Inequalities:

Inequalities:

Introduction to inequalities:

In mathematics, an inequality is a statement about the relative size or order of two objects, or about whether they are the same or not.

* The notation a < b means that a is less than b.

* The notation a > b means that a is greater than b.

* The notation a ≠ b means that a is not equal to b, but does not say that one is greater than the other or even that they can be compared in size.

In each statement above, a is not equal to b. These relations are known as strict inequalities. The notation a < b may also be read as "a is strictly less than b".

In contrast to strict inequalities, there are two types of inequality statements that are not strict.

* The notation a ≤ b means that a is less than or equal to b (or, equivalently, not greater than b)

* The notation a ≥ b means that a is greater than or equal to b (or, equivalently, not less than b).

This was a very general view about the meaning of Inequalities,now let us understand the look at a more definite explanation,An inequalities is a statement, that relates the size or order of two objects or about whether they are the same or not.Solving linear inequalities is very much similar to solving linear equations, except for one small but important change: you flip the inequality sign whenever you multiply or divide the inequality by a negative number. In linear equation we don’t do that.

Procedure for Solving Inequalities:


As in the case of solving inequalities equations, there are certain rules for the inequality problems which do not change the solutions. Here is a list of "permissible'' manipulations:

Step1: Adding/subtracting the same number on both sides.

Step2: Switching sides and changing the orientation of the inequality sign.

Step3: Multiplying/dividing by the same positive number on both sides.

Step4: Multiplying/dividing by the same negative number on both sides and changing the orientation of the inequality sign.

* The notation a < b means a is less than b.
* The notation a > b means a is greater than b.
* The notation a ≠ b means a is not equal to b,
* The notation a ≤ b means a is less than or equal to b (or, equivalently, not greater than b)
* The notation a ≥ b means a is greater than or equal to b

Inequalities - Example Problems:

Solving inequalities for Addition and Subtraction:

1. 4x+2 < -2x+14

Solution:

4x+2-2 < -2x+14-2

4x < -2x+12

4x+2x < -2x+2x+12

6x < 12

6x /6< 12/6

x < 2

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Data Analysis and Interpretation

Data Analysis and Interpretation:
Let us understand how we can analyse the data and interpret it,in this blog let us understand the basic meaning of Data analysis and Interpretation.Next let us understand the meaning of Interpretation,the basic meaning of Interpretation is The power or explaining.
Analysis of data is a process of inspecting, cleaning, transforming, and modeling data with the goal of highlighting useful information, suggesting conclusions, and supporting decision making. Data analysis has multiple facets and approaches, encompassing diverse techniques under a variety of names, in different business, science, and social science domains.But the statistical data analysis is the data analysis plan to examine the research questions. Statistics, Data Analysis, and Probability introduce statistics as a problem-solving process. Different ways to organize, represent data, describe and analyze variation in data. In this introduction, we will briefly discuss those statistical concepts that provide the necessary foundations for more specialized expertise in any area of statistical data analysis.

Statistical Data Analysis:

To find what statistical data analyses are, first we go to define statistics. Statistics is a group of methods that are used to collect, analyze, present, and interpret data. Statistical data analysis gives hands to promote the use of statistical techniques to apply in order to make decisions in the problem solving in data analysis.

Steps for Statistical Data Applications:

In generally statistical data analysis involves four basic steps:

Step1. Defining the problem is step 1.

Step2. Collecting the data is the step 2.

Step3. Analyzing the data is step 3.

Step4. Reporting the results is last step.

Defining the Problem:

A correct definition of the problem is imperative in order to obtain exact correct data about it. It is very difficult to getting data without clear explanation of the problem.

Collecting data:

When the data are collecting and statistical methods have become easy.

Analyzing the Data:

Statistical data analyses are divides methods for analyzing data into two categories: 1. Eploratory methods, nd 2. Confirmatory methods.

Exploratory methods:

Exploratory methods are used for cover what the data seems to be simple arithmetic and draw pictures to summarize the data.

