FUNDAMENTAL CONCEPTS IN INFERENTIAL STATISTICS

Inferential statistics is the nucleus around which quantitative empirical research revolves (Kpolovie, 2011). There are specific definitions for terms that are crucial to statistical inference in this chapter. Such fundamental concepts that underlie research are examined here so that the application of different inferential statistical techniques will better be understood and learned with the greatest ease. Examples of the concepts include inferential statistics, population and sample, hypothesis, central limit theorem, normal curve, the level of significance, decision errors and two-tailed cum one-tailed tests.

INFERENTIAL STATISTICS

Inferential statistics is the science of using measurements of a sample to make dependable generalizations about important aspects of the measurement of a population. While a measure based on the observation of sample characteristics is called statistic and is usually symbolized with Roman alphabet like X with a bar (X) and S; a corresponding measure on the basis of a population characteristics is termed parameter and is usually symbolized with Greek alphabet such as ^ and a for mean and standard deviation, respectively (Kpolovie, 2014). Inferential statistics is employed for drawing of reasonable conclusions about the large group, called population by collecting and studying a portion of the population, called sample (Kpolovie, 2016). Unlike descriptive statistics that is concerned with collecting and describing data on only the observed individuals or events, inferential statistics is used for drawing of tentatively acceptable conclusions deductively about an entire population that may be infinite on the basis of data collected from a representative sample of that population. In other words, statistical inference refers to the conclusions drawn about a population, based on the information in a representative sample from the population. Thus, representative sample is a sine qua non criterion of inferential statistics because the accuracy of statistical inference can only be ensured or guaranteed when the data are drawn randomly from the population under investigation in such a meticulous way that every segment of the population is adequately and proportionally represented without bias in the sample (Cohen, 2008; Howell, 2002; McQueen and Knussen, 2006; Keller and Warrack, 2003; Aczel, 1999; Meltzoff, 2007; Crowl, 1996; Best, 1981; Blaxer, 2009; Kvanli, Pavur and Guynes, 2000; Bowerman, O'Connell and Hand, 2001; Breakwell, Hammond and Fife-Schaw, 2001 and Frank and Althoen, 1994).

The use of inferential statistics is of absolute importance in the social or behavioural sciences that deal with human beings who are composed of very complex interaction of emotions, motivations (intrinsic and extrinsic), learned behaviours and genetically determined actions to the extent that one cannot even be sure that the same individual will always behave in the same way (Kpolovie, 2016b). Under such circumstance, the exact situation about an entire population with respect to a variable of interest cannot be known ordinarily without quantitative, empirical and inferential investigations. It is even more so with the fact that humans are characterized by a great deal of imperfectly remembered experiences. It is only with research that adopts inferential statistics that data-based credible information about the actual situation of any variable in the whole population can be validly known. With inferential statistics, a manageable sample that is truly representative of the population is thoroughly studied with regard to a given variable; and whether or not a significant difference or relationship exists in both the sample and population is mathematically determined. The probability or likelihood that conclusions about the requisite characteristics of the sample are validly applicable to the population is only determinable with inferential statistics. In other words, inferential statistics employs inferential tests for determination of the level of confidence that can be placed in the results of the sample data, if the study had covered the population. In most cases, such statistical tests use a probability of or less than 0.05 (5 out of 100) or 0.01 (1 out of 100) to show that the apparent difference or relationship in the sample data is due merely to chance alone or purely to the manipulation or condition of the independent variable (Furr & Bacharach, 2014; Groth-Marnat & Wright, 2016; Cohen, Swerdlik & Sturman, 2013).

Inferential statistics can be satisfactorily defined as the entire body of quantitative techniques that enable the researcher to make appropriate generalizations from limited observations of a randomly drawn sample to the totality of the particular population from which the sample was taken. Research entails far more than collection, organization and quantitative description of data as presented in Chapter 2 to center on explanation, control, prediction and generalization (Kpolovie, 2011b; 2012). Research is more concerned with or places much greater emphasis on statistical inference than mathematical description because it is only with the statistical inference that conclusions, far-reaching interpretations, explanations, predictions, generalizations and control of phenomena for the satisfaction of human needs are done.

