Monday 10 October 2011

Sampling techniques And Sampling Distributions

1:Population:-

Population is the totality of the observations made on all the objects(under investigation) possessing some

common specific characteristics, which are of particular interest to researchers.

The population is the aggregate of the elements and these elements are the basic units that comprise and

define a population.The population must be defined in terms of

(1) Content, (2) Unit, (3) Extent, (4) Time

For instance,the students of the first year class at a given college,the characteristics to be investigated

may be score received by each student in a college entrance examinations,in a given year.

Population may be finite or infinite.

(i)Finite Population:-

A population is said to be finite if it includes a limited number of elementary units.

Examples of a finite population are:

The heights of all the students enrolled at a college in a given

year,the wages of all employees of a steell mill in a given year,the amount of money spend by each

student in an engineering university in a given academic year,or the grading of items as defective and

non-defective that are produced by an industry on a given day.

(ii):Infinite Population:-

A population is said to be infinite if it consist of unlimited number of elementary units.Atleast hypothetically,

there is no limit to the no of units it can include.

Examples of a infinite population are:

The weights at birth of all human beings,the result obtained by rolling of

die,the lifetime of all the bulbs produced in a production process that operates indefinitely under given

manufacturing conditions.

2: Sample:-

A sample is the part of production which is selected eith the expectation that it will represent

the characteristics of the population.

(i):Sampling:-

Sampling is a procedure of selecting a representative sample from a given population.

(ii):Sample survey versus complete enumeration:-

The collection of information from a part of the population is called making a sample survey.

The collection of information from all elements in a population is called taking a census or making a complete enumeration.

(iii):Purpose of sampling:-

There are basic purposes of sampling are:

(a) To obtain maximum information about the characteristics of the population with minimum cost,time and effort.

(b) To find the reliability of the estimates derived from the sample.

(iv):Advantages of sampling:-

(a) Time saving:-

A sample involves lesser amount of time and energy than a complete enumeration both in the execution and the

analysis of data.This is a vital consideration ehen the information is urgently needed as the results from a sample

survey are more readily available.

(b) Economic:-

A sample survey requires less expenses and labour as compared to a complete census because the cost

of covering only a fraction will be lower than that of covering the whole population.

(c) Accuracy:-

A sample survey provides the results which are almost as accurate as those obtained by complete census.

A properly designed and carefully executed sample survey will provide even better result.

(d) Feasibility:-

Sometimes the data are obtained by tests that are destructive.For example,to tkhow the average life of

certain type of electric bulb,we shall take a sample of these bulbs and keep them on until they burn out.

we cannot think of testing the whole lot.In testing blood of a patient we donot drain the entire blood out

of him but examine just a few drops.Sampling may be the only means available for obtaining the desired

information when the population is infinite or inaccessible.In such cases complete enumeration would

neither be physically possible nor practically feasible..

Whatever be the merits of sampling,it cannot totally replace a complete census.A census is a record of

nation's history and its importance has to be given due acknowledgement.

(v): Limitations of sampling:-

If the basic fact of each and every unit in the production are needed,census become indispensable.For Example,

the list of income tax payers is prepared very carefully,the list of voters is prepared to include the name of each and

every voter,or an inventoryof all goods and stocks necessary to know the total amount of stocks of firm.

