% level
(Mean of the distribution of the score concerning the difference
between sample means) figure 1.2

 Thus 5% level of significance we will have 5% of the area. all in
one tail (at the 1.64 standard deviation unit above the mean rather
than having it equally divided into two tails as shown in Figure
1.1, for a two-tailed test.
 Consequently, in a one-tailed test for testing the difference
between large sample means, z score of 1.64 will be taken as a
yardstick at the 5% level of significance for the rejection of the
null hypothesis instead of 1.96 in the case of a two-tailed test.
 In the case of at distribution of small sample means, in making
use of the one-tailed test we have to look for the critical / values,
written in Table C of the Appendix, against the row (N1 + N2 - 2)
degrees of freedom under the columns labelled 0.10 and 0.02
(instead of 0.05 and 0.01 as in the case of two-tailed test) to test
the significance at 0.05 and 0.01% levels of significance,
respectively.

7. Two-way Analysis of Variance
Analysis Of Variance:
 we know the use of z and t tests for testing the
significance of the difference between the means of two
samples or groups. In practice where we have several
samples of the same general character drawn from a
given population and wish to know whether there are
any significant differences between the means of these
samples.
 In such situations, several differences between
several means are determined in a statistical technique
known by the name analysis of variance in which all
the data are treated together and a general null
hypothesis of no difference among the means of the
various samples or groups is tested.

Meaning of the Term 'Analysis of Variance :
A composite procedure for testing simultaneously the
difference between several sample means is known as the
analysis of variance.
It helps us to know whether any of the differences between
the means of the given samples are significant.
If the answer is 'yes', we examine pairs (with the help of the t
test) to see just where the significant differences lie. If the
answer is 'no', we do not proceed further.
The term 'analysis of variance' deals with the task of analysing
of breaking up the total variance of a large sample or a
population consisting of a number of equal groups or sub-
samples into components (two kinds of variances).

Two kinds of variances, given as follows:
1. "Within-groups" variance. This is the average variance of the
members of each group around their respective group
means, i.e. the mean value of the scores in a sample (as
members of each group may vary among themselves).
2. "Between-groups" variance. This represents the variance of
group means around the total or grand mean of all groups, i.e. the
best estimate of the population mean (as the group means may
vary considerably from each other).
For deriving more useful results, we can use variance as a measure
of dispersion (deviation from the mean) in place of some useful
measures like standard deviation. Consequently, the variance of an
individual's score from the grand mean may be broken into two
parts (as pointed out earlier), viz. within-group variance and
between groups variance.

In this way, the technique of analysis of variance as a single
composite test of significance, for the difference between
several group means demands the derivation of two
estimates of the population variance, one based on variance
of group means (between groups variance) and the other on
the average variance within the groups (within groups
variance.) Ultimately, the comparison of the size of between
groups variance and within-groups variance called F-ratio
denoted by between-groups variance within-groups variances
used as a critical ratio for determining the significance of
the difference between group means at a given level of
significance.

Procedure For Calculating The Analysis Of Variance:
As already pointed out the deviation of an individual's score
belonging to a sample or a group of population from the grand mean
can be is divided into two parts:
(i) deviation of the individual's score from group mean; and
(ii) deviation of the group mean from the grand mean.
Consequently, the total variance of the scores of all individuals
included in the study may be partitioned into within-group
variance and between-groups variance.
The formula used for the computation of variance ( ) is Σ X/N, i.e.
sum of the squared deviation from the mean value divided by
total frequencies.

Hence, by taking N as the common denominator, the total sum of
the squared deviation of scores around the grand or general mean
for all groups combined can be made equal to the sum of the two
partitioned, between groups and within-groups sum of squares.
Mathematically,
Total sum of squares (around the general mean)= between-groups
sum of squares + within-groups sum of squares or
Hence the procedure for the analysis of variance involves the
following main tasks:
Computation of total sum of squares ( )
Computation of between-groups sum of squares ( )
Computation of within-groups sum of squares ( )
Computation of F-ratio
Use of t test (if the need for further testing arises).

We understand these steps by adopting the following
terminology: X = Raw score of any individual included in the
study (any score entry in the given table)
 ΣX = Grand Sum
Grand or general mean
 X1 , X2 denote scores within first group, second group. ... n1,
n2, n3 =
 n1 , n2 , n3 = No. of individuals in first, second and third
groups
….. denote means of the first group, second group
N = Total No. of score or frequencies
 = n1+n2+n3+……..

