In statistics, a sample is a subset of a population. Typically, the population is very large, making a census or a complete enumeration of all the values in the population impractical or impossible. The sample represents a subset of manageable size. Samples are collected and statistics are calculated from the samples so that one can make inferences or extrapolations from the sample to the population. This process of collecting information from a sample is referred to as sampling.

A complete sample is a set of objects from a parent population that includes ALL such objects that satisfy a set of well-defined selection criteria. For example, a complete sample of Australian men taller than 2m would consist of a list of every Australian male taller than 2m. But it wouldn't include German males, or tall Australian females, or people shorter than 2m. So to compile such a complete sample requires a complete list of the parent population, including data on height, gender, and nationality for each member of that parent population. In the case of human populations, such a complete list is unlikely to exist, but such complete samples are often available in other disciplines, such as complete magnitude-limited samples of astronomical objects.

An unbiased sample is a set of objects chosen from a complete sample using a selection process that does not depend on the properties of the objects. For example, an unbiased sample of Australian men taller than 2m might consist of a randomly sampled subset of 1% of Australian males taller than 2m. But one chosen from the electoral register might not be unbiased since, for example, males aged under 18 will not be on the electoral register. In an astronomical context, an unbiased sample might consist of that fraction of a complete sample for which data are available, provided the data availability is not biased by individual source properties.

The best way to avoid a biased or unrepresentative sample is to select a random sample,...

...technique for optimizing search under some constraints • Express the desired solution as an n-tuple (x1 , . . . , xn ) where each xi ∈ Si , Si being a ﬁnite set • The solution is based on ﬁnding one or more vectors that maximize, minimize, or satisfy a criterion function P (x1 , . . . , xn ) • Sorting an array a[n] – Find an n-tuple where the element xi is the index of ith smallest element in a – Criterion function is given by a[xi ] ≤ a[xi+1 ] for 1 ≤ i < n –Set Si is a ﬁnite set of integers in the range [1,n] • Brute force approach – Let the size of set Si be mi – There are m = m1 m2 · · · mn n-tuples that satisfy the criterion function P – In brute force algorithm, you have to form all the m n-tuples to determine the optimal solutions • Backtrack approach – Requires less than m trials to determine the solution – Form a solution (partial vector) and check at every step if this has any chance of success – If the solution at any point seems not-promising, ignore it – If the partial vector (x1 , x2 , . . . , xi ) does not yield an optimal solution, ignore mi+1 · · · mn possible test vectors even without looking at them • All the solutions require a set of constraints divided into two categories: explicit and implicit constraints Deﬁnition 1 Explicit constraints are rules that restrict each xi to take on values only from a given set. – Explicit constraints depend on the particular instance I...

...Arrays
An array is a series of elements of the same type placed in contiguous memory locations that can be individually referenced by adding an index to a unique identifier.
That means that, for example, we can store 5 values of type int in an array without having to declare 5 different variables, each one with a different identifier. Instead of that, using an array we can store 5 different values of the same type, int for example, with a unique identifier.
For example, an array to contain 5 integer values of type int called billy could be represented like this:
[pic]
where each blank panel represents an element of the array, that in this case are integer values of type int. These elements are numbered from 0 to 4 since in arrays the first index is always 0, independently of its length.
Like a regular variable, an array must be declared before it is used. A typical declaration for an array in C++ is:
type name [elements];
where type is a valid type (like int, float...), name is a valid identifier and the elements field (which is always enclosed in square brackets []), specifies how many of these elements the array has to contain.
Therefore, in order to declare an array called billy as the one shown in the above diagram it is as simple as:
| |int billy [5]; |
NOTE: The elements field within brackets [] which represents the number of elements the array is going to hold, must be a constant value, since arrays are blocks of...

...NATIONAL BOARD FOR HIGHER MATHEMATICS
AND
HOMI BHABHA CENTRE FOR SCIENCE EDUCATION
TATA INSTITUTE OF FUNDAMENTAL RESEARCH
Pre-REGIONAL MATHEMATICAL OLYMPIAD, 2013
Mumbai Region
October 20, 2013
QUESTION PAPER SET: A
• There are 20 questions in this question paper. Each question carries 5 marks.
• Answer all questions.
• Time allotted: 2 hours.
QUESTIONS
1. What is the smallest positive integer k such that k(33 + 43 + 53 ) = an for some positive
integers a and n, with n > 1?
n
√
2. Let Sn =
k=0
1
√ . What is the value of
k+1+ k
99
1
?
n=1 Sn + Sn−1
3. It is given that the equation x2 + ax + 20 = 0 has integer roots. What is the sum of all
possible values of a?
4. Three points X, Y, Z are on a striaght line such that XY = 10 and XZ = 3. What is the
product of all possible values of Y Z?
5. There are n − 1 red balls, n green balls and n + 1 blue balls in a bag. The number of ways of
choosing two balls from the bag that have different colours is 299. What is the value of n?
6. Let S(M ) denote the sum of the digits of a positive integer M written in base 10. Let N be
the smallest positive integer such that S(N ) = 2013. What is the value of S(5N + 2013)?
7. Let Akbar and Birbal together have n marbles, where n > 0.
Akbar says to Birbal, “ If I give you some marbles then you will have twice as many marbles
as I will have.” Birbal says to Akbar, “ If I give you some marbles then you will have thrice
as many marbles as I will have.”
What is the minimum...