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Wednesday 17 August 2011

Algebra: Verbal Expressions to Variable Expressions

In questions of Algebra, it is often asked first to convert the verbal expressions into variable expressions and then find the answer of the algebraic expressions. But for most of the students it becomes difficult to convert the verbal expressions and therefore it leads them to wrong solution. An algebraic expression is a number, variable or combination of the two connected by some mathematical operation like addition, subtraction, multiplication, division, exponents, and/or roots.2x + y, a/5, and 10 - r are all examples of algebraic expressions.
In order to convert the verbal expression into variable expression, you want to
1. read the problem carefully,
2. pick out key words and phrases and determine their equivalent mathematical meaning,
3. replace any unknowns with a variable, and
4. Put it all together in an algebraic expression.
Now here are some of the keywords used for the basic operators. i.e., +,-,*,/.
Addition: plus, more than, the sum of, the total of, increased by, added to, etc
Subtraction: minus, less than, the difference of, less, decreased by, subtracted from, etc
Multiplication: multiplied by, times, the product of, twice; double, of, etc
Division: divided by, quotient of, the ratio of, etc.
The really best way to translate verbal expressions into variable expressions is to think about what the verbal expression means, and then think about how you would compute that if you were given numbers. Now here are some examples to clarify the concept.
Example:
Problem: Translate '3 less than x' to a variable expression.
Solution: if you translate it word for word then you get 3-x, which is wrong. Try to understand the context of the verbal expression and then writing the algebraic expression for it. So, think about how you would compute the number that is 3 less than 10. You wouldn't compute 3-10, you would compute 10-3. Now if you write the same thing down with x, you get the right answer of x-3.
Examples:
1. the sum of the product of five and a number and the product of seven and another number
2. a number plus the product of the number and nine
3. the difference between a number and the total of three times the number and six
1. the sum of the product of five and a number and the product of seven and another number.
Now by reading the context cut the expression into parts which make sense, first take product of five and a number. It can be written as 5 multiplied by a variable say x that is
5x.
Now the other part of the expression is product of seven and a number, similarly, it can be written as 7y, where y is another unknown number.
Now, if we combine the two parts of the expression we get, 5x+7y.
2. a number plus the product of the number and nine
first take product of the number and nine. It can be written as 9 multiplied by a number say x or it becomes 9x. so, the whole expression becomes x+9x.
3. the difference between a number and the total of three times the number and six.
The above statement can be written as: the difference between a number and the total of (three times the number) and six. The total of (three times the number) and six' can stand alone, so put a parentheses around it. the difference between a number and (the total of (three times the number) and six).Now work from the inside out. 'three times the number', that's 3x. 'the total of 3x and six', that's 3x+6. 'the difference between a number and (3x+6)', that must be x-(3x+6). Simplifying this we get x-3x-6=-2x-6.
In this way the verbal expressions can be converted into algebraic expressions.

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Frequency Distribution

A method of organizing and considering the raw data is the conversion of raw data into a grouped data.Here, the various items of data are classified into certain groups or classes and the no of items lying in each group or class is put against that group or class. The data organized and summarized in this way is known as Frequency Distribution.

Example:

The following are the marks out of 100 obtained by 25 students in the subject of physics at their final examination.

45, 50, 51, 51, 54, 53, 53, 52, 55, 57, 54, 58, 60, 62, 59, 61, 61, 62, 64, 63, 48, 65, 53, 66, 65

The frequency distribution for the above data may be like given below:

Marks	Frequency	Class boundaries	Mid point (x)
45-49	2	44.5-49.5	47
50-54	9	49.5-54.5	52
55-59	4	54.5-59.5	57
60-64	7	59.5-64.5	62
65-69	3	64.5-69.5	67
Total	25

Formation of Frequency Distribution (Tally Bar Method)

In this method the raw data or the ungrouped data are presented into a grouped data. Choice is yours to select the size of class interval. Generally, the size of class interval determined on the basis of total number of observations and the desired number of groups.

Following are the steps in tally method.

(i) Find range of data.

(ii) Find the size of class interval by dividing the range by the number of classes you wish to make. For example, the largest observation is 126 and the smallest observation is 20 and if we have to make 10 groups or classes, then

Size of each class interval = 126 - 20

10(groups)

= 106

= 10.6 or 11 approximately

(iii) Prepare three columns

(a) Class interval

(b) Tally marks

(iv) Select the data element one by one.

(v) Look for the class in which each element of ungrouped data falls. Draw a small tally mark (/) against that class and also tick the element concerned with a tick mark sign. In this way you can remember that you have counted for the element. Continue this way with the next element unto the last element of the data set. If 5 or more tallies appear in any class, mark every fifth tally diagonally as(

)

(vi) Count the number of tally marks against each class and write that number in the frequencies column.

Example:

Following are the numbers of telephone cells made in a week to 30teachers of a high school.

5, 8, 11, 25, 13, 16, 20, 17, 15, 16, 30, 21, 14, 18, 19, 6, 22, 26, 15, 19, 35, 29, 31, 23, 25, 20, 10, 9, 7, 26.

Using 5 as class interval make a frequency distribution by tally method.

Solution:

Class Interval	Tally Marks	Frequency (f)
5-9		5
10-14	////	4
15-19	///	8
20-24		5
25-29		5
30-34	//	2
35-39	/	1
Total	---	30

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