A problem always arises
Same operation as adding natural numbers; and if we call the elements in our model ‘fractional numbers’, we must be equally clear that these may or may not have the same properties as natural numbers. We use the same words as for natural numbers because we are hoping to generalize the corresponding ideas, in the manner described in Chapter 4, page 61.
In particular, we would like a notation for fractional numbers which (i) is based on same numerals as are used for natural numbers; (ii) allows us to use the same method for adding as those which we have learnt for natural numbers; either as they stand, or extending them by learning a few extra procedures, as we did when we developed our short-multiplication schema into long multiplication.
Fraction
The model we are going to develop will be the same, except for the nature of the units, for all the physical qualities for which we are using it. So when we refer to ‘cutting up’, we are using it in the generalized sense of any way of breaking up into parts. (we want to keep ‘division’ for the mathematical operation: compare ‘uniting’ sets and ‘adding’ numbers.)
This represents any standard object,
And this represents it cup up into five.
Clearly this way of cutting up is of no use for measurement, however, since we do not know how big* the bits are; and whether or not we get a match with our given object will depend on which bit we choose.
If we cut up our standard object into bits which match each other, according to whatever quality we are trying to measure, this gets over the second problem, of which one we choose. How big the bits are will then depend on how many of them there are. A cutting-up of this kind we will call sharing; bits which are equivalent (i.e. which match in the way described) we will call parts; and we will describe that size of the parts by saying into how many of these parts we have shared our standard object. So this represents a standard object** shared into fifth parts.
This, eighth parts.
This, third parts.
This represents the result of sharing into eighth parts, and then combining*** three of these parts.
We call this fractional parts of an object, or for short ‘three eighths’ of it. A fractional parts is thus a part which is obtained by a double action of sharing and combining. Abstracting what is common to all these double actions, we get in realm 3 a mathematical double operation which is called a fraction.
The mathematical notation for this double operation is (read this as ‘three over eight’). Since the numeral below the line tells us the name of the parts represented, - whether they are fifth…………..
* In the generalized sense; i.e. how heavy, how long, how great an e.m.f.,etc
** Thought it has been emphasized that this is a generalized object on which generalized action of sharing and combining are to be done, it is nevertheless a help to one’s thinking to imagine it as something more concrete; such as a cake which we are literally cutting up into fair shares – i.e. equal volumes of cake.
*** In whatever way is appropriate to the physical quality concerned. If weight, by putting in same scale pan. If volume of cake, the same person eats it.
Part, eighth part, thirrd part, etc,- this is called the denominator of the fraction. The numeral above the line tells us how many such parts are combined, and is called the numerator.
The nation might appear to suggest, both from our habit of reading downwards and from its often being written as for convenience in printing or typing, that the combining is done first, whereas in the foregoing description we first share into 8 part and then combine 3 of these. Hewever, we shell see that this actions are commutative- we get the same result whichever we do first. So the notation may be taken as representing simultancously booth of the two possibel orders of the mathematical double operation.
Start with a standart object.
Share into 8 parts.
Coimbine 3 of these eighy parts: result. Three eighth-parts of an object.
Now the other way about. Start with a standard object.
Combine 3 of these standard object.
Share this into 8 parts: result, one eight part of three objects.
Except for their arragement(which does not affect the quantity), the shaded part is the same as before. So the fraction represents(÷, 8 x 3), as embodied in the first set of diagrams above, and (×, 3 : 8), as embodied in the second set of diagrams. This is one reason for reading as ‘three over eight’, rathere than ‘three eighths’, which implies only the first of these alternative orders.
Equivalent fractions
We shall now reverse the process. By using embodiments of these doble operations which we call fractions and equivalence relation between fractions.
Fraction Embediment
And so on; the pattern is clear.
Though the fraction themselves are different, they correspond to same amount of whatever physical quality we are concorned with. If we applied the corresponding actions of sharing and combining to a standard object, the resulting part-objects would match. Whit units attached, the fraction represent equal measures. (Equal amounts of cake, in the concret example). In this respect they are therefore equifalent; and we may collect them together into the equivalent class { .
In the same way we can find other sets of equivalent fractions. For example:
Set of equivalent fractions: { .
Another example, this time without diagrams: {
Not only is the pattern of each equiovalence class clear, but a general method of forming them is beginning to energe.
Start with any fraction,
Double both upper and lower numbers
Treble both upper and lower numbers,
Etc.
Equivalence class
And, in general, if a, b, kare natural numbers, then the fraction (is equivalent to) .
This works both ways; since also , we can get another fraction equivalent to any given fraction by either multiplying or dividing the numerator by the same natural number. The former we can always do; the latter, which is the well-known ‘cancelling’ rule, sometimes.
Example: .
We also have: .
So these also belong to the last named equivalence class, which we can now write:
{ .
Fractional numbers
The caracteristic property of any set ot equifalent fraction we call a fractional number. Whit a unit attached, each fraction in an equivalence class represent the same measure; and, without the unit, it represent the same number. This means that we can use any fraction from the set as a name for the number of that set; and, although this invites confosion if we do not know what is going on, if we do know, it has considerable advantages for purposes of calculation.
So if we are talking about fractions, which are double operations,
If we are talking about fractional numbers,
For each denotes the same equivalence class. The sign in the middle therefore indicates which of the two is meant.
Adding fractional numbers. We want this matematical operation to correspond to combining part-objects. This is straightforward if the numbers are represented by fractions having the same denominator, for we are then combining part-objects of the same kind; an essential already note on page 187. but we have to remember that adding does not mean quite the same for fractional numbers as for natural numbers. To remind ourselves of this we use for the new kind of addition, and + for the old kind.
Example:
If the denominators are not equal, this is where the interchange-ability principle within equivalences sets (page 176) comes to our help. Since all the fractions in a equivalence set stand for the same number, we can choose whichever ones suit us best for some other purpose, in this case a calculation.
Suppose that we want to add (say)
Replace by these equivalent fractions
Which stand for the same numbers
As before. For denominator, we choose
4 9=36.
Now we can add.
=
It should, of course, make no difference which fractions we us as replacements, provided that they stand for the original numbers and have the same denominators. Let us try the calculation by a different route.
First we will replace
The original fraction by =
Equivalent ones using =
The canceling rule.