 Step 1. Arrangement of the given table and computation of
some initial values.
 In this step, the following values needed in computation
are calculated from the experimental data arranged in proper
tabular form:
 Sum of square …and the grand sum , ΣX
 Group means ,…and the grand sum mean
 Correlation term C computed by the formula
 C =

Source of
variation
Sum of
squares
df Mean square
variance
Between-
groups
K-1
Within-groups N-K

Let us illustrate now the whole process of using the analysis of
variance technique with the help of an example,
Example 1.1: The aim of an experimental study was to determine
the effect of three different techniques of training on the learning
of a particular skill. Three groups, each consisting of seven
students of class PG in Adola college, assigned randomly, were
given training through these different techniques. The scores
obtained on a performance test were recorded as follows:
Group I Group II Group III
3 4 5
5 5 5
3 3 5
1 4 1
7 9 7
3 5 3
6 5 7

Test the difference between groups by adopting the analysis of
variance technique.
Solution:
Step 1. Original table computation
Rating of the coaching experts
Groups I Groups I Groups I Groups I
3 4 5 12
5 5 5 15
3 3 5 11
1 4 1 6
7 9 7 23
3 5 3 11
6 5 7 18

Here n1= n2 = n3 = 7 , N= n1+n2+n3 = 21
Group means =
Correction term C =

Step 2. Squared-table computation
Total
9 16 25 50
25 25 25 75
9 9 25 43
1 16 1 18
49 81 49 179
9 25 9 43
36 25 49 110

Step 7. Calculation of F-ratio
Source of
variation
Sum of squares df Mean square
variance/df
Between-groups 2 3.72/2 = 1.86
Within-groups 18 75.43/18 = 4.19
= 1.86/4.19 = 0.444

Step 8. Interpretation of F-ratio. The F-ratio table (Table R given in
the Appendix) is referred to for 2 degrees of freedom for smaller
mean square variance on the left-hand side, and for 18 degrees of
freedom for greater mean square variance across the top. The
critical values of F obtained by interpolation are as follows:
Critical value of F = 19.43 at 0.05 level of significance
Critical value of F = 99.44 at 0.01 level of significance
Our computed value of F (0.444) is not significant at both the levels
of significance and hence, the null hypothesis cannot be rejected
and we may confidently say that the differences between means are
not significant and therefore, there is no need for further testing
with the help of t test.

Two-way analysis of variance:
In one-way analysis of variance involving one experimental
variable(independent variable). If two experimental variables are .
Such experiments involve two-way classification based on two
experimental variables.
In this way, we have to simultaneously study the impact of two
experimental variables, each having two or more levels,
characteristics or classifications and hence we have to carry out
the two-way analysis of variance.
So far, we have dealt with one-way analysis of variance involving
one experimental variable. However, experiments may be
conducted simultaneous of two experimental variables. Such
experiments involve two-way classification based on two
experimental variables. Let us make some distinction between the
need for carrying out one-way and two-way analyses of variance
through some illustrations.

Suppose that we want to study the effect of four different
methods of teaching.
Here, the method of teaching is the experimental variable
(independent variable which is to be applied at four levels). We
take four groups of students randomly selected from a class.
These four groups are taught by the same teacher in the same
school but by different methods. At the end of the session, all the
groups are tested through an achievement test by the teacher.
The mean scores of these four groups are computed. If we are
interested in knowing the significance of the differences between
the means of these groups, the best technique is the analysis of
variance.
Since only one experimental variable (effect of the method of
teaching) is to be studied, we have to carry out one-way analysis
of variance.

Let us assume that there is one more experimental or independent
variable in the study, in addition to the method of teaching. Let it
be the school system at three levels which means that three
school systems are chosen for the experiment. These systems can
be government school, government-aided school and public school.
Now the experiment will involve the study of 4 x 3 groups. We have
groups each in the three types of schools (all randomly selected).
The achievement scores of these groups can then be compared by
the method of analysis of variance by establishing a null
hypothesis that neither the school system nor the methods have
anything to do with the achievement of pupils.

