**CHI-SQUARE TEST OF INDEPENDENCE OF STUDENTS’ PERFORMANCE IN UME AND POST-UME**

**abstract**

The general introduction and key definitions were clearly stated in Chapter One. The chi-square distribution and chi-square goodness-of-fit test were covered in Chapter 2. The chi-square test of independence was highlighted in Chapter 3.

Finally, in Chapter 4, we discussed the application of the chi-square test of independence to real-world data and reached a conclusion.

**CHAPTER ONE**

**INTRODUCTION**

**1.1 Background of the Study**

The Federal Government introduced the policy of post-UME (University Matriculation Examination) screening by universities in 2005, through the former Minister of Education, Mrs. Chinwe Obaji. This policy required all tertiary institutions to screen candidates further after their UME results before admitting them. According to Obaji, candidates with a score of 200 or higher will be short-listed by Jamb and their names and scores will be sent to their preferred universities, which will then do or undertake another screening test in the form of aptitude tests, oral interviews, or even another examination. Obaji measured the success of her policy by appearing on national television to show cases of students who had scored 280 or higher but were unable to score 20% in the final exam. the post-UME screening. According to her, these students must have cheated during the Jamb examination and thus were unable to pass post-UME screening because there was no room for them to cheat or impersonate.

Based on the then-Minister of Education’s policy, Prof. E.A.C. Nwanze, the former Vice-Chancellor, implemented the policy by instituting university post-UME screening. Since then, the policy has proven to be extremely effective at the university. The screening exercise continues to require 200 or higher scores.

The Chi-square test is used to determine whether or not more than two population proportions are equal. The chi-square test is discussed in two parts: chi-square goodness-of-fit and chi-square test of independence. Chi-square test of goodness of fit determines whether a specific theoretical probability, such as the binomial distribution, is a close approximation to a sample frequency distribution. While the test of independence is a method for determining whether the hypothesis of independence between different variables is viable. This procedure tests the equality of more than two population proportions. Both X2 tests provide a conclusion as to whether a set of observed frequencies differs so significantly from a set of theoretical frequencies that the hypothesis from which the theoretical frequencies were derived should be rejected.

The chi-square distribution theorem is as follows: Assume X1, X2,… Xv are independent normally distributed random variables with mean V zero and variance s2 = 1. Then X21 + X22, +… + X2v = X2j equals X2. j=1 is randomly distributed with v degrees of freedom.

In statistics, hypothesis testing is required before making any decision. As a result, Kreyszig (1988) defined hypothesis as any reasonable assumption about a distribution’s parameters. It is always difficult to carry out hypothesis testing on the entire population in statistics; therefore, a sample is drawn from the population and used to make inferences about the population. The hypothesis is rejected if the inference made does not agree with the set assumptions; otherwise, it is not rejected.

Furthermore, the procedure of hypothesis testing in parameter statistics assists us in deciding whether to reject or not to reject a hypothesis or determining whether observed samplediffers significantly from expected result. Thus, hypothesis testing is the process of making decisions based on the sample.

Various attempts, according to Rao (1952, 1970), have been made to construct a consistent theory from which all tests of significance can be deduced as solutions to precisely stated mathematical problems. It is difficult to argue whether such a theory exists, but formal theories that lead to a clear understanding of the problems are still important. One such theory, contributed by Neyman and Pearson (1928, 1933), is significant because it unraveled the various complex problems in hypothesis testing and led to the construction of general theories in problems of discrimination, sequential test, and so on.

Many questions arise in relation to the cause. of hypothesis testing, such as when a hypothesis should be rejected or not rejected. What is the likelihood that we will make the wrong decision, resulting in a monetary loss? We might also wonder if two variables are independent or if a distribution follows a specific pattern. All of these are likely questions that will arise during the decision-making process. However, the chi-square statistical test can provide answers to the above questions.

**1.2 Objectives of the Study**

1. To assess students’ ability to conduct independent research.

2. To demonstrate the use of chi-square tests and their application to real-world problems to students.

3. To assist students in understanding the decision-making process.

4. Assist students in understanding hypothesis testing and making statistical inferences based on sample data.

**Significance of the Study**

This study is extremely important for students in mathematics, social and management sciences, and any other managerial discipline or field interested in understanding the nitty-gritty of the decision-making process using chi-square independence tests.

