Manual Modes of Parametric Statistical Inference

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Clear thinking is always more important than the ability to do arithmetic. Java, once an esoteric programming language for animating Web pages, is now a full-fledged platform for building JavaScript E-labs' learning objects with useful applications. As you used to do experiments in physics labs to learn physics, computer-assisted learning enables you to use any online interactive tool available on the Internet to perform experiments.

The purpose is the same; i. The appearance of computer software, JavaScript, Statistical Demonstration Applets, and Online Computation are the most important events in the process of teaching and learning concepts in model-based, statistical decision making courses. These e-lab Technologies allow you to construct numerical examples to understand the concepts, and to find their significance for yourself.

Unfortunately, most classroom courses are not learning systems. The way the instructors attempt to help their students acquire skills and knowledge has absolutely nothing to do with the way students actually learn. Many instructors rely on lectures and tests, and memorization. All too often, they rely on"telling. Certainly, we learn by doing, failing, and practicing until we do it right. The computer assisted learning serves this purpose. A course in appreciation of statistical thinking gives business professionals an edge.

Professionals with strong quantitative skills are in demand. This phenomenon will grow as the impetus for data-based decisions strengthens and the amount and availability of data increases. The statistical toolkit can be developed and enhanced at all stages of a career. Decision making process under uncertainty is largely based on application of statistics for probability assessment of uncontrollable events or factors , as well as risk assessment of your decision. For more statistical-based Web sites with decision making applications, visit Decision Science Resources , and Modeling and Simulation Resources sites.

The main objective for this course is to learn statistical thinking; to emphasize more on concepts, and less theory and fewer recipes, and finally to foster active learning using the useful and interesting Web-sites. It is already a known fact that"Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write. Further Readings: Chernoff H. Churchman C.

Early in the book he stated that knowledge could be considered as a collection of information, or as an activity, or as a potential. He also noted that knowledge resides in the user and not in the collection. Rustagi M. The Birth of Probability and Statistics The original idea of"statistics" was the collection of information about and for the"state". The word statistics derives directly, not from any classical Greek or Latin roots, but from the Italian word for state.

The birth of statistics occurred in mid th century. A commoner, named John Graunt, who was a native of London, began reviewing a weekly church publication issued by the local parish clerk that listed the number of births, christenings, and deaths in each parish. These so called Bills of Mortality also listed the causes of death. Graunt who was a shopkeeper organized this data in the form we call descriptive statistics, which was published as Natural and Political Observations Made upon the Bills of Mortality.

Probability and statistics EBook - Socr

Shortly thereafter he was elected as a member of Royal Society. Thus, statistics has to borrow some concepts from sociology, such as the concept of Population. It has been argued that since statistics usually involves the study of human behavior, it cannot claim the precision of the physical sciences. Probability has much longer history. Probability is derived from the verb to probe meaning to"find out" what is not too easily accessible or understandable.

The word"proof" has the same origin that provides necessary details to understand what is claimed to be true. Probability originated from the study of games of chance and gambling during the 16 th century. Probability theory was a branch of mathematics studied by Blaise Pascal and Pierre de Fermat in the seventeenth century. Currently in 21 st century, probabilistic modeling is used to control the flow of traffic through a highway system, a telephone interchange, or a computer processor; find the genetic makeup of individuals or populations; quality control; insurance; investment; and other sectors of business and industry.

New and ever growing diverse fields of human activities are using statistics; however, it seems that this field itself remains obscure to the public. Professor Bradley Efron expressed this fact nicely: During the 20 th Century statistical thinking and methodology have become the scientific framework for literally dozens of fields including education, agriculture, economics, biology, and medicine, and with increasing influence recently on the hard sciences such as astronomy, geology, and physics.

In other words, we have grown from a small obscure field into a big obscure field. Further Readings: Daston L. The book points out that early Enlightenment thinkers could not face uncertainty. A mechanistic, deterministic machine, was the Enlightenment view of the world. David H. Offers a general historical collections of the probability and statistical literature. Gillies D. Covers the classical, logical, subjective, frequency, and propensity views. Hacking I. A philosophical study of early ideas about probability, induction and statistical inference. Hald A. Peters W. It teaches the principles of applied economic and social statistics in a historical context.