All the inferences about the population are from a representative sample, drawn randomly. But in spite of the randomization, a sample does not reflect characteristics of the parent population completely or with absolute fidelity because the characteristics of another similarly drawn sample from the population might not be perfectly the same with those of the first. Since the sample from which inferences are made about the population is usually not a perfect carbon copy of the population, the statistical techniques that are appropriate for data description are insufficient for the crucial task of making inferences about the population. Therefore, the leap of drawing inferences from sample to its population crosses the chasm from what is observed to what is unobserved. The chasm is bridged by unobservable probability distributions and application of probability theory as well as higher mathematical models or methods, known as inferential statistics (Goldstein, 2011; Gay, 2010; Snijders & Bosker, 2012).

There are two types of statistical operations to which research data can be subjected for making appropriate inferences about the population from the sample. They are parametric statistics and nonparametric statistics. It must be reiterated emphatically that of these two, the parametric statistics is incomparably more powerful, sensitive, appropriate, accurate and suitably desirable classically. A more powerful statistical test is that which can detect a small but real difference or relationship in the sample while simultaneously still being able to reject non-real difference or relationship that might be apparent. Parametric statistics are the most widely or commonly used and known methods of statistical inference employed in educational, behavioural or social science research. Another major reason for the superiority of parametric statistics is that the population parameters which are inferred from the sample characteristics are based on a type of distribution that only the assumptions of parametric statistics adequately meet. Such assumptions are:

(a) The observations have to be exclusively independent. That is, selection of one case is neither influenced by nor determines the chances for any of the other cases to be included in the sample.

(b) The observations should be drawn from normally distributed populations with respect to the variables under investigation.

(c) The two or more samples should have homoscedasticity. Homoscedasticity demands that the different samples should be having about equal variance.

(d) The data must be collected at the interval and ratio scales of measurement to accommodate arithmetic operations of addition, division, subtraction, and multiplication that they will be subjected to.

Examples of parametric statistical tests include t-tests, z-tests, analysis of variance (ANOVA), analysis of covariance (ANCOVA), multivariate analysis of variance (MANOVA), product moment correlation, regression and multiple regression.

Nonparametric statistics, on the other hand, use data that are collected at lower levels of measurement (nominal and ordinal scales). They have lower efficiency and do not require homogeneity of variance as well as normally distributed population. In short, nonparametric statistical tests are distribution-free statistical procedures and are less powerful. Also, they do not require the stringent conditions for parametric tests. Examples of nonparametric tests include Mann-Whitney, Wilcoxon test, Kruskal-Wallis test, sign test, median test, rank order correlation, Friedman test, Hodges-Lehmann test for two samples and chi-square (%).

POPULATION

The population of an investigation can be defined as the entire set of all possible measurements, events, characteristics or elements that pertain to an existing group of people, objects or units that are of interest to the researcher, and to which the investigator would like to generalize the findings of the study. Any distinct or identifiable total set of elements of interest that is being investigated by the researcher, and to which results of the survey can be validly generalized, is a population.

The population can be a relatively small set in number, all of which can easily be collected; a large set but finite in number, the entirety of which would be impractical to collect; or still a set of infinite events like the distance in kilometers covered by all the vehicles in a continent, annually. Irrespective of the size, a population must have certain distinct qualities or characteristics that make it unique or different from every other population. The features that distinguish a population from all others must be made very clear for easy identification. For instance, a population can be:

a. All graduates of 2009 from Department of Psychology in University of Port Harcourt (Uniport)

b. All who graduated in 2005 to 2009 from Federal universities in Nigeria

c. All university graduates in Nigeria.

The number of persons that constitute the first example is relatively small, and data on the variables of interest can be collected from all of the members. The features that distinguish them from other people are that each of them must have graduated in 2009 from Department of Psychology in Uniport. Graduates from that department in other years, graduates from Psychology departments in other universities in 2009 and graduates from any other department in Uniport during the period under consideration, 2009; do not and cannot be part of the population.