Confirmatory methods:

Confirmatory methods use ideas from probability theory for answer the specific questions.

Reporting the Results:

Through inferences, an estimate the characteristics of a population can be obtained from a sample. The results may report in the form a graph or a set (or) group of percentages.


Statistical Data Analysis Applications are Given Below:

* Statistical methods are used to help people for identify, study, and solve many complex problems.

* The statistical data analysis is used for so many applications; like that the statistical data analysis section will discuss the assumptions of regression, which include the removing of outliers from the data set, the examining of the linearity, and constant variance.

* The statistical data analysis section should know about how the regression will be interpreted. The statistical data analysis plan should state that the linear regression is being evaluated for the model fit,

* And then it is the part of the statistical data analysis is to discuss the R-square value of the linear regression.

* Then the next part of the statistical data analysis plan would be to examine the beta coefficient, the t-value,

* The last part of the statistical data analysis would discuss the significant beta in terms of the criterion variable happiness

* The statistical data analysis also approximates the sample size needed for the analysis.

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Correlation Theory and Regression Analysis

Correlation Theory and Regression Analysis:

The term correlation deals with the relationship between two or more variables. If a change in one variable effect a change in other variable, the variables are said to be correlated.

There are basically three types of correlation, namely,

* Positive correlation
* Negative correlation
* Zero correlation.

In this Blog, let us also learn about Regression.

Regression Definition:

A regression is defined as a statistical analysis assessing the association between two variables. It is used to find the relationship between two variables.


Regression Formula:

Regression Equation(y) = a + bx
Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX2 - (ΣX)2)
Intercept(a) = (ΣY - b(ΣX)) / N

where
x and y are the variables.
b = the slope of the regression line
a = the intercept point of the line and the y axis.
N = Number of values or elements
X = First Score
Y = Second Score
ΣXY = Sum of the product of first and Second Scores
ΣX = Sum of First Scores
ΣY = Sum of Second Scores
ΣX 2 = Sum of square First Scores.

Correlation study are frequently used in psychology research to explain about relationships between the variables. There is a necessary relationship between two variables that can be study about correlation, one variable causes a change in another variable does not proved by finding a correlation relationship. we can study that no equal causation in correlation.

The Purpose of Correlation Study:

Correlation study used for the purpose of representing the relationships between variables. Correlation has three possible way of study: a negative correlation, a positive correlation and no correlation and no correlation. The measurement of correlation co-efficient of correlation strength and the range can be measured from –1 to +1.

*

Positive Correlations: Both variables are increase or decrease at the same time. A correlation coefficient nearer to +1 represents a strong positive correlation.
*

Negative Correlations: Indicates the increase in the amount of one variable, the other decreases (and vice versa). A correlation coefficient nearer to -1 represents a strong negative correlation.
*

No Correlation: It Indicates no relationships between two variables. The correlation coefficient of 0 represents no correlation.

Types of Correlation Study:

1. Naturalistic Observation:

Naturalistic observation explains about recording and observing the number of variables interested in the natural environment without manipulation or interference by the experimenter.

2. The Survey Method:

The most common methods for surveying and questionnaries used in psychological research. In this survey method, the participants can be selected by random sampling can complete a survey, test, or question that is related to variables of interest. Ensuring generalizability of survey results is one of the vital part for random sampling.

3. Archival Research:

Archival research can be performed by the method of analyzing study and this is conducted by researchers of other area or they looking for historical patient records.
Purpose

The correlation is a different way to measure how that is associated with two related variables. The research looks at things that already exist and determines if and in what way those things are related to each other. By doing correlations the purpose is to allow us to make a what we know about one variable based on prediction for another variable.

Hope you like the above example of Correlation Theory and Regression Analysis.Please leave your comments, if you have any doubts.