In this way, we have to simultaneously study the impact of two
experimental variables, each having two or more levels,
characteristics or classifications and hence we have to carry out
the two-way analysis of variance. In the two-way classification
situation, an estimate of population variance, i.e. total variance is
supposed to be broken into:
(i) variance due to methods alone,
(ii) variance due to school alone, and (1) S residual variance in
the groups called interaction variance (M x S; M= methods, S =
schools) which may exist on account of the following factors:
Chance, Uncontrolled variables like lack of uniformity in
teachers. Relative merits of the methods (which may differ from
one school to another).

In other words, there may be interaction between methods and
schools which means that although no method may be regarded
as good or bad in general, yet it may be more suitable or
unsuitable for a particular school system. Hence, the presence of
interaction variance is unique feature with all the experimental
studies involving two or more experimental variables.
Example 7 : In a research study, there were two experimental or
independent variables: a seven-member group of player and three
coaches who were asked to rate the players in terms of a particular
trait on a ten-point scale. The data were recorded as under:
Rating by
three
Coaches
Players
1 2 3 4 5 6 7
A 3 5 3 1 7 3 6
B 4 5 3 4 9 5 5
C 5 5 5 1 7 3 7

Apply the technique of analysis of variance for analyzing these data
Step 1. Arrangement of the data in a proper table and computation of
essential initial values.
Rating of coach
Players Total of
Rows
Square of
the total of
rows
1 3 4 5 12 144
2 5 5 5 15 225
3 3 3 5 11 121
4 1 4 1 6 36
5 7 9 7 23 529
6 3 5 3 11 121
7 6 5 7 18 324
Total of
Columns

Total
9 16 25 50
25 25 25 75
9 9 25 43
1 16 1 18
49 81 49 179
9 25 9 43
36 25 49 110

Step 7 Computation of F-ratio. Table illustration the computation
of mean square variance value.
Source of
variation
df Sum of square Mean square
variance
Rows (Players ) (r -1) = 6 61.15
Columns
(Coaches)
(c -1) = 2 3.72
Interaction or
residual
(r -1) (c -1) = 12 14.28

Step 8 Interpretation of F-ratio. Table gives the critical value of F:
Kind of f df for
greater
mean
square
variance
df for
smaller
mean
square
variance
Critical
value of
F at
0.05
level
Critical
value of
F at
0.01
level
Judgem
ent
about
the
significa
nce of
compute
d F
Conclusi
on
F (for
rows)
2 12 3.88 6.93 Significa
nt at
both
level
Null
hypothe
sis
rejected
F (for
column)
6 12 3.00 4.82 Not
significa
nt both
levels
Null
Hypothe
sis not
rejected

Here, F (for rows) is highly significant and hence null
hypothesises rejected. It indicates that the coaches did
discriminate among the players.
The second F (for columns) is insignificant and hence, the null
hypothesis cannot be rejected.
It indicates that the coaches did not differ significantly among
themselves in their ratings of the players.
In other words, their ratings may be taken as trustworthy and
reliable.

Underlying Assumptions In Analysis Of Variance:
The following are the fundamental assumptions for the use
of analysis of variance technique
1. The dependent variable which is measured should be
normally distributed in the population.
2. The individuals being observed should be distributed
randomly in the groups.
3. Within-groups variances must be approximately equal.
4. The contributions to variance in the total sample must
be additive.

8. Correlation and Simple Linear Regression
Regression is a method for studying the relationship
between two or more quantitative variables
Simple linear regression (SLR):
One quantitative dependent variable
 response variable
 dependent variable
 Y
One quantitative independent variable
 explanatory variable
 predictor variable
 X

 Multiple linear regression:
One quantitative dependent variable
Many quantitative independent variables
 SLR Examples:
 predict salary from years of experience
 estimate eect of lead exposure on school
testing performance
 predict force at which a metal alloy rod
bends based on iron content
 Example: Health data
Variables:
 Percent of Obese Individuals
 Percent of Active Individuals

Sl No Percent
Obesity
Percent Active
1 29.7 55.3
2 28.9 51.9
3 35.9 41.2
4 24.7 56.3
5 21.3 60.4
6 26.3 50.9
.
.
.
.