Now we can find a =
Smaller common =
Denominator, namely =
2 3=6. =
This answer looks different, but of course represents the same fractional number as , since = = . So we have verified that the interchangeability principle works in this case. A general proof is not difficult, but requires the use algebra. Multiplying fractional numbers. As yet we have no meaning for ‘multiplying’ in the new context of fractional numbers. We could of course decide to do without a meaning – there are plenty of mathematical system which have only one operation. But we shall then not have generalized the natural number system completely, so we ought to try. We can either look for a meaning for ‘multiplication’ which is satisfactory in the realm of pure mathematics, and then see whether it provides a useful working model for realm 1; or we can use the requirement of a satisfactory working models to suggest a meaning, and then check whether it is mathematically acceptable. Both approaches have their merits. The latter, being less abstract, is the one we shall use here.
Start as usual with a standard object.
Then this object represents the fraction
In natural numbers, 3 4 when embodied in physical objects means : start with a 3 set
And combine 4 of these.
So in fractional numbers, might reasonably mean : start with two third-parts of an object,
And take four fifth-parts of this.
In natural numbers ‘calculate 3 4’ means ‘find the number of the resulting set’. In fractional numbers, ’calculate ’ might therefore reasonably mean ‘find what fractional part of the object the resulting part-object is’. The resulting part-object is shown by the cross-hatched area. The original object has now been shared into 15 parts (3 5), and the cross-hatched area combines 8 (2 4) of these.
This suggests that
Would be a reasonable way to multiply these fractions ; reasonable, in the sense that it gives a good working model for part-objects. It also satisfies requirements (i) and (ii) on page 186 very well.
These two methods, for addition and multiplication of fractional numbers, are of course those which have been agreed by mathematicians – we have been pretending we did not know in order to try to see how they were arrived at. Stated generally, if a, b, c, d are natural numbers, then the method for adding is:
And the method for multiplying is
Where and refer to operations on fractional numbers, and + and to those on natural numbers.
There is still much unsaid about fractional numbers. Techniques for manipulating them have not been systematized, and decimal notation – which can greatly simplify some of these manipulations – has not been introduced. Neither of these will be done here, since the present aim is comprehension rather than skill at computation. Also, we have not checked that the fractional numbers have the five properties of a number system which we found in Chapter 9 to be so important. This we must certainly do. Since the treatment is algebraic, is has been put into an appendix to this chapter. The reader who does not think easily in algebraic terms may take it on trust, since he already has the ideas, and only requires to be assured that they also hold good for fractional numbers. There is also a third matter of importance, which is whether and to what extent natural and fractional numbers can be intermixed. This last point will be discussed in Chapter 12, with the help of the ideas of isomorphism and mathematical generalization.
APPENDIX
Fractional numbers have the five properties of a number system
Let a, b, c, d, e, f, x, y ,stand for any natural numbers.
Than , etc. will represent fractional numbers.
ADDITION IS COMMUTATIVE
We can only add if the denominators are equal.
This property follows immediately from the corresponding property for natural numbers, and the same is true for all the other properties.
ADDITION IS ASSOCIATIVE
MULTIPLICATION IS COMMUTATIVE
MULTIPLICATION IS ASSOCIATION
MULTIPLICATION IS DISTRIBUTIVE OVER ADDITION
Showing posts with label Matematich. Show all posts
Showing posts with label Matematich. Show all posts
QUADRATIC EQUATIONS
Any equation of type ax2 + bx + c = 0 where a, b, and c are constants and a <> 0, is in standard form for a quadratic equation.
Quadratic equations of type ax2 + bx + c = 0 and ax2 + bx = 0 (c is 0) can be factored to solve for x. Examples:
1. Problem: Solve 3x2 + x - 2 = 0 for x.
Solution: Factor.
(3x - 2)(x + 1) = 0
Use the principle of zero products,
which says, if ab = 0, either
a, b, or both must be
equal to zero.
3x - 2 = 0, x + 1 = 0
3x = 2 , x = -1
x = (2/3)
x = -1, (2/3)
2. Problem: Solve 3x2 + 5x = 0 for x.
Solution: Factor.
x(3x + 5) = 0
Use the principle of zero products.
x = 0, 3x + 5 = 0
3x = -5
x = -(5/3)
x = 0, -(5/3)
Quadratic equations of type ax2 + c = 0 can be solved by solving for x. Example:
3. Problem: Solve 3x2 = 6 for x.
Solution: Recognize that the equation is quadratic because it is the same as 3x2 - 6 = 0.
Divide each side by 3.
x2 = 2
Take the square root of each side.
x = SQRT(2), -(SQRT(2))
Many times you will come across quadratic equations that are not easy to factor or solve. In those cases, there is a special formula called the quadratic formula that you can use to solve any quadratic equation.
The solutions of any quadratic equation, ax2 + bx + c = 0 is given by the following formula, called the quadratic formula:
-b ± SQRT(b2 - 4ac)
x = -------------------
2a
Example:
1. Problem: Solve 3x2 + 5x = -1 for x.
Solution: First find the standard form of
the equation and determine a, b,
and c.
3x2 + 5x + 1 = 0
a = 3
-5 ± SQRT(52 - 4(3)(1))
x = -----------------------
2 * 3
b = 5
c = 1
Plug the values you found for a, b, and c into the quadratic formula. Perform any indicated operations.
-5 ± SQRT(25 - 12)
x = ------------------
6
-5 ± SQRT(13)
x = -------------
6
The solutions are as follows:
-5 + SQRT(13) -5 - SQRT(13)
x = -------------, -------------
6 6
Some equations are not quadratic equations, but are in the same form, such as x4 - 9x2 + 8 = 0. To solve equations such as that, you make a substitution, solve for the new variable, and then solve for the original variable. Example:
1. Problem: Solve x4 - 9x2 + 8 = 0 for x.
Solution: Let u = x2. Then substitute u for every x2 in the equation.
u2 - 9u + 8 =0
Factor.
(u - 8)(u - 1) = 0
Utilize the principle of zero products.
u - 8 = 0, u - 1 = 0
u = 8 , u = 1
Now substitute x2 for u and solve the equations.
x2 = 8, x2 = 1
x = ±SQRT(8), x = ± 1
x = ±2(SQRT(2))
x = ±2(SQRT(2)), ±1
Another way to solve systems of equations is by substitution. In this method, you solve on equation for one variable, then you substitute that solution in the other equation, and solve. Example:
1. Problem: Solve the following system:
x + y = 11
3x - y = 5
Solution: Solve the first equation for y (you could solve for x - it doesn't matter).
y = 11 - x
Now, substitute 11 - x for y in the second equation. This gives the equation one variable, which earlier algebra work has taught you how to do.