Featured topics include public opinion polls, industrial quality control, factor analysis, Bayesian methods, program evaluation, non-parametric and robust methods, and exploratory data analysis. Porter T. The author states that statistics has become known in the twentieth century as the mathematical tool for analyzing experimental and observational data. Enshrined by public policy as the only reliable basis for judgments as the efficacy of medical procedures or the safety of chemicals, and adopted by business for such uses as industrial quality control, it is evidently among the products of science whose influence on public and private life has been most pervasive.

Statistical analysis has also come to be seen in many scientific disciplines as indispensable for drawing reliable conclusions from empirical i. This new field of mathematics found so extensive a domain of applications. Stigler S. It covers the people, ideas, and events underlying the birth and development of early statistics.

Tankard J. This work provides the detailed lives and times of theorists whose work continues to shape much of the modern statistics. Statistical Modeling for Decision-Making under Uncertainties: From Data to the Instrumental Knowledge In this diverse world of ours, no two things are exactly the same. A statistician is interested in both the differences and the similarities ; i. The actuarial tables published by insurance companies reflect their statistical analysis of the average life expectancy of men and women at any given age.

From these numbers, the insurance companies then calculate the appropriate premiums for a particular individual to purchase a given amount of insurance. Exploratory analysis of data makes use of numerical and graphical techniques to study patterns and departures from patterns. The widely used descriptive statistical techniques are: Frequency Distribution ; Histograms; Boxplot; Scattergrams and Error Bar plots; and diagnostic plots. In examining distribution of data, you should be able to detect important characteristics, such as shape, location, variability, and unusual values.

From careful observations of patterns in data, you can generate conjectures about relationships among variables. The notion of how one variable may be associated with another permeates almost all of statistics, from simple comparisons of proportions through linear regression. The difference between association and causation must accompany this conceptual development.

Data must be collected according to a well-developed plan if valid information on a conjecture is to be obtained. The plan must identify important variables related to the conjecture, and specify how they are to be measured. From the data collection plan, a statistical model can be formulated from which inferences can be drawn. As an example of statistical modeling with managerial implications , such as "what-if" analysis , consider regression analysis. Regression analysis is a powerful technique for studying relationship between dependent variables i.

Summarizing relationships among the variables by the most appropriate equation i. This is an important and common statistical decision, which should be given due consideration, since an inadequate sample size invariably leads to wasted resources. The sample size determination section provides a practical solution to this risky decision.

Statistical models are currently used in various fields of business and science. However, the terminology differs from field to field. For example, the fitting of models to data, called calibration, history matching, and data assimilation, are all synonymous with parameter estimation. Your organization database contains a wealth of information, yet the decision technology group members tap a fraction of it. Employees waste time scouring multiple sources for a database. The decision-makers are frustrated because they cannot get business-critical data exactly when they need it.

Therefore, too many decisions are based on guesswork, not facts. Many opportunities are also missed, if they are even noticed at all. Knowledge is what we know well. Information is the communication of knowledge. In every knowledge exchange, there is a sender and a receiver. The sender make common what is private, does the informing, the communicating. Information can be classified as explicit and tacit forms. The explicit information can be explained in structured form, while tacit information is inconsistent and fuzzy to explain. Know that data are only crude information and not knowledge by themselves.

Data is known to be crude information and not knowledge by itself. The sequence from data to knowledge is: from Data to Information, from Information to Facts, and finally, from Facts to Knowledge. Data becomes information, when it becomes relevant to your decision problem. Information becomes fact, when the data can support it. Facts are what the data reveals. However the decisive instrumental i. Fact becomes knowledge, when it is used in the successful completion of a decision process. Once you have a massive amount of facts integrated as knowledge, then your mind will be superhuman in the same sense that mankind with writing is superhuman compared to mankind before writing.