Curve Fitting

Curve fitting equation:

Introduction to curve fitting equation:

Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data.
The study of ways of constructing functions whose graphs are curves that "best" approximates a given collection of points.We consider approximations by linear functions, polynomials of degree 2 or 3 and exponential functions using the method of least squares. After working through these materials, the student should be able

* to calculate using a graphing calculator or computer software the least squares line, parabola or third degree equation which best fits data.


Curve Fitting Equation:

Built-in curves are able to be use as an assist for data revelation, near assume value of a purpose wherever no data are obtainable; also to go over the associations with two or else further variables. Extrapolation illustrates to the make use of a fixed curve away from the collection of the practical data, with is the topic to a larger degree of indecision given that it might reproduce the way use to make the curve as greatly when it reproduce the experiential data.



Equation for Curve Fitting:

Permit establish through a 1st degree polynomial equation:

y=ax+b

This be a line through slope a. We identify to a line determination attach whichever two points. Thus, a 1st degree polynomial equation is an accurate fit during whichever two points.

Condition we raise the order of the equation toward a next degree polynomial, we obtain:

y=ax2+bx+c

Determination accurately fit an easy curve toward three points.

But we enlarge the order of the equation toward a 3rd degree polynomial, we obtain:

y=ax3+bx2+cx+d

These resolve accurately well four points.

A further common statement would exist toward declare it resolve accurately well four constraints. Every constraint be able to be a point, angle, otherwise curvature. Angle with curving constraints is mainly a lot additional toward the split ends of a curve; also during such belongings be call end conditions.

The curve fit for the polynomial equation is

Y (x) = P(x, n) = a0 + a1*x + a2*x2 ... + ... an*x n

The curve fit for the nonlinear approximation is

F(y) = a0*f0(x) + a1*f1(x) + ... + an*fn (x)

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Standard Probability:

Standard Probability:

Introduction:

A standard probability normal table is also known as “Unit Normal Table" is a numerical table for the values of Φ, the cumulative distribution task of the sharing. They are used to locate the probability that a statistic is practical below, above, or connecting values on the standard normal distribution, and by addition, any normal distribution.Let us now learn about the uses of standard Probability spaces,standard probability spaces are used routinely in ergodic theory, which cannot be said on probability theory. Some probabilists hold the following opinion: only standard probability spaces are pertinent to probability theory, thus, it is a pity that the standardness is not included into the definition of probability space. Others disagree, however:

Arguments against standardness:

* the definition of standardness is technically demanding;
* the same about the theorems based on that definition;
* it is possible (and natural) to build all the probability theory without the standardness;
* events and random variables are essential, while probability spaces are auxiliary and should not be taken too seriously.

Arguments in favour of standardness:

* conditioning is easy and natural on standard probability spaces, otherwise it becomes obscure;
* the same for measure-preserving transformations between probability spaces, group actions on a probability space, etc.;
* ergodic theory uses standard probability spaces routinely and successfully;
* being unable to eliminate these (auxiliary) probability spaces, we should make them as useful as possible.


Normal Distributions:

Normal distributions are symmetrical, bell-figure distributions that are useful in describing actual-earth information. The standard probability normal distribution, represent by the text Z, is the normal distribution has a denote of 0 and a standard deviation of 1. Since probability table cannot be printed for every normal distribution, (there are infinite), it is general practice to change a normal to a standard normal, and use a Z table to find probabilities.

Reading the Table:

Tables use at least 3 different conventions, depending on the understanding of the meaning of an entry such as 1.58:

Cumulative: This is nearly all common, and gives Prob (Z ≤ 1.58)

Complementary cumulative: The complement (1–x) of more than: Prob (Z ≥ 1.58)

Cumulative from zero: The cumulative probability, initial from 0: Prob (0 ≤ Z ≤ 1.58)

These can easily be check by inspect a number like 2.99:

* If this is just about 1 , next it display cumulative probabilities;
* If this is just about 0 , next it display complementary probabilities;
* If this is just about 0.5 , next it display cumulative from 0 probabilities.



Printed tables typically give cumulative probabilities, the possibility that a statistic takes a number less than or the same to a number, from at smallest amount.


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