**1.4 Scope of the Study**

The University of Benin Post-UME, chi-square (c2) distribution, test, and procedure for implementing c2-test are all covered in this study.

**1.5 Important Definitions**

Some of the key concepts in statistical analysis are as follows:

Ibrahim (2009) defined population as a set of existing units (typically people, objects, transactions, or events) that we want to study. A population can be both finite and infinite. For example, the population of students in the mathematics department is finite, whereas the population of all possible outcomes (heads, tails) in successive coin tosses is infinite.

Population parameter: According to Dass (1988), population parameters are statistical constraints of the population such as mean (), standard deviation (s). Greek letters are used to represent parameters.

Ibrahim (2009) defined a sample as a subset of a population’s units – that is, a portion or part of the population. The population of interest. The sampling procedure or sampling plan refers to the method of selecting the sample.

Statistical Hypothesis: A statistical hypothesis, according to Spiegel (1961), is a statement about the value of a population parameter that may or may not be true. They are general statements about the population’s probability distribution.

The term “null hypothesis” refers to the assumption that there is “no significant difference between the value of the universe parameter being tested and the value of the statistic computed from a sample drawn from that universe,” according to Clark and Schkade (1969). In other words, the null hypothesis assumes that the difference between the parameter specified in the hypothesis and the statistic is due to sampling error. Ho is commonly used to represent it.

If statistical testing results in the rejection of the null hypothesis, the alternative hypothesis will be accepted (Clark and Schkade, 1969). It is denoted by the letters H1 or HA.

Variable: According to Ibrahim (2009), a variable is any quantity or attribute whose value varies from one unit of investigation to the next. A random variable is one that takes on different numerical values as a result of chance. A random variable is best defined as a number (Standing for some event) that has not yet been observed but will be chosen by chance.

One-tailed Population Variance Test: According to Frank and Althoen (1994), the tail of a test is easily determined by the statement of alternative hypothesis. The alternative hypothesis could be directional, and if it is, there are two possible test situations: or >. The hypothesis for the right tail test is as follows: Ho: s2 = so2 vs H1: s2 > so2.

The hypothesis for the left tail test is as follows:

H1: s2 s2o Ho: s2 = s20

Two-tailed population variance test: The alternative hypothesis, according to Frank and Althoen (1994), also determines this tail. In this case, the alternative hypothesis is non-directional, and there is only one possible test situation; such hypothesis is given as:

H1: s 2 1 s2o vs Ho: s2 = s2o

Eerors of Type I and II: Spiegel (1961) asserts that if we reject a hypothesis when it If it is accepted, we say that a type I error has occurred. If, on the other hand, we accept a hypothesis when it should be rejected, we have committed a type II error. In either case, a bad decision or a lapse in judgment occurred.

Level of Significance: According to Bluman (1992), this is the likelihood of making a type I error. That is, the likelihood of rejecting a true null hypothesis rather than accepting it. It is commonly denoted by or 100% level.

Dass (1988) defines degree of freedom as the number of independent constraints “in a set of data”.

Acceptance and Rejection Regions: According to Devore (2000), the rejection region is the region where

The calculated statistics will show that the null hypothesis is rejected. It is represented by the diagram below in the case of X2 – distribution:

Y

Rejection zone

0 c2n-1 µ c2

While the acceptance region is the range of calculated statistics within which the null hypothesis will be accepted. The diagram below also represents it:

Y

Acceptance range 0 c2n-1 c2

According to Ibrahim (2009), this is very similar to the frequency distribution, which shows how many times a given value occurs in a range of values. As a result, probability distribution is defined as a listing of all the results of an experiment and the probability associated with each result.

The moment generating function was first defined by Guttman (1980). Mx(t) of random variable x by – etxF(x) dx, if X is continuous Mx(t) = E(etx) = etxp(x), if X is discrete x provided the integral and series converge.

**1.6 Limitation of the Study**

This project is limited to UME and Post-UME data from the Faculty of Physical and Life Sciences at the University of Benin. Two sessions were considered: 2008/2009 and 2009/2010.

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