3x - (11 - x) = 5
3x - 11 + x = 5
4x = 16
x = 4
Now, substitute 4 for x in either equation and solve for y. (We use the first equation below.)
4 + y = 11
y = 7
The solution is the ordered pair,
(4, 7).
The last method, addition, is probably the most complicated, but is necessary when dealing with more complex systems, such as systems with three or more variables. The idea behind the addition method is to replace an equation with a combination of the equations in the system. To obtain such a combination, you multiply each equation by a constant and add. You choose the constants so that the resulting coefficient of one of the variables will be 0. Example:
2. Problem: Solve the following system:
5x + 3y = 7
3x - 5y = -23
Solution: Multiply the second equation by 5 to make the x-coefficient a multiple of 5. (This works because it does not change the equation (see the multiplication property).)
15x - 25y = -115
Next, multiply the first equation by -3 and add it to the second equation. This gets rid of the x-term.
-15x - 9y = -21
15x - 25y = -115
-----------------
- 34y = -136
Now, solve the second equation for y. Then substitute the result into the first equation and solve for x.
-34y = -136
y = 4
5x + 3(4) = 7
5x + 12 = 7
5x = -5
x = -1
The solution is the ordered pair, (-1, 4).
Since you would need a three-dimensional coordinate system to solve systems in three variables, solving graphically is not an option. Substitution would work, but is usually unmanageable. Therefore, we will use the addition method, which is basically the same process as it is with systems in two variables.
1. Problem: Solve the following system:
x + y + z = 4
x - 2y - z = 1
2x - y - 2z = -1
Solution: Start out by multiplying the first equation by -1 and add it to the second equation to eliminate x from the second equation.
-x - y - z = -4
x - 2y - z = 1
----------------
-3y - 2z = -3
Now eliminate x from the third equation by multiplying the first equation by -2 and add it to the third equation.
-2x - 2y - 2z = -8
2x - y - 2z = -1
------------------
-3y - 4z = -9
Next, eliminate y from the third equation by multiplying the second equation by -1 and adding it to the third equation.
3y + 2z = 3
-3y - 4z = -9
--------------
-2z = -6
Solve the third equation for z.
-2z = -6
z = 3
Substitute 3 for z in the second equation and solve for y.
-3y - 2z = -3
-3y - 2(3) = -3
-3y - 6 = -3
-3y = 3
y = -1
Lastly, substitute -1 for y and 3 for z in the first equation and solve for x.
x + (-1) + 3 = 4
x + 2 = 4
x = 2
The answer is (2, -1, 3).
The easiest way to solve systems of inequalities is to solve them by graphing. Therefore, it is best if you know how to graph inequalities in two variables (5x - 4y < 13, for example). Example:
1. Problem: Graph y < x.
Solution: First graph the equation y = x. However, the line must be drawn dashed because the less than sign tells us the line is not included in the solution. Next, test a point that is located above the line and one that is below the line. Any point you pick above the line, such as (0, 2), y is greater than x, so points above the line are not in-cluded in the solution. Points below the line, such as (3, -3) have a y value that is less than the x value, so all points below the line are included in the solution.
To solve a system or conjunction of inequalities, it is easiest to graph each of the inequalities and then find their intersection. Example:
1. Problem: Graph the following system:
2x + y >= 2
4x + 3y <= 12
(1/2) <= x <= 2
y >= 0
Solution: See the figure below.
Quadratic equations of type ax2 + bx + c = 0 and ax2 + bx = 0 (c is 0) can be factored to solve for x. Examples:
1. Problem: Solve 3x2 + x - 2 = 0 for x.
Solution: Factor.
(3x - 2)(x + 1) = 0
Use the principle of zero products,
which says, if ab = 0, either
a, b, or both must be
equal to zero.
3x - 2 = 0, x + 1 = 0
3x = 2 , x = -1
x = (2/3)
x = -1, (2/3)
2. Problem: Solve 3x2 + 5x = 0 for x.
Solution: Factor.
x(3x + 5) = 0
Use the principle of zero products.
x = 0, 3x + 5 = 0
3x = -5
x = -(5/3)
x = 0, -(5/3)
Quadratic equations of type ax2 + c = 0 can be solved by solving for x. Example:
3. Problem: Solve 3x2 = 6 for x.
Solution: Recognize that the equation is quadratic because it is the same as 3x2 - 6 = 0.
Divide each side by 3.
x2 = 2
Take the square root of each side.
x = SQRT(2), -(SQRT(2))
Many times you will come across quadratic equations that are not easy to factor or solve. In those cases, there is a special formula called the quadratic formula that you can use to solve any quadratic equation.
The solutions of any quadratic equation, ax2 + bx + c = 0 is given by the following formula, called the quadratic formula:
-b ± SQRT(b2 - 4ac)
x = -------------------
2a
Example:
1. Problem: Solve 3x2 + 5x = -1 for x.
Solution: First find the standard form of
the equation and determine a, b,
and c.
3x2 + 5x + 1 = 0
a = 3
-5 ± SQRT(52 - 4(3)(1))
x = -----------------------
2 * 3
b = 5
c = 1
Plug the values you found for a, b, and c into the quadratic formula. Perform any indicated operations.
-5 ± SQRT(25 - 12)
x = ------------------
6
-5 ± SQRT(13)
x = -------------
6
The solutions are as follows:
-5 + SQRT(13) -5 - SQRT(13)
x = -------------, -------------
6 6
Some equations are not quadratic equations, but are in the same form, such as x4 - 9x2 + 8 = 0. To solve equations such as that, you make a substitution, solve for the new variable, and then solve for the original variable. Example:
1. Problem: Solve x4 - 9x2 + 8 = 0 for x.
Solution: Let u = x2. Then substitute u for every x2 in the equation.
u2 - 9u + 8 =0
Factor.
(u - 8)(u - 1) = 0
Utilize the principle of zero products.
u - 8 = 0, u - 1 = 0
u = 8 , u = 1
Now substitute x2 for u and solve the equations.
x2 = 8, x2 = 1
x = ±SQRT(8), x = ± 1
x = ±2(SQRT(2))
x = ±2(SQRT(2)), ±1
Another way to solve systems of equations is by substitution. In this method, you solve on equation for one variable, then you substitute that solution in the other equation, and solve. Example:
1. Problem: Solve the following system:
x + y = 11
3x - y = 5
Solution: Solve the first equation for y (you could solve for x - it doesn't matter).
y = 11 - x
Now, substitute 11 - x for y in the second equation. This gives the equation one variable, which earlier algebra work has taught you how to do.