The following figure illustrates the statistical thinking process based on data in constructing statistical models for decision making under uncertainties. Click on the image to enlarge it and THEN print it. The Path from Statistical Data to Managerial Knowledge The above figure depicts the fact that as the exactness of a statistical model increases, the level of improvements in decision-making increases. That's why we need Business Statistics. Statistics arose from the need to place knowledge on a systematic evidence base. This required a study of the rules of computational probability, the development of measures of data properties and relationships, and so on.

Statistical inference aims at determining whether any statistical significance can be attached that results after due allowance is made for any random variation as a source of error. Intelligent and critical inferences cannot be made by those who do not understand the purpose, the conditions, and applicability of the various techniques for judging significance. Considering the uncertain environment, the chance that"good decisions" are made increases with the availability of"good information.

The above figure also illustrates the fact that as the exactness of a statistical model increases, the level of improvements in decision-making increases. Knowledge is more than knowing something technical. Knowledge needs wisdom. Wisdom is the power to put our time and our knowledge to the proper use. Wisdom comes with age and experience. Wisdom is the accurate application of accurate knowledge and its key component is to knowing the limits of your knowledge.

Wisdom is about knowing how something technical can be best used to meet the needs of the decision-maker. Wisdom, for example, creates statistical software that is useful, rather than technically brilliant. For example, ever since the Web entered the popular consciousness, observers have noted that it puts information at your fingertips but tends to keep wisdom out of reach. The notion of "wisdom" in the sense of practical wisdom has entered Western civilization through biblical texts.

In the Hellenic experience this kind of wisdom received a more structural character in the form of philosophy.

A Parametric Approach to Nonparametric Statistics

In this sense philosophy also reflects one of the expressions of traditional wisdom. Business professionals need a statistical toolkit. Statistical skills enable you to intelligently collect, analyze and interpret data relevant to their decision-making. Statistical concepts enable us to solve problems in a diversity of contexts. Statistical thinking enables you to add substance to your decisions. That's why we need statistical data analysis in probabilistic modeling. Statistics arose from the need to place knowledge management on a systematic evidence base.

This required a study of the rules of computational probability, the development of measures of data properties, relationships, and so on. The purpose of statistical thinking is to get acquainted with the statistical techniques, to be able to execute procedures using available JavaScript, and to be conscious of the conditions and limitations of various techniques. Statistical Decision-Making Process Unlike the deterministic decision-making process, such as linear optimization by solving systems of equations , Parametric systems of equations and in decision making under pure uncertainty , the variables are often more numerous and more difficult to measure and control.

However, the steps are the same. They are: Simplification Building a decision model Testing the model Using the model to find the solution: It is a simplified representation of the actual situation It need not be complete or exact in all respects It concentrates on the most essential relationships and ignores the less essential ones. It is more easily understood than the empirical i.

It can be used again and again for similar problems or can be modified. Fortunately the probabilistic and statistical methods for analysis and decision making under uncertainty are more numerous and powerful today than ever before. The computer makes possible many practical applications. A few examples of business applications are the following: An auditor can use random sampling techniques to audit the accounts receivable for clients.

A plant manager can use statistical quality control techniques to assure the quality of his production with a minimum of testing or inspection. A financial analyst may use regression and correlation to help understand the relationship of a financial ratio to a set of other variables in business. A market researcher may use test of significace to accept or reject the hypotheses about a group of buyers to which the firm wishes to sell a particular product. A sales manager may use statistical techniques to forecast sales for the coming year.

Further Readings: Corfield D. Lapin L. Pratt J. Raiffa, and R. What is Business Statistics? The main objective of Business Statistics is to make inferences e. The condition for randomness is essential to make sure the sample is representative of the population. It provides knowledge and skills to interpret and use statistical techniques in a variety of business applications. A typical Business Statistics course is intended for business majors, and covers statistical study, descriptive statistics collection, description, analysis, and summary of data , probability, and the binomial and normal distributions, test of hypotheses and confidence intervals, linear regression, and correlation.

