eChapters

TEST OF SIGNIFICANCE FOR CORRELATION COEFFICIENT

Coefficient of correlation is a numeric characteristic of a population and is estimated from a sample of the bi-variate population. If ρ be the coefficient of correlation in the population and r be its sample estimate based on n pairs of observations, then standard error of r is given

SKEWNESS

Skewness means lack of symmetry. Skewness denotes the tendency of a distri- bution to depart from symmetry. A frequency distribution is said to be skewed if the frequencies decrease with markedly greater rapidity on one side of the central maximum than on the other side. This characteristic of a frequency distribution is known as skewness and the measures of asymmetry are usually called measures of skewness.

The main object of measuring skewness is to know the direction of the varia- tion from an average and to compare the frequency distribution and the shape of their curves.

There are two types of skewness

(i) Positive Skewness (ii) Negative Skewness

FOR GROUPED DATA

Objective: The following data shows the marks of students and their frequencies find the standard deviation and its coefficient.

STANDARD DEVIATION AND THEIR COEFFICIENT

The standard deviation is a measure of the spread of scores within a set of data. Usually, we are interested in the standard deviation of a population. However, as we are often presented with data from a sample only, we can estimate the population standard deviation from a sample standard deviation. These two standard deviations - sample and population standard deviations - are calculat- ed differently. In statistics, we are usually presented with having to calculate sample standard deviations.

STANDARD NORMAL DISTRIBUTION

If μ = 0 and σ = 1, the distribution is called the standard normal distribution or the unit normal distribution, and a random variable with that distribution is a standard normal deviate. The approximate test is given by

t TEST FOR PAIRED OBSERVATION

This test is used for testing whether two series of paired observations are gen- erated from the same population on the basis of the difference in their sample means.

t TEST FOR SINGLE MEAN

A t-test is any statistical hypothesis test in which the test statistic follows a Student’s t distribution if the null hypothesis is supported. It can be used to determine if two sets of data are significantly different from each other, and is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known. When the scaling term is unknown and is replaced by an estimate based on the data, the test statistic (under certain conditions) follows a Student’s t distribution.

t TEST FOR TWO SAMPLE MEAN:

Comparison of two sample means  and assumed to have been obtained on the basis of random samples of sizes n₁ and n₂ from the same population which is assumed to be normal.

The two-way ANOVA compares the mean differences between groups that have been split on two independent variables (called factors). The primary purpose of a two-way ANOVA is to understand if there is an interaction be- tween the two independent variables on the dependent variable. For example, you could use a two-way ANOVA to understand whether there is an interac- tion between gender and educational level on test anxiety amongst university students, where gender (males/females) and education level (undergraduate/ postgraduate) are your independent variables, and test anxiety is your depen- dent variable. Alternately, you may want to determine whether there is an in- teraction between physical activity level and gender on blood cholesterol con- centration in children, where physical activity (low/moderate/high) and gender (male/female) are your independent variables, and cholesterol concentration is your dependent variable.

The interaction term in a two-way ANOVA informs you whether the effect of one of your independent variables on the dependent variable is the same for all values of your other independent variable (and vice versa). For example, is the effect of gender (male/female) on test anxiety influenced by educational level (undergraduate/postgraduate)? Additionally, if a statistically significant interaction is found, you need to determine whether there are any “simple main effects”, and if there are, what these effects are (we discuss this later in our guide).

The chi square is a continuous distribution based on the assumption that the ul- timate class frequency is not less than 5 otherwise the value of χ₂ will be over- estimated and may cause too many rejections of null hypothesis. Therefore, to maintain the continuity of χ₂ and to draw the correct inference in 2×2 we apply a correction which is known as Yates’s correction for continuity.

Raster sub-setting is done to select the desired area or bands of the raster image on which the user wants to perform analysis. Raster sub-setting is done in two ways:

Spectral subset: Spectral sub-setting is done to extract few desired bands from the multiband raster image which are required for analysis.

Spatial subset: Spatial sub-setting is done to minimize the area or extent of raster image to the desired area or extent for further analysis.

Raster sub-setting in R is done by functions available in the ‘raster’ package. The following packages are required in this chapter:

True/False
 
Q.1.    The set of observations or measurements collected on specific characteristics from each item (individual) constitutes data.
Q.2.    Each characteristic that is measured or observed, varies from item to item and is called “Variable”.
Q.3.    “Blood type”, “stage of disease” and “weight” observed on a group of cancer patients, are three distinct variables.
Q.4.    A variable that can be expressed in numbers such as age, height, blood pressure, is said to be “Quantitative variable”.

