eChapters

REGRESSION

Regression means to revert or return back. The term was first introduced by Sir Francis Galton in1877. Regression technique is applicable in all those fields where two or more relative variables have the tendency to go back to the mean. Regression analysis refers to the methods by which estimates are made of the values of a variable from the knowledge of the values of one or more other variables and to the measurement of the errors involved in this estimation pro- cess.

In statistical modeling, regression analysis is a statistical process for estimat- ing the relationships among variables. It includes many techniques for mod- eling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. Regres- sion analysis helps one understand how the typical value of the dependent variable (or ‘criterion variable’) changes when any one of the independent variables is varied, while the other independent variables are held fixed. Most commonly, regression analysis estimates the conditional expectation of the dependent variable given the independent variables – that is, the average value of the dependent variable when the independent variables are fixed.

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.

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.

F TEST (VARIANCE RATIO TEST)

F distribution is applied in several tests of significance relating to the equality of two sampling variances drawn on the basis of independent samples from a normal population.

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

CHI- SQUARE TEST (χ₂ TEST)

The chi square test is used to determine if the two attributes are independent of each other. Chi square is a measure to evaluate the difference between observed frequencies and expected frequencies to examine whether the difference so obtained is due to a chance factor or due to sampling error.

In a contingency table if each attribute is divided into two classes it is known as 2×2 contingency table. When one attribute is divided into two classes and another into r or c resultant contingency table is known as r×2 or 2×c contingency table. Here, r denotes the number of rows and c the number of columns. For such data, the statistical hypothesis under test is that the two attribute are independent of one another. To test this hypothesis we use the test statistic

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.

ANALYSIS OF VARIANCE

Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality of two or more population (or treatment) means by examining the variances of samples that are taken. ANOVA allows one to determine whether the differences between the samples are simply due to random error (sampling errors) or whether there are systematic treatment effects that cause the mean in one group to differ from the mean in another.

Most of the time ANOVA is used to compare the equality of three or more means, however when the means from two samples are compared using ANOVA it is equivalent to using a t-test to compare the means of independent samples.

ANOVA is based on comparing the variance (or variation) between the data samples to variation within each particular sample. If the between variation is much larger than the within variation, the means of different samples will not be equal. If the between and within variations are approximately the same size, then there will be no significant difference between sample means.

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).

A simple random sample (SRS) of size n consists of n individuals from the population chosen in such a way that every set of n individuals has an equal chance to be the sample actually selected.

A simple random sample is a subset of a statistical population in which each member of the subset has an equal probability of being chosen. An example of a simple random sample would be the names of 25 employees being chosen out of a hat from a company of 250 employees. In this case, the population is all 250 employees, and the sample is random because each employee has an equal chance of being chosen.

1.1. Definition

A tractor is a specific vehicle designed to deliver a high tractive torque at slow speeds, for the purposes of haulage or for pulling an agricultural implement or equipment or machinery. This vehicle is also used as a source of power for stand-alone agricultural machineries. The word tractor is taken from Latin being the agent noun of trahere “to pull”.

1.2. History of evolution

The first engine-powered farm tractors used steam and were introduced in 1868. These engines resembled small road locomotives. They were used for general road haulage and in particular for timber trade. The most popular steam tractor was the Garrett 4CD.

In 1887, gasoline-fuelled engines were tried as an alternative to steam engines. Later, Henry Ford produced his first experimental gasoline powered tractor in 1907, under the direction of his chief engineer Joseph Galamb. It was referred to as an “automobile plow”. While unpopular at first, these gasoline powered machines began to catch on. In the 1910, the size of the tractor was smaller than the earlier introductions and also became more affordable. In 1917, Henry Ford introduced the first mass-produced tractor. These were built in the U.S.A., Ireland, England and Russia. The unique feature of these tractor consisted of the strength of the engine holding the entire machine together. By 1920, tractors with a gasoline powered internal combustions engine had become a norm.

1.3. History in India

The level of mechanization in Agriculture was very low at the time of independence in 1947. But, the initial few five-year plans implemented by the Government of India gave an impetus to the improvement in the production and productivity of major crops including mechanization via joint ventures and tie-ups between local industrialists and international tractor manufacturers. It was three decades after independence that the domestic production of 4-wheel tractors grew slowly. In 1990, the tractor production was nearly 2,70,000 units per year and by the 2013, the volume approached to about 6,96,000 units per year. India has presently overtaken the United States as the world’s largest producer of four-wheel tractors with over 16 national and 4 multinational corporations engaged in its production. But still the Food and Agriculture Organization estimates that from out of the total agricultural area, less than 50% is under-mechanized land preparation. This indicates that potential for large tractorization still exists. Credit facilities for farmers have been improving steadily over the years. This is one of the prime drivers of the sale of tractor in India.