Statistics is a science of making decisions with respect to the characteristics of a group of persons or objects on the basis of numerical information obtained from a randomly selected sample of the group. Statisticians refer to this numerical observation as realization of a random sample. However, notice that one cannot see a random sample. A random sample is only a sample of a finite outcomes of a random process. At the planning stage of a statistical investigation, the question of sample size n is critical. Clearly, a larger sample provides more relevant information, and as a result a more accurate estimation and better statistical judgement regarding test of hypotheses.

Under-lit Streets and the Crimes Rate: It is a fact that if residential city streets are under-lit then major crimes take place therein. Activities Associated with the General Statistical Thinking and Its Applications The above figure illustrates the idea of statistical inference from a random sample about the population. The major task of Statistics is the scientific methodology for collecting, analyzing, interpreting a random sample in order to draw inference about some particular characteristic of a specific Homogenous Population. For two major reasons, it is often impossible to study an entire population: The process would be too expensive or too time-consuming.

The process would be destructive. In either case, we would resort to looking at a sample chosen from the population and trying to infer information about the entire population by only examining the smaller sample.

Very often the numbers, which interest us most about the population, are the mean m and standard deviation s , any number -- like the mean or standard deviation -- which is calculated from an entire population, is called a Parameter. If the very same numbers are derived only from the data of a sample, then the resulting numbers are called Statistics. Frequently, Greek letters represent parameters and Latin letters represent statistics as shown in the above Figure.

The uncertainties in extending and generalizing sampling results to the population are measures and expressed by probabilistic statements called Inferential Statistics. Therefore, probability is used in statistics as a measuring tool and decision criterion for dealing with uncertainties in inferential statistics. An important aspect of statistical inference is estimating population values parameters from samples of data. An estimate of a parameter is unbiased if the expected value of sampling distribution is equal to that population.

The sample mean is an unbiased estimate of the population mean. The sample variance is an unbiased estimate of population variance. This allows us to combine several estimates to obtain a much better estimate. The Empirical distribution is the distribution of a random sample, shown by a step-function in the above figure.

The empirical distribution function is an unbiased estimate for the population distribution function F x. Given you already have a realization set of a random sample, to compute the descriptive statistics including those in the above figure, you may like using Descriptive Statistics JavaScript. To reduce this uncertainty and having high confidence that statistical inferences are correct, a sample must give equal chance to each member of population to be selected which can be achieved by sampling randomly and relatively large sample size n. Given you already have a realization set of a random sample, to perform hypothesis testing for mean m and variance s 2 , you may like using Testing the Mean and Testing the Variance JavaScript, respectively.

Statistics is a tool that enables us to impose order on the disorganized cacophony of the real world of modern society. The business world has grown both in size and competition. Corporate executive must take risk in business , hence the need for business statistics. Business statistics has grown with the art of constructing charts and tables! It is a science of basing decisions on numerical data in the face of uncertainty.

Business statistics is a scientific approach to decision making under risk. In practicing business statistics, we search for an insight, not the solution. Our search is for the one solution that meets all the business's needs with the lowest level of risk. Business statistics can take a normal business situation, and with the proper data gathering, analysis, and re-search for a solution, turn it into an opportunity.

While business statistics cannot replace the knowledge and experience of the decision maker, it is a valuable tool that the manager can employ to assist in the decision making process in order to reduce the inherent risk, measured by, e. Among other useful questions, you may ask why we are interested in estimating the population's expected value m and its Standard Deviation s?

Here are some applicable reasons. That is, what is a good estimate for m? That is, what is a good estimate for s? That is, comparing several m 's, and several s 's. Common Statistical Terminology with Applications Like all profession, also statisticians have their own keywords and phrases to ease a precise communication. However, one must interpret the results of any decision making in a language that is easy for the decision-maker to understand. This lack of communication between statisticians and the managers is the major roadblock for using statistics. Population: A population is any entire collection of people, animals, plants or things on which we may collect data.