3x - (11 - x) = 5
3x - 11 + x = 5
4x = 16
x = 4
Now, substitute 4 for x in either equation and solve for y. (We use the first equation below.)
4 + y = 11
y = 7
The solution is the ordered pair,
(4, 7).
The last method, addition, is probably the most complicated, but is necessary when dealing with more complex systems, such as systems with three or more variables. The idea behind the addition method is to replace an equation with a combination of the equations in the system. To obtain such a combination, you multiply each equation by a constant and add. You choose the constants so that the resulting coefficient of one of the variables will be 0. Example:
2. Problem: Solve the following system:
5x + 3y = 7
3x - 5y = -23
Solution: Multiply the second equation by 5 to make the x-coefficient a multiple of 5. (This works because it does not change the equation (see the multiplication property).)
15x - 25y = -115
Next, multiply the first equation by -3 and add it to the second equation. This gets rid of the x-term.
-15x - 9y = -21
15x - 25y = -115
-----------------
- 34y = -136
Now, solve the second equation for y. Then substitute the result into the first equation and solve for x.
-34y = -136
y = 4
5x + 3(4) = 7
5x + 12 = 7
5x = -5
x = -1
The solution is the ordered pair, (-1, 4).
Since you would need a three-dimensional coordinate system to solve systems in three variables, solving graphically is not an option. Substitution would work, but is usually unmanageable. Therefore, we will use the addition method, which is basically the same process as it is with systems in two variables.
1. Problem: Solve the following system:
x + y + z = 4
x - 2y - z = 1
2x - y - 2z = -1
Solution: Start out by multiplying the first equation by -1 and add it to the second equation to eliminate x from the second equation.
-x - y - z = -4
x - 2y - z = 1
----------------
-3y - 2z = -3
Now eliminate x from the third equation by multiplying the first equation by -2 and add it to the third equation.
-2x - 2y - 2z = -8
2x - y - 2z = -1
------------------
-3y - 4z = -9
Next, eliminate y from the third equation by multiplying the second equation by -1 and adding it to the third equation.
3y + 2z = 3
-3y - 4z = -9
--------------
-2z = -6
Solve the third equation for z.
-2z = -6
z = 3
Substitute 3 for z in the second equation and solve for y.
-3y - 2z = -3
-3y - 2(3) = -3
-3y - 6 = -3
-3y = 3
y = -1
Lastly, substitute -1 for y and 3 for z in the first equation and solve for x.
x + (-1) + 3 = 4
x + 2 = 4
x = 2
The answer is (2, -1, 3).
The easiest way to solve systems of inequalities is to solve them by graphing. Therefore, it is best if you know how to graph inequalities in two variables (5x - 4y < 13, for example). Example:
1. Problem: Graph y < x.
Solution: First graph the equation y = x. However, the line must be drawn dashed because the less than sign tells us the line is not included in the solution. Next, test a point that is located above the line and one that is below the line. Any point you pick above the line, such as (0, 2), y is greater than x, so points above the line are not in-cluded in the solution. Points below the line, such as (3, -3) have a y value that is less than the x value, so all points below the line are included in the solution.
To solve a system or conjunction of inequalities, it is easiest to graph each of the inequalities and then find their intersection. Example:
1. Problem: Graph the following system:
2x + y >= 2
4x + 3y <= 12
(1/2) <= x <= 2
y >= 0
Solution: See the figure below.
ANALYSIS OF IMPLEMENTATION OF VARIOUS
(An approach in the field of educational research)
Abstraction
Analysis of variance is part of the statistics was first introduced by Sir Ronald & Fisher. This analysis has been utilized in all areas of research that uses quantitative data. In the implementation, analysis of variance using ANOVA (Analysis of Variance). In the education sector analysis of variance was used to test whether there are similarities or differences between the learning methods and sebagianya.
Preliminary
In a study of quantitative data obtained must be processed and analyzed to become an information that we want. The process of acquisition, processing, and data analysis are usually referred to as the statistical method. According to Hadi Sutrisno (1996: 3) statistic is knowledge related to the ways of data collection, processing or penganalisisannya and conclusion based on the collection and analyzing data that are made. While the analysis of the data itself, according to Hasan, Iqbal (2004: 29) can be interpreted as follows: (1) to compare two things or two variable value / more to know the difference / ratio is then drawn a conclusion, (2) regulate the process of sequence data, mengorganisaikannya into a pattern, categories, and the basic outline of the unit, (3) detailing the process of formal efforts to find a theme and to formulate hypotheses as suggested by the data and in an effort to provide assistance on themes and hypotheses. And according Setyowati, Eni (2008: 1) Statistics is the science of collecting, organizing, presenting, analyzing, and interpreting the data into information for effective decision making emmbantu. The excess of data with statistical analysis are: (1) allows an exact description of something. Verbal symbols is more efficient than verbal language, (2) allows a person to work exactly in the process and ways of thinking. Although not absolutely true, but it can be set up to which level of the conclusion is correct, (3) researchers provide summaries of research results in a more meaningful and more compact because it provides certain rules, (4) can draw general conclusions, and (5) possible to conduct forecasts (Hasan, Iqbal, 2004: 30). One of the quantitative data with statistical analysis is the analysis of variance.
Before we perform analysis of variance we need to first understand the concept of experimental design, because the analysis of variance was an experiment or experimental research.
Experimental design
Experiments are generally carried out to find something. Therefore, theoretically, be interpreted as a test or experiment planned inquiry to obtain new facts (Steel and Torrie, in Herawati Nita, 2007: Online). And experimental design can be interpreted as a test or series of tests in which significant changes made to the variables of a process or system so that we can observe and identify the reasons for the change in output response. While according to the Herawati Miliken and Johnson, Nita (2007: Online), the experimental design is very related to the planning of research to obtain the maximum information from materials that are available. And can also be interpreted as a set of rules / way / procedure to apply the treatment to the experimental units.
From the various definitions of the above, it is clear that the purpose of the experiment is similar to that answer one or more questions to get the maximum information by: (1) Determine which variables most affect the response (response), (2) determine how sets of independent variables that influence The dependent variable thus approaching the desired nominal value, (3) Determine how the set of independent variables that influence the dependent variable range of small, (4) Determine how to set up an independent variable so that the uncontrollable variables as small as possible.