... |20 |
| | |
|TOTAL |50 |
Sampling Plan:
● Population: The study aimed to include all the borrowers from Banks, from cooperatives and from individual lenders
● Sample Size: A sample size of 50 respondents will be taken for this study out of the massive counts of clients of Microfinancing and the available allotted time and the confined location we took the survey. The 50 different borrowers from different financial providers who’s age lies between 20 years old up to 50 years will be our respondents. The sample will be taken in the form of strata based on name, age, gender, civil status, location, educational attainment, monthly salary and borrowing capacity.
Sampling Technique:
The sampling technique will be probability sampling
Instrumentation:
This study will use questionnaire checklist to determine the data, process the data and to come up with a strategy with the calculated and studied data. Through this questionnaire we can determine the problems and satisfactory rate of...

...65055_22_ch22_p915-952.qxd 9/12/06 5:16 PM Page 22-1
CHAPTER
Sample Survey
CONTENTS
STATISTICS IN PRACTICE: DUKE
ENERGY
22.1 TERMINOLOGY USED
IN SAMPLE SURVEYS
22.2 TYPES OF SURVEYS AND
SAMPLING METHODS
22.3 SURVEY ERRORS
Nonsampling Error
Sampling Error
22.4 SIMPLE RANDOM SAMPLING
Population Mean
Population Total
Population Proportion
Determining the Sample Size
22.5 STRATIFIED SIMPLE
RANDOM SAMPLING
Population Mean
Population Total
Population Proportion
Determining the Sample Size
22.6 CLUSTER SAMPLING
Population Mean
Population Total
Population Proportion
Determining the Sample Size
22.7 SYSTEMATIC SAMPLING
22
65055_22_ch22_p915-952.qxd 9/12/06 5:16 PM Page 22-2
22-2
Chapter 22
STATISTICS
Sample Survey
in PRACTICE
DUKE ENERGY*
CHARLOTTE, NORTH CAROLINA
Duke Energy is a diversified energy company with a portfolio of natural gas and electric businesses and an affiliated real estate company. In 2006, Duke Energy merged
with Cinergy of Cincinnati, Ohio, to create one of North
America’s largest energy companies, with assets totaling
more than $70 billion. Today, Duke Energy serves more
than 5.5 million electric and gas customers in North
Carolina, South Carolina, Ohio, Kentucky, Indiana, and
Ontario, Canada.
To improve service to its customers, Duke Energy
continually looks for emerging customers needs. In the
following...

...right 2.5 cm, top 2.5 cm and bottom 4cm
Double spacing. Times new roman size 12 Chapter title (as in chapter one , two, three) in the center underlined.
Remember to put table of contents,
1.1 Background of the study
Respondents – (From your journals)
Sample –
Location –
DV – (Consider as Title too)
IV –
IV –
IV –
Before ending this part write the following paragraph
“Therefore, the purpose of this study aims to explore the TITLE among the respondents and the correlation IV IV IV and DV.”
1.2 Problem Identification
5 Research Specific Objectives
Each objective 5 paragraph with a research question at the end.
Research question you can make up from your objective.
For this part use your main reference journals
1.3 Research Objectives
1.3.1 General Objective
To study the (TITLE), to specify the study of (RESPONDENTS) will be chosen as the subjects of this research.
1.3.2 Specific Objectives
1) To profile the Respondent’s characteristics among the “SAMPLE” in “LOCATION”
2) To investigate the “IV” among the “SAMPLE” in “LOCATION”
3) To determine the “IV” among the “SAMPLE” in “LOCATION”
4) To examine the “IV” among the “SAMPLE” in “LOCATION”
5) To identify the correlation between “DV” with selected variables, IV IV and IV
1.4 Hypothesis of the study
Ho
Statements
Type of analysis
Ho1
There is no...

...or universe, is the entire set people data or things that is the subject of exploration.
A census involves obtaining information, not from a sample, but rather from the entire population or universe.
A sample (as opposed sampling) is a subset of the population/universe.
For Marketing Research purposes, sampling usually involves people, not data or things.
Sampling Plans are strategies and mechanics for selecting members of thesample from the population:
1. Define the population. It is usually limited based on some set of characteristics, e.g., males, aged 21-39, who have consumed alcoholic beverages within the past 3 months for a beer study.
2. Choose data collection methodology. What kind of information do you require from the sample, how will they be identified, where are they available, etc.
3. Set sampling frame. This is as exhaustive a list as operationally and economically possible that represents the population and is also accessible utilizing the selected methodology.
4. Choose sampling method.
• Probability samples are those that allow all members of the sampling frame an equal opportunity of selection. Probability samples include Simple Random, Systematic, Stratified and Cluster sampling
• Nonprobability samples do not allow all members of the sampling frame an equal opportunity of...

...students in the junior classes.
2.2 Sample size& Technique
Four hundred students will be selected in the study sample since among the three camps the population will be limited to two camps. From the two camps, fifty students will be selected from each class and five teachers from each school.
2.2.0 Sampling techniques
The following techniques will be employed:
1. Convenient sampling will be used to select four schools out of the nine secondary schools based on the accessibility of the schools due to security impediments as movement from one camp to another is enabled only by means of police escort.
2. Stratified random sampling would then be used to select students and teachers according to their respective classes, so form three and four classes would be selected together with teachers who teach them.
3. Purposive sampling will be used to ensure that both female and male students are selected and at equal numbers to ensure representativeness.
2.2.1 The sample size
From the four secondary schools, four hundred form three and four students will be sampled. A total of fifty form three and fifty form four students from each of the four schools will form the sample. A total of twenty teachers from the chosen schools will participate. The sample size will be derived from the sample size formulae.
SS=ZZ×P×(1-P)÷CC
Where: Z=Z value (e. g for 95% confidence level)
P=Percentage...