In the second example, the population is large but finite in number, as the number of Federal Universities in Nigeria is known and the number of individuals who graduated from each of them in 2005, 2006, 2007, 2008 and 2009 can be obtained from the institutions and added. But it is not practically possible to directly collect data on the variables under investigation from all of the persons that constitute this population. The unique characteristics of this population are that each member must have graduated within 2005 and 2009 from a Federal university in Nigeria, and definitely not from any of the State or privately owned universities. Also, all those who graduated from the Federal universities before and after 2005 to 2009 can never be part of the population. Nigerians who graduated within 2005 and 2009 from Federal universities of other countries cannot be part of the population. Since the population is large, all of the members cannot be reached for purposes of data collection on the variables under investigation. Only a sample of the population can be studied directly and the results can be generalized validly to graduates from Nigerian Federal universities within the period under consideration.

The population in the last example is infinite as it incorporates all university graduates in Nigeria without any frame of time. All those who graduated from schools in other parts of the world and are in Nigeria form part of this population. All individuals who graduated in the last one hour up to people who graduated from university 100 or more years ago are part and parcel of this population. But again, there is a characteristic that makes members of the population different from every other person. All individuals in Nigeria who have not graduated from a university cannot be part of the population. Data on the variables of interest can only be obtained from a sample that is carefully drawn from the population to adequately represent all its facets that can be studied and the results generalized to the population, and not beyond that population.

The population for every research should be clearly defined realistically regarding what the researcher can actually access and not ideally on the basis of the vague target. The smaller a population and the more precisely defined the population, the greater the accuracy of generalization of findings from a representative sample to the population. In most cases, when the population of a research is not too large and is realistically defined money, effort and time are saved on the part of the researcher and simultaneously increases the validity of his/her conclusions and generalizations from the sample studied to the population. Therefore, the unique characteristics of both the population and sample of every research should be vividly specified in sufficient details to allow for easy replicability of the work by subsequent researchers.

SAMPLE

Inferential statistics entails a process of moving from the part to the whole, making observations of the sample and deductively drawing conclusions or generalizations from it to the entire population. It is in most cases not possible to observe the whole population in research. Hence knowledge of the whole comes from inferences or conclusions drawn validly from the study of its representative sample.

The sample is a portion of the research population that is meticulously drawn to adequately and proportionally represent all characteristics of the population, and from which data on the variables under investigation are collected and analyzed for making of valid inferences to the population. A portion drawn from the population must be truly representative of the parent population by proportionally covering all the facets of the population before it can be considered and used as a sample of any investigation. Only results from the observation of a representative sample can be rightly generalized to the population from which it was selected randomly (Kpolovie, Ewansiha & Esara, 2016).

It is practically impossible in most circumstances for a researcher to directly study every member of the population. The use of sample becomes crucially indispensable in all such situations. Actually, it is not even necessary to observe all the possible cases for a phenomenon to be understood. One does not have to eat all the fried peanuts in a basin to ascertain whether they were very well and tastefully cooked. Sampling allows for detailed observation of a portion of the population about the phenomenon of study and the conclusions or results arrived at are mathematically generalized to the population with adequate degree of certainty through proper inferential statistical tests.

Qualities of a good sample

There are three qualities that a good sample must possess, namely, representativeness, accuracy and precision. Emphatically, drawing of valid inferences from the sample to the parent population demands that the sample be obtained carefully and randomly for it to be a true representative cross section of individuals, events, elements, objects or characteristics in the population. McQueen and Knussen (2006) have therefore averred that if the population under investigation contains both males and females of varying ages, the sample must also be made up of men and women of varying ages proportionally. The sample should necessarily be composed of the same cross section of persons or elements that make up the population in proportional distribution in terms of relevant characteristics such as age, gender, educational level, occupational status, cultural background and marital status. The sample has to be a true representative of the population from which it was drawn to guarantee equivalence of the characteristics of the persons that constitute them. It is on this basis that findings of the sample can correctly be generalized validly to cover the population. It is however worthy of note that in many areas of cognitive psychology such as testing of visual acuity, or response to audio signals, or memory; and in some medical research, it might be safe to assume that some measures will not vary as a result of gender and personality, or regional location and culture (Waltz, Strickland & Lenz 2016). This accounts for why the Paracetamol, for instance, for males and females do not differ in their chemical composition.