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ABSTRACT

In chapter one, the general introduction and important definitions were well stated. In chapter two, the chi-square distribution and chi-square goodness-of-fit test were treated. In chapter three, the chi-square test of independence was highlighted. Finally, in chapter four, we considered the application of chi-square test of independence to real life data and conclusion was drawn.

CHAPTER ONE

INTRODUCTION

1.1 Background to the Study

The Federal Government in the year 2005 through the former Minister of Education, Mrs. Chinwe Obaji, introduced the policy of post-UME (University Matriculation Examination) screening by Universities. This policy mandated all tertiary institutions to carry out the task of further screening candidates after their UME result before giving admission. According to Obaji, candidates with a score of 200 and above will be short-listed by Jamb and their names and scores sent to their universities of choice which should then do or undertake another screening test in form of aptitude tests, oral interview or even another examination. Obaji measured the success of her policy by coming on national television to show cases of students who had scored 280 and above but could not score 20% in the post-UME screening. According to her, these students must have been engaged in cheating during Jamb examination and so could not pass post-UME screening because there was no room for them to cheat or be impersonated. Based on this policy of the then Minister of Education, the former Vice-Chancellor, Prof. E.A.C. Nwanze implemented the policy by introducing the university post-UME screening. Since then, the policy has been highly effective in the university. 200 scores and above remain the benchmark of sitting for the screening exercise.

Chi-square test provides the basis for judging whether more than two population proportions may be considered to be equal. Chi-square test is discussed by considering two aspects, such as chi-square goodness-of-fit and chi-square test of independence. Chi-square test of goodness of fit provides a means for deciding whether a particular theoretical probability such as the binomial distribution is a close approximation to a sample frequency distribution. While test of independence constitutes a method for deciding whether the hypothesis of independence between different variables is tenable. This procedure provides test for the equality of more than two population proportions. Both X2 tests furnish a conclusion on whether a set of observed frequencies differs so greatly from a set of theoretical frequencies that the hypothesis under which the theoretical frequencies were derived should be rejected. Theorem of chi-square distribution is given as: Let X1, X2, … Xv be independent normally distributed random variables with mean
V
zero and variance s2 = 1. Then, X21 + X22, + … + X2v = ∑ X2j is X2 –
j=1 distributed with v degree of freedom. For any decision to be made in statistics, there is need to carry out hypothesis testing. Therefore, Kreyszig (1988) defined hypothesis as any reasonable assumption about the parameters of a distribution. In testing hypothesis in statistics, it is always difficult to carry out testing of hypothesis on the whole population, in this case, a sample is drawn from the population, and it is used to make inference concerning the population. If the inference made is not in agreement with the set assumptions, the hypothesis is rejected, otherwise it is not rejected. Moreover, the procedure of testing hypothesis in parameter statistic helps us to decide whether to reject or not to reject a hypothesis or determine whether observed sample differs significantly from expected result. Hypothesis testing is therefore the process of making decision based on the sample.

According to Rao (1952, 1970), various attempts have been made to build up a consistent theory from which all tests of significance can be deduced as solution to precisely stated mathematical problems. It is difficult to argue whether such a theory exists or not, but formal theories leading to a clear understanding of the problems are nonetheless important. One such theory, contributed by Neyman and Pearson (1928, 1933) is an important development because it unfold the various complex problems in testing of hypothesis and led to the construction of general theories in problems of discrimination, sequential test, etc. There are many questions that arise in the cause of hypothesis testing such as; when should a hypothesis be rejected or not rejected? What is the probability that we will make the wrong decision which can be led to a consequential loss? We may also ask if two variable are independent or interested to know whether if a distribution follows a specific pattern. All these are likely questions that arise in decision making. However, with chi-square statistical test, it is possible to provide answers to the above questions.

1.2 Objectives of the Study

1. To test students’ ability in carrying out a study independently.

2. To show students the use of chi-square tests and its application to real life problems.

3. To help students understand the process involves in decision making.

4. To assist students to understand hypothesis testing on the basis of sample data and make a statistical inference.

Significance of the Study

This study is highly significant to students in mathematics, social and management sciences and all other managerial discipline or field of understanding the nitty-gritty of the process involved in decision making using chi-square independence tests.