In designing a study, researchers often control the specific influences such as treatment, population, or a combination of treatments. Therefore, before the research goes it raises some questions that must be answered: (1) How many treatments that must be applied, (2) How many times each treatment must be observed, (3) What are the units of parole, (4) How to apply to the treatment unit experiment and observing the response, (5) Can the design was analyzed and compared?
To be able to answer these questions directly and do not have can not be answered in general. Here, the experimental design is used so that it can play an important role in the development process and the process of finding and solving problems in order to enhance research.
Three main principles in experimental design are: (1) Deuteronomy. Deuteronomy is applied one treatment to more than one experimental unit. Apling Deuteronomy is of importance in an investigation and has the function to provide experimental error, increase precision by reducing the standard deviation, improve generalization, (2) randomization. Randomization is the underlying statistical methods in experimental design. Randomization is the application of treatments to experimental units so that all / any single experiment has an equal opportunity to emnerima a treatment. The concept of randomization is valid also for decision or determination of a sub sample of observation units. Randomization function to avoid bias, ensure the existence of freedom between observations, and overcome the sources of variability that are known but unpredictable effects; (3) Grouping. Grouping is a technique used to improve the accuracy of the experiment. Grouping done if there is a source of diversity that can be known and its influence can be estimated. Experimental material arranged in groups of experimental units are relatively uniform (Herawati, Nita, 2007: Online)
Variety Analysis
During this time, analysis of variance was used in the biometric field (agriculture, biology, medicine, engineering, etc.). In this paper we will focus on the use of analysis of variance in education. Analysis of variance was first introduced by Sir Ronald & Fisher and is basically an arithmetic process for dividing the total sum of squares into components associated with a known source of variability (Steel and Torrie, 1991: 168). In Hadi, Sutrisno (1988: 367), analysis of variance was also known as variance analysis is a tool to test hypotheses about differences in the mean zero value of more than two samples simultaneously. According Suharyadi (2004: 440), analysis of variance (ANOVA) was used to distinguish three or more with the middle value assumptions underlying ANOVA are (a) sample obtained from a normal population, (b) each population has the same standard deviation, and ( c) all populations are independent of each other. From bebarapa definition above we can conclude that the analysis of variance is a method of describing the total diversity of the data into components that measure various sources of diversity with the aim of testing the similarity some central values in one go.
Analysis of variance is the separation of the total number of kuadarat into components associated with a known source of variability. In the analysis of variance involving real test basic assumptions are: (1) Effect of additive treatment and environment, (2) are random experimental error, to spread freedom and normal dis ekitar middle value range of zero and common (same). These basic assumptions is an absolute thing that must be filled in using analysis of variance. Non-fulfillment of one or more assumptions can affect both the real level (level of significance) and the sensitivity of F or t the deviation from the null hypothesis is true.
In the case of abnormalities, the actual real rate is usually larger than those stated. This resulted in rejection of the null hypothesis opportunity hypothesis is true when larger, in other words too often be said when in fact not significantly different. Researchers may think that he used the real level 5 percent when in fact 7a tau 8 percent. If the assumption is not fulfilled aditifitas heterogeneity will cause an error. Various error components contributed by various observations did not expect the same variety. This resulted in a general range of assumptions automatically become not met as well. While the assumption of freedom of error will didapats ecara done directly if the randomization.
Because of these basic assumptions is absolute, after the data obtained from the research, then the first step is to test the crew if the data meet the assumptions there. Recently many statistical software that can help researchers to test these assumptions more easily. And if these assumptions are not met after testing proved one way around that is to perform data transformation.
Basically the analysis of variance was divided into five categories, namely (1) Analysis of Variety I (Classification of One Way / One Way Anova), (2) Variety Analysis II (Classification Many Directions), (3) Analysis of Variety III (Factorial Experiments), ( 4) Variety Analysis IV (Split plot design), and (5) V Variety Analysis (Number of Child-Class is not the same) (Steel and Torry, 1991: 168). In this paper only be described analysis of variance was I, II, and III.
Variety Analysis I (One-Way ANOVA)
According to Santoso, Singgih (2007: xxvi), Analysis of variance I used to completely randomized design. This design is the simplest and only examine the contents of the data column. Statistical procedures used are one-way anova. Now this with the help of existing software such as SPSS, statistical and sebagainyakita will be easier to do anova test without doing the calculation manually. In this discussion we will try to use the aid of statistical software SPSS 16.0 application.
Guide to do a one-way ANOVA test were as berikuit:
1. Determine and Hi Ho
2. Determine the confidence level or a significant level; in general level of confidence (confidence level) is 95% so that the significant level (significant level) is 100% - 95% = 5%
3. Determine the statistical procedures that will be used:
SSB formula above can be calculated by:
SSW formula above can be calculated by:
Where SST (total sum of square) is a quadratic toatal all values of data, which can be expressed by the formula:
4. Take the conclusion to accept or reject H0
If statistics count
Example application: A school melakukan research to determine whether a variety of teaching methods baru give different results, or not with the academic ability of students to examine the effectiveness of both methods, in this study included standard methods that have been used in the schools as a comparison, while two The new method is the method of X and Y. methods Each method is applied to 5 students with different students. After three months of training, academic ability is measured through a test score. Data from the study are as follows:
Student Name X Y Standard
A 49 71 83
B 60 60 87
C 57 65 89
D 59 59 92
E 55 69 95
(Kurniawan, 2007: Online)
Analysis: In the case of so-called factor (independent variable) is a teaching method while the level / treatment are three methods of teaching and the score was the dependent variable.
Testing procedures are:
1. Create a hypothesis:
Ho: score values using three methods of teaching the same relative to each other.
(Μ1 = μ2 = μ3)
Hi: at least one score value of a different teaching methods with other methods.
2. 95% confidence level and 5% significance level.
3. With SPSS 16.0 statistical procedures
STUDENT'S NAME SCORE METHOD
1 1 49
2 1 60
3 1 57
4 1 59
5 1 55
6 2 71
7 2 60
8 2 65
9 2 59
10 2 69
11 3 83
12 3 87
13 3 89
14 3 92
15 3 95
SPSS output is as follows:
Obtained:
F count = 64.877
F table (0:05; 2.12) = 3.88
Conclusion:
Therefore, calculated F> F table, so Ho rejected
It was concluded that teaching methods are applied to produce at least one score in three different teaching methods or in other words, these three methods of teaching have a different impact on students' academic abilities.