A scatterplot or scatter diagram can give us a general idea of
the relationship between obe-sity and activity...
Graph:
The points are plotted as the pairs (xi; yi)
for i = 1; : : : ; 25

Inspection suggests a linear relationship between obesity and
activity (i.e. A straight line would go through the bulk of the
points, and the points would look randomly scattered around this
line).
Simple Linear Regression
The model
 The basic model
Yi = β0 + β1xi + Єi
 Yi is the observed response or dependent variable for
observation i
 xi is the observed predictor, regressor, explanatory
variable, independent variable, covariate
 Єi is the error term
 Єi are iid N(0; 2)
(iid means independently and identically distributed)

 So, E[Yi/xi] = β0 + β1xi + β0 = β0 + β1xi
The conditional mean (i.e. the expected value of Yi given xi, or
after conditioning on xi) is “β0 + β1xi" (a point on the estimated
line).
Some interpretation of parameters:
 β0 is conditional mean when x=0
 β1 is the slope, also stated as the change
in mean of Y per 1 unit change in x
 σ2 is the variability of responses about the
conditional mean

 Simple Linear Regression Assumptions
Key assumptions
 linear relationship exists between Y and x
*we say the relationship between Y and x is linear if the
means of the conditional distributions of Y jx lie on a
straight line
 independent errors
(this essentially equates to independent observations
in the case of SLR)
 constant variance of errors
 normally distributed errors

9. CHI-SQUARE
Chi-square tests are often used in hypothesis testing.
The chi-square statistic compares the size any discrepancies
between the expected results and the actual results, given
the size of the sample and the number of variables in the
relationship.
What Is a Chi-Square Statistic?
A chi-square (χ2) statistic is a test that measures
how a model compares to actual observed data. The
data used in calculating a chi-square statistic must
be random, raw, mutually exclusive, drawn from
independent variables, and drawn from a large
enough sample. For example, the results of tossing
a fair coin meet these criteria.

Chi-square tests are often used in hypothesis testing. The chi-
square statistic compares the size any discrepancies between the
expected results and the actual results, given the size of the
sample and the number of variables in the relationship. For these
tests, degrees of freedom are utilized to determine if a
certain null hypothesis can be rejected based on the total number
of variables and samples within the experiment. As with any
statistic, the larger the sample size, the more reliable the results.
chi-square (χ2) theme
 A chi-square (χ2) statistic is a measure of
the difference between the observed (fo) and
expected frequencies(fe) of the outcomes of a
set of events or variables.

 χ2 depends on the size of the difference between actual
and observed values, the degrees of freedom, and the
samples size.
Goodness of fit
 As a test of goodness of fit, Pearson tried to make use of
the chi square distribution for devising a test for determining
how well experimentally obtained results fit in the results
expected theoretically on some hypothesis.
 χ2 can be used to test whether two variables are related
or independent from one another or to test the goodness-
of-fit between an observed distribution and a theoretical
distribution of frequencies.

What Does a Chi-Square Statistic Tell You?
 There are two main kinds of chi-square tests: the test of
independence, which asks a question of relationship, such as, "Is
there a relationship between student sex and course choice?"; and
the goodness-of-fit test, which asks something like "How well does
the coin in my hand match a theoretically fair coin?
 When considering student sex and course choice, a χ2 test for
independence could be used. To do this test, the researcher
would collect data on the two chosen variables (sex and courses
picked) and then compare the frequencies at which male and
female students select among the offered classes using the
formula a χ2.

 If there is no relationship between sex and course selection
(that is, if they are independent), then the actual frequencies
at which male and female students select each offered course
should be expected to be approximately equal, or conversely,
the proportion of male and female students in any selected
course should be approximately equal to the proportion of
male and female students in the sample.
 A χ2 test for independence can tell us how likely it is that
random chance can explain any observed difference between
the actual frequencies in the data and these theoretical
expectations.

 χ2 provides a way to test how well a sample of data matches
the (known or assumed) characteristics of the larger population
that the sample is intended to represent. If the sample data do
not fit the expected properties of the population that we are
interested in, then we would not want to use this sample to
draw conclusions about the larger population.
 For example consider an imaginary coin with exactly 50/50
chance of landing heads or tails and a real coin that you toss
100 times. If this real coin has an is fair, then it will also have
an equal probability of landing on either side, and the expected
result of tossing the coin 100 times is that heads will come up
50 times and tails will come up 50 times.

 In this case, χ2 can tell us how well the actual results of 100
coin flips compare to the theoretical model that a fair coin will
give 50/50 results. The actual toss could come up 50/50, or
60/40, or even 90/10. The farther away the actual results of the
100 tosses is from 50/50, the less good the fit of this set of tosses
is to the theoretical expectation of 50/50 and the more likely we
might conclude that this coin is not actually a fair coin.
Null hypothesis:
 Null hypothesis is set up No actual difference between the
observed frequencies (derived from experimental results and
expected frequencies, derived on the basis of some hypothesis of
equal probability or chance factor or hypothesis of normal
distribution).
 Computation of value of chi square.