Supervised classification is the technique to classify raster image in which prior information about the classes in the form of training sites is provided by the user. Training site represents a homogeneous area in a particular class. The spectral response pattern of training sites is used to classify the raster image.The pixels of raster imagewhich have spectral response pattern similar to that of training sites of a particular class are grouped in that class. As there is some variability in the spectral response pattern of a particular class, therefore, multiple training sites per class are taken for supervised classification. Supervised classification of raster images can be performed by various methods. Conventional methods include maximum likelihood classifier, minimum distance to mean classifier, and parallelepiped classifier. Now a days, machine learning algorithms are also gaining importance in this context. These classifying algorithms are first trained or calibrated using the data of training sites.This trained model is then validated on the independent datasets for accuracy assessment. If the accuracy of the developed model is of acceptable level, then that model is used for the classification of raster data.

In this chapter, we are going to discuss the execution of maximum likelihood classifier and few machine learning techniques for supervised classification of raster image in R software. The training sites used in this exercise are in the form of points shapefile. The raster image is classified into six classesi.e. built- up, fallow soil, riverine sand, salt-affected soil, vegetation, and water. For performing this exercise, first of all,set the working directory and import the required packages.

Not all the scientific problems demand the detection and establishment of causal relationship among natural phenomenon. Many of the social research problems seek to study the status quo rather than the relation between two or more natural phenomenon. Few of the impact assessment studies (which are kind of causal relationship studies to an extent) also demand for knowing the status quo before and after implementation of interventions. Many other studies demand for benchmark assessment to make policy decisions. Under such circumstances, survey research is a useful research method.

True/False

Q.1.    In many epidemiological studies with substantial follow up time, it is very often an interest to study the time taken for a particular event to occur.
Q.2.    The time taken for death or graft failure since transplant, time taken for relapse after treatment, time to death after treatment etc. are examples for survival time.

Introduction

A Brief Overview of Industry 4.0

It is widely agreed that foundational technologies such as 5G, IoT, Robotics, Nano-Tech, Brain-Computer Interface (BCI), Neural Networks, Quantum Computing, Cost-efficient and low-emission Transportation, and Sustainable Power have coalesced to give birth to industry 4.0.

Industry 4.0, sometimes also designated as the fourth wave of Industrial Revolution, is all about transforming traditional manufacturing and related industrial practices through various digital technologies. It represents a paradigm shift characterized by the amalgamation of multiple technologies such as computing, the Internet of Things (IoT), convergence, artificial intelligence (AI), big data, and automation into a unified protocol for application in the industrial processes.

At its core, Industry 4.0 emphasizes the interconnectedness and intelligent communication between machines, products, and people. It revolutionizes production and operational methods by enabling machines to operate semi autonomously and make decentralized decisions. This allows for increased f lexibility, efficiency, and customization within manufacturing processes.

Key elements of Industry 4.0 include the digitization of manufacturing, smart factories, and the use of data-driven technologies to optimize production. It incorporates concepts such as predictive maintenance, where machines predict their own maintenance needs, and the Industrial Internet of Things (IIoT), allowing devices and systems to communicate, analyze, and optimize processes in real-time.

6.1 Introduction

So far we have considered the methods of sampling in which the successive units (elements or clusters) were selected with the help of random numbers. Now, we shall consider a new method of sample selection in which only the first unit of the sample is selected with the help of random number, the rest being selected automatically according to a predetermined pattern. The method is known as systematic sampling. It is operationally more convenient than simple random sampling and which at the same time ensures for each unit, equal probability ofinclusion in the sample. The systematic sampling can also be termed as systematic random sampling because of the first unit of the sample being chosen randomly. The pattern usually followed in selecting a systematic sample is a simple pattern involving regular spacing of units.

Q.1 Systematic random sampling or systematic sampling, both are different sampling schemes.

Q.2 The pattern usually followed in selecting a systematic sample is a simple pattern involving a constant spacing of units in the sample.

Experimental unit : An experimental unit is the material to which is applied the treatment and on which the variable under study is measured. In an agricultural field experiment, the plot of land, not the individual plant, will be the experimental unit; in a feeding experiment of cows,the whole cow is the experimental unit; in human experiments in which the treatment affects the individual, the individual will be the experimental unit.

Treatment : The different procedures under comparison in an experiment are the different treatments, i.e in an agricultural experiment, the different varieties of a crop or the different manures will be the treatments. In a dietary or medical experiment, the different diets or medicines, etc. are the treatments.

9.1 Introduction

In the preceeding chapter, we have studied the estimation part of statistical decision (inference) infered about the parameters of the population on the basis of information available in the sample observations drawn from the parent population under study. In the present chapter we discuss the problems related with the tests of statistical hypothesis and its significance about the population parameters under study. For understanding of the subject matter considered in this chapter, very clearly and easily, one must be familier with the statistical terminologies used in the subject matter. Hence we discuss the various notable terminologies with some suitable examples in a simple way for better understanding.