Variety Analsisi II (Classification Many Directions)
In the analysis of variance II we not only use a single classification, but many classifications. Entered into this dual classification is not only possibly be done in many investigations but is also very useful to get more information and more accurate. According to Santoso, Singgih (2007: xxviii) II analysis of variance was used for the randomized block design. This design to test the contents of columns and rows of data. Same statistical procedures used with a one-way anova.
Example applications: the same problem but variable analysis of variance was first included in the testing of students to determine whether the student 1 to 5 resulted in a score of academic ability. A school conducted a study to determine whether the various new teaching methods give different results, or not with the academic ability of students to examine the effectiveness of both methods, in this study included standard methods that have been used in schools as a comparison, while the two new methods The method is a method of X and Y. Each method is applied to five students. After three months of training, academic ability is measured through a test score. Data from the study are as follows:
Student Name X Y Standard
A 49 71 83
B 60 60 87
C 57 65 89
D 59 59 92
E 55 69 95
(Kurniawan, 2007: Online)
In this model there is a block variable, namely STUDENT. Now there will be two tests, namely the influence of the method and influence of students, in the language of the test statistics are columns and rows.
To test the column that contains the variable method of teaching procedures performed:
1. Create a hypothesis:
Ho: score values using three methods of teaching the same relative to each other.
(Μ1 = μ2 = μ3)
Hi: at least one score value of a different teaching methods with other methods.
2. 95% confidence level and 5% significance level.
3. With SPSS 16.0 statistical procedures
Obtained:
F count = 64.877
F table (0:05; 2.12) = 3.88
Conclusion:
Therefore, calculated F> F table, so Ho rejected
It was concluded that teaching methods are applied to produce at least one score in three different teaching methods or in other words, these three methods of teaching have a different impact on students' academic abilities.
The procedure to test the line that contains the variable students is as follows:
Ho: no significant differences between the academic ability of students in existing schools
Hi: at least one school with different academic scores than other schools
In SPSS, use the menu GENERAL LINEAR MODEL
STUDENT'S NAME SCORE METHOD
1 1 49
2 1 60
3 1 57
4 1 59
5 1 55
1 2 71
2 2 60
3 2 65
4 2 59
5 2 69
1 3 83
2 3 87
3 3 89
4 3 92
5 3 95
P value (sig) for variable p-value method = 0.000 (sig) for the variable name = 0.796. The value of having a probability value above 5% so thank Ho.
It can be concluded that the average score of academic ability was not significantly different for students who are there, the average score of academic ability relative the same four methods for both students A, B, C, D, and E. While the variables of teaching methods are analyzed with the conclusion remains that there are clear differences in scores of academic abilities seen from the method of teaching provided.
Variety Analysis III (Factorial Experiments)
In the analysis of variance III there are a number of treatment in each of several categories that form a grid treatment. Selection of decomposition of the design of treatment leads to the treatment sum of squares into components of the additive-konponen following hypothesis test (Steel and Torrie, 1993: 403).
Factor is a kind of treatment, and in the factorial experiment, each factor has several treatments. Factorial experiment is an experiment so that treatment consists of all possible combinations of levels of several factors. This experiment gives a huge advantage in explanatory research, which to our knowledge about the optimum level of each factor is still very minimal. This experiment is testing the contents of columns and rows of data, as well as whether there is any interaction between the contents of columns and rows. Satatistik procedure used is TWO-WAY ANOVA.
Example applications:
School Name X Y Standard
A 49 71 83
A 60 60 87
B 57 65 89
B 59 59 92
In this model the interaction between variables was tested columns and rows.
1. Create a hypothesis
Ho: There is no interaction between the methods with these schools
Hi: There is an interaction between the methods with these schools
2. Procedures with SPSS 16.0
REFERENCES
Steel and Torrie, Principles and Procedures of Statistics A Biometrics Approach, Jakarta, Gramedia Pustaka Utama, 1993.
Hasan, Iqbal, Research Analysis With Statistics, Jakarta, Earth Literacy, 2006.
Hadi, Sutrisno, Statistics 3, Yogyakarta, Andi Offset, 1993.
Suharyadi, Statistics for Modern Economy and Finance, Jakarta, Salemba Four, 2004.
Setyowati, Eni, Diktat Maetode Statistics, Tulungagung, STAIN, 2008.
Sudjana. Statistical Methods. Bandung, Tarsito, 1996.
Setyowati, Eni, Application Module Statistics with SPSS 16.0. Tulungagung, STAIN, 2008.
Hamang, Abdul, Statistical Methods, Jakarta, Graha Science, 2005.
Santoso, Singgih, Questions answered Statistics with SPSS and EXCEL, Jakarta, Gramedia, 2007.
Herawati, Nita, 2007, Experimental Design (Online) http://lemlit.unila.ac.id/file/makalah% 20pdf/BAHANMETODOL.DOSEN.pdf, accessed on January 28, 2008
Kurniawan, Deni, 2007. Analysis of variance of a Direction (One Way ANOVA) ineddeni.files.wordpress.com/2007/11/oneway.pdf - Similar pages, accessed on 28 January 2008
Abstraction
Analysis of variance is part of the statistics was first introduced by Sir Ronald & Fisher. This analysis has been utilized in all areas of research that uses quantitative data. In the implementation, analysis of variance using ANOVA (Analysis of Variance). In the education sector analysis of variance was used to test whether there are similarities or differences between the learning methods and sebagianya.
Preliminary
In a study of quantitative data obtained must be processed and analyzed to become an information that we want. The process of acquisition, processing, and data analysis are usually referred to as the statistical method. According to Hadi Sutrisno (1996: 3) statistic is knowledge related to the ways of data collection, processing or penganalisisannya and conclusion based on the collection and analyzing data that are made. While the analysis of the data itself, according to Hasan, Iqbal (2004: 29) can be interpreted as follows: (1) to compare two things or two variable value / more to know the difference / ratio is then drawn a conclusion, (2) regulate the process of sequence data, mengorganisaikannya into a pattern, categories, and the basic outline of the unit, (3) detailing the process of formal efforts to find a theme and to formulate hypotheses as suggested by the data and in an effort to provide assistance on themes and hypotheses. And according Setyowati, Eni (2008: 1) Statistics is the science of collecting, organizing, presenting, analyzing, and interpreting the data into information for effective decision making emmbantu. The excess of data with statistical analysis are: (1) allows an exact description of something. Verbal symbols is more efficient than verbal language, (2) allows a person to work exactly in the process and ways of thinking. Although not absolutely true, but it can be set up to which level of the conclusion is correct, (3) researchers provide summaries of research results in a more meaningful and more compact because it provides certain rules, (4) can draw general conclusions, and (5) possible to conduct forecasts (Hasan, Iqbal, 2004: 30). One of the quantitative data with statistical analysis is the analysis of variance.