Application of this formula requires the following
computation:
1.Computation of expected frequencies based on some
hypothesis.
2.Finding the square of the differences between observed
and expected frequencies and dividing it by the expected
frequency in each case to determine the sum of these
quotients.
Determining the Degrees of Freedom:
Determining the Number of Degrees of Freedom
 The data in chi square problems are usually available or may be
arranged in the form of a contingency table showing observed and
expected frequencies in some specific rows and columns.

The formula for the computation of the number of degrees of
freedom in a chi square problem usually runs as follows:
Where
 df= (r - 1) (c -1)
 r = No. of rows in the contingency table
 c = No. of columns in the contingency table
Determining the Critical Value of χ2
 As for the z and / tests of significance, there exists a table
for the critical value of x required for significance at a
predetermined significance level (5% or 1%) for the computed
degrees of freedom. The desired critical value of x thus, may be
read from Table F given in the Appendix of this text.

Heads 40 50 -10 100 2
Tails 60 50 10 100 2
Total 100 100

Example: Chi Square test of Independence between two means
100 boys and 60 girls were asked to selected one of
the five elective subjects. The choice of the two
sexes were tabulated separately.
sex subjects
A B C D E Total
Boys 25 30 10 25 10 100
Girls 10 15 5 15 15 60

Solution:
Step 1. Establishing a null hypothesis-Null hypothesis. The choice
of the subject is independent of sex, i.e. there exists no significant
difference between the choices of the two sexes.
Step 2. Computation of the expected frequencies. For this purpose,
we first present the given data in the contingency table form. The
expected frequencies after being computed, are written within
brackets just below the respective observed frequencies. The
contingency table and the process of computation of expected
frequencies is given in Table.
Categories
Sex A B C D E Total
Boys 25 30 10 25 10 100
(21.9) () () () ()
Girls 10 15 5 15 15 60
() () () () ()
Total 35 45 15 40 25 160

Categories
Sex A B C D E Total
Boys 25 30 10 25 10 100
(21.9) (28.1) (9.4) (25) (15.6)
Girls 10 15 5 15 15 60
(13.1) (16.9) (5.6) (15) (9.4)
Total 35 45 15 40 25 160

100x35/160=21.9 60x35/160=13.1
100x45/160-28.1 60x45/160=16.9
100x15/160=9.4 60x15/160=5.6
100x40/160=25 60x40/160=15
100x25/160=15.6 60x25/60=9.4
Computation of expected frequencies

fo fe fo-fe (fo-fe)2 (fo-fe)2/fe
25 21.9 3.1 9.61 0.4388
30 28.1 1.9 3.61 0.1284
10 9.4 0.6 0.36 0.038
25 25 00 00 0000
10 15.6 -5.6 31.36 2.010
10 13.1 -3.1 9.61 0.733
15 16.9 -1.9 3.61 0.214
5 5.6 -0.6 0.36 0.064
15 15 00 00 0000
15 9.4 5.6 31.36 3.336
Total 160 160 X2 =6.962

fo fe fo-fe (fo-fe)2 (fo-fe)2/fe
25 21.9 3.1 9.61 .439
30 28.1 1.9 3.61 .128
10 9.4 .6 .36 .038
25 25 0 0 .000
10 15.6 -5.6 31.36 2.010
10 13.1 -3.1 9.61 .733
15 16.9 -1.9 3.61 .214
5 5.6 -.6 .36 .064
15 15 0 0 .000
15 9.4 5.6 31.36 3.336
Total 160 160 X2 =6.962

Step 4: Testing the null hypothesis
 No of degree of freedom=(r-1)(c-1)=(2-1)(5-1)=4 For 4 degrees of
freedom , from Table F of the x2 distribution.
 Critical value of chi (x2) =9.48 at 5% level of significance.
 Critical value of chi (x2) = 13.277 at 1% level of significance.
 The computed value of x2 is much less than that of critical
value of x2 at 5% and 1% levels of significance.
 Hence it is to be taken as non-significant. Consequently , the
null hypothesis cannot be rejected it is quite independent of sex.

Module 8-S M & T C I, Regular.pptx

Module 8-S M & T C I, Regular.pptx

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