Before we perform analysis of variance we need to first understand the concept of experimental design, because the analysis of variance was an experiment or experimental research.
Experimental design
Experiments are generally carried out to find something. Therefore, theoretically, be interpreted as a test or experiment planned inquiry to obtain new facts (Steel and Torrie, in Herawati Nita, 2007: Online). And experimental design can be interpreted as a test or series of tests in which significant changes made to the variables of a process or system so that we can observe and identify the reasons for the change in output response. While according to the Herawati Miliken and Johnson, Nita (2007: Online), the experimental design is very related to the planning of research to obtain the maximum information from materials that are available. And can also be interpreted as a set of rules / way / procedure to apply the treatment to the experimental units.
From the various definitions of the above, it is clear that the purpose of the experiment is similar to that answer one or more questions to get the maximum information by: (1) Determine which variables most affect the response (response), (2) determine how sets of independent variables that influence The dependent variable thus approaching the desired nominal value, (3) Determine how the set of independent variables that influence the dependent variable range of small, (4) Determine how to set up an independent variable so that the uncontrollable variables as small as possible.
In designing a study, researchers often control the specific influences such as treatment, population, or a combination of treatments. Therefore, before the research goes it raises some questions that must be answered: (1) How many treatments that must be applied, (2) How many times each treatment must be observed, (3) What are the units of parole, (4) How to apply to the treatment unit experiment and observing the response, (5) Can the design was analyzed and compared?
To be able to answer these questions directly and do not have can not be answered in general. Here, the experimental design is used so that it can play an important role in the development process and the process of finding and solving problems in order to enhance research.
Three main principles in experimental design are: (1) Deuteronomy. Deuteronomy is applied one treatment to more than one experimental unit. Apling Deuteronomy is of importance in an investigation and has the function to provide experimental error, increase precision by reducing the standard deviation, improve generalization, (2) randomization. Randomization is the underlying statistical methods in experimental design. Randomization is the application of treatments to experimental units so that all / any single experiment has an equal opportunity to emnerima a treatment. The concept of randomization is valid also for decision or determination of a sub sample of observation units. Randomization function to avoid bias, ensure the existence of freedom between observations, and overcome the sources of variability that are known but unpredictable effects; (3) Grouping. Grouping is a technique used to improve the accuracy of the experiment. Grouping done if there is a source of diversity that can be known and its influence can be estimated. Experimental material arranged in groups of experimental units are relatively uniform (Herawati, Nita, 2007: Online)
Variety Analysis
During this time, analysis of variance was used in the biometric field (agriculture, biology, medicine, engineering, etc.). In this paper we will focus on the use of analysis of variance in education. Analysis of variance was first introduced by Sir Ronald & Fisher and is basically an arithmetic process for dividing the total sum of squares into components associated with a known source of variability (Steel and Torrie, 1991: 168). In Hadi, Sutrisno (1988: 367), analysis of variance was also known as variance analysis is a tool to test hypotheses about differences in the mean zero value of more than two samples simultaneously. According Suharyadi (2004: 440), analysis of variance (ANOVA) was used to distinguish three or more with the middle value assumptions underlying ANOVA are (a) sample obtained from a normal population, (b) each population has the same standard deviation, and ( c) all populations are independent of each other. From bebarapa definition above we can conclude that the analysis of variance is a method of describing the total diversity of the data into components that measure various sources of diversity with the aim of testing the similarity some central values in one go.
Analysis of variance is the separation of the total number of kuadarat into components associated with a known source of variability. In the analysis of variance involving real test basic assumptions are: (1) Effect of additive treatment and environment, (2) are random experimental error, to spread freedom and normal dis ekitar middle value range of zero and common (same). These basic assumptions is an absolute thing that must be filled in using analysis of variance. Non-fulfillment of one or more assumptions can affect both the real level (level of significance) and the sensitivity of F or t the deviation from the null hypothesis is true.
In the case of abnormalities, the actual real rate is usually larger than those stated. This resulted in rejection of the null hypothesis opportunity hypothesis is true when larger, in other words too often be said when in fact not significantly different. Researchers may think that he used the real level 5 percent when in fact 7a tau 8 percent. If the assumption is not fulfilled aditifitas heterogeneity will cause an error. Various error components contributed by various observations did not expect the same variety. This resulted in a general range of assumptions automatically become not met as well. While the assumption of freedom of error will didapats ecara done directly if the randomization.
Because of these basic assumptions is absolute, after the data obtained from the research, then the first step is to test the crew if the data meet the assumptions there. Recently many statistical software that can help researchers to test these assumptions more easily. And if these assumptions are not met after testing proved one way around that is to perform data transformation.
Basically the analysis of variance was divided into five categories, namely (1) Analysis of Variety I (Classification of One Way / One Way Anova), (2) Variety Analysis II (Classification Many Directions), (3) Analysis of Variety III (Factorial Experiments), ( 4) Variety Analysis IV (Split plot design), and (5) V Variety Analysis (Number of Child-Class is not the same) (Steel and Torry, 1991: 168). In this paper only be described analysis of variance was I, II, and III.
Variety Analysis I (One-Way ANOVA)
According to Santoso, Singgih (2007: xxvi), Analysis of variance I used to completely randomized design. This design is the simplest and only examine the contents of the data column. Statistical procedures used are one-way anova. Now this with the help of existing software such as SPSS, statistical and sebagainyakita will be easier to do anova test without doing the calculation manually. In this discussion we will try to use the aid of statistical software SPSS 16.0 application.
Guide to do a one-way ANOVA test were as berikuit:
1. Determine and Hi Ho
2. Determine the confidence level or a significant level; in general level of confidence (confidence level) is 95% so that the significant level (significant level) is 100% - 95% = 5%
3. Determine the statistical procedures that will be used:
SSB formula above can be calculated by:
SSW formula above can be calculated by:
Where SST (total sum of square) is a quadratic toatal all values of data, which can be expressed by the formula:
4. Take the conclusion to accept or reject H0
If statistics count
Example application: A school melakukan research to determine whether a variety of teaching methods baru give different results, or not with the academic ability of students to examine the effectiveness of both methods, in this study included standard methods that have been used in the schools as a comparison, while two The new method is the method of X and Y. methods Each method is applied to 5 students with different students. After three months of training, academic ability is measured through a test score. Data from the study are as follows:
Student Name X Y Standard
A 49 71 83
B 60 60 87
C 57 65 89
D 59 59 92
E 55 69 95
(Kurniawan, 2007: Online)
Analysis: In the case of so-called factor (independent variable) is a teaching method while the level / treatment are three methods of teaching and the score was the dependent variable.
Testing procedures are:
1. Create a hypothesis:
Ho: score values using three methods of teaching the same relative to each other.
(Μ1 = μ2 = μ3)
Hi: at least one score value of a different teaching methods with other methods.
2. 95% confidence level and 5% significance level.
3. With SPSS 16.0 statistical procedures
STUDENT'S NAME SCORE METHOD
1 1 49
2 1 60
3 1 57
4 1 59
5 1 55
6 2 71
7 2 60
8 2 65
9 2 59
10 2 69
11 3 83
12 3 87
13 3 89
14 3 92
15 3 95
SPSS output is as follows:
Obtained:
F count = 64.877
F table (0:05; 2.12) = 3.88
Conclusion:
Therefore, calculated F> F table, so Ho rejected
It was concluded that teaching methods are applied to produce at least one score in three different teaching methods or in other words, these three methods of teaching have a different impact on students' academic abilities.
Variety Analsisi II (Classification Many Directions)
In the analysis of variance II we not only use a single classification, but many classifications. Entered into this dual classification is not only possibly be done in many investigations but is also very useful to get more information and more accurate. According to Santoso, Singgih (2007: xxviii) II analysis of variance was used for the randomized block design. This design to test the contents of columns and rows of data. Same statistical procedures used with a one-way anova.
Example applications: the same problem but variable analysis of variance was first included in the testing of students to determine whether the student 1 to 5 resulted in a score of academic ability. A school conducted a study to determine whether the various new teaching methods give different results, or not with the academic ability of students to examine the effectiveness of both methods, in this study included standard methods that have been used in schools as a comparison, while the two new methods The method is a method of X and Y. Each method is applied to five students. After three months of training, academic ability is measured through a test score. Data from the study are as follows:
Student Name X Y Standard
A 49 71 83
B 60 60 87
C 57 65 89
D 59 59 92
E 55 69 95
(Kurniawan, 2007: Online)
In this model there is a block variable, namely STUDENT. Now there will be two tests, namely the influence of the method and influence of students, in the language of the test statistics are columns and rows.
To test the column that contains the variable method of teaching procedures performed:
1. Create a hypothesis:
Ho: score values using three methods of teaching the same relative to each other.
(Μ1 = μ2 = μ3)
Hi: at least one score value of a different teaching methods with other methods.
2. 95% confidence level and 5% significance level.
3. With SPSS 16.0 statistical procedures
Obtained:
F count = 64.877
F table (0:05; 2.12) = 3.88
Conclusion:
Therefore, calculated F> F table, so Ho rejected
It was concluded that teaching methods are applied to produce at least one score in three different teaching methods or in other words, these three methods of teaching have a different impact on students' academic abilities.
The procedure to test the line that contains the variable students is as follows:
Ho: no significant differences between the academic ability of students in existing schools
Hi: at least one school with different academic scores than other schools
In SPSS, use the menu GENERAL LINEAR MODEL
STUDENT'S NAME SCORE METHOD
1 1 49
2 1 60
3 1 57
4 1 59
5 1 55
1 2 71
2 2 60
3 2 65
4 2 59
5 2 69
1 3 83
2 3 87
3 3 89
4 3 92
5 3 95
P value (sig) for variable p-value method = 0.000 (sig) for the variable name = 0.796. The value of having a probability value above 5% so thank Ho.
It can be concluded that the average score of academic ability was not significantly different for students who are there, the average score of academic ability relative the same four methods for both students A, B, C, D, and E. While the variables of teaching methods are analyzed with the conclusion remains that there are clear differences in scores of academic abilities seen from the method of teaching provided.
Variety Analysis III (Factorial Experiments)
In the analysis of variance III there are a number of treatment in each of several categories that form a grid treatment. Selection of decomposition of the design of treatment leads to the treatment sum of squares into components of the additive-konponen following hypothesis test (Steel and Torrie, 1993: 403).
Factor is a kind of treatment, and in the factorial experiment, each factor has several treatments. Factorial experiment is an experiment so that treatment consists of all possible combinations of levels of several factors. This experiment gives a huge advantage in explanatory research, which to our knowledge about the optimum level of each factor is still very minimal. This experiment is testing the contents of columns and rows of data, as well as whether there is any interaction between the contents of columns and rows. Satatistik procedure used is TWO-WAY ANOVA.
Example applications:
School Name X Y Standard
A 49 71 83
A 60 60 87
B 57 65 89
B 59 59 92
In this model the interaction between variables was tested columns and rows.
1. Create a hypothesis
Ho: There is no interaction between the methods with these schools
Hi: There is an interaction between the methods with these schools
2. Procedures with SPSS 16.0
REFERENCES
Steel and Torrie, Principles and Procedures of Statistics A Biometrics Approach, Jakarta, Gramedia Pustaka Utama, 1993.
Hasan, Iqbal, Research Analysis With Statistics, Jakarta, Earth Literacy, 2006.
Hadi, Sutrisno, Statistics 3, Yogyakarta, Andi Offset, 1993.
Suharyadi, Statistics for Modern Economy and Finance, Jakarta, Salemba Four, 2004.
Setyowati, Eni, Diktat Maetode Statistics, Tulungagung, STAIN, 2008.
Sudjana. Statistical Methods. Bandung, Tarsito, 1996.
Setyowati, Eni, Application Module Statistics with SPSS 16.0. Tulungagung, STAIN, 2008.
Hamang, Abdul, Statistical Methods, Jakarta, Graha Science, 2005.
Santoso, Singgih, Questions answered Statistics with SPSS and EXCEL, Jakarta, Gramedia, 2007.
Herawati, Nita, 2007, Experimental Design (Online) http://lemlit.unila.ac.id/file/makalah% 20pdf/BAHANMETODOL.DOSEN.pdf, accessed on January 28, 2008
Kurniawan, Deni, 2007. Analysis of variance of a Direction (One Way ANOVA) ineddeni.files.wordpress.com/2007/11/oneway.pdf - Similar pages, accessed on 28 January 2008
Subscribe to:
Posts (Atom)
Posts