It is the entire group of interest, which we wish to describe or about which we wish to draw conclusions. In the above figure the life of the light bulbs manufactured say by GE, is the concerned population. Qualitative and Quantitative Variables: Any object or event, which can vary in successive observations either in quantity or quality is called a"variable. A qualitative variable, unlike a quantitative variable does not vary in magnitude in successive observations. The values of quantitative and qualitative variables are called"Variates" and"Attributes", respectively.

Variable: A characteristic or phenomenon, which may take different values, such as weight, gender since they are different from individual to individual. Randomness: Randomness means unpredictability. The fascinating fact about inferential statistics is that, although each random observation may not be predictable when taken alone, collectively they follow a predictable pattern called its distribution function. For example, it is a fact that the distribution of a sample average follows a normal distribution for sample size over In other words, an extreme value of the sample mean is less likely than an extreme value of a few raw data.

Sample: A subset of a population or universe. An Experiment: An experiment is a process whose outcome is not known in advance with certainty. Statistical Experiment: An experiment in general is an operation in which one chooses the values of some variables and measures the values of other variables, as in physics. A statistical experiment, in contrast is an operation in which one take a random sample from a population and infers the values of some variables.

For example, in a survey, we"survey" i. A random sample from the relevant population provides information about the voting intentions. In order to make any generalization about a population, a random sample from the entire population; that is meant to be representative of the population, is often studied. For each population, there are many possible samples.

A sample statistic gives information about a corresponding population parameter. For example, the sample mean for a set of data would give information about the overall population mean m. It is important that the investigator carefully and completely defines the population before collecting the sample, including a description of the members to be included.

Example: The population for a study of infant health might be all children born in the U. The sample might be all babies born on 7 th of May in any of the years. An experiment is any process or study which results in the collection of data, the outcome of which is unknown. In statistics, the term is usually restricted to situations in which the researcher has control over some of the conditions under which the experiment takes place. Example: Before introducing a new drug treatment to reduce high blood pressure, the manufacturer carries out an experiment to compare the effectiveness of the new drug with that of one currently prescribed.

Newly diagnosed subjects are recruited from a group of local general practices. Half of them are chosen at random to receive the new drug, the remainder receives the present one. So, the researcher has control over the subjects recruited and the way in which they are allocated to treatment. Design of experiments is a key tool for increasing the rate of acquiring new knowledge. Knowledge in turn can be used to gain competitive advantage, shorten the product development cycle, and produce new products and processes which will meet and exceed your customer's expectations. Primary data and Secondary data sets: If the data are from a planned experiment relevant to the objective s of the statistical investigation, collected by the analyst, it is called a Primary Data set.

However, if some condensed records are given to the analyst, it is called a Secondary Data set. Random Variable: A random variable is a real function yes, it is called" variable", but in reality it is a function that assigns a numerical value to each simple event. You may assign any other two distinct real numbers, as you wish; however, non-negative integer random variables are easy to work with.

Random variables are needed since one cannot do arithmetic operations on words; the random variable enables us to compute statistics, such as average and variance. Any random variable has a distribution of probabilities associated with it. Probability: Probability i. Random phenomena are not haphazard: they display an order that emerges only in the long run and is described by a distribution.

The mathematical description of variation is central to statistics. The probability required for statistical inference is not primarily axiomatic or combinatorial, but is oriented toward describing data distributions. Sampling Unit: A unit is a person, animal, plant or thing which is actually studied by a researcher; the basic objects upon which the study or experiment is executed.

For example, a person; a sample of soil; a pot of seedlings; a zip code area; a doctor's practice. Parameter: A parameter is an unknown value, and therefore it has to be estimated. Parameters are used to represent a certain population characteristic. For example, the population mean m is a parameter that is often used to indicate the average value of a quantity. Within a population, a parameter is a fixed value that does not vary. Each sample drawn from the population has its own value of any statistic that is used to estimate this parameter. For example, the mean of the data in a sample is used to give information about the overall mean m in the population from which that sample was drawn.

Statistic: A statistic is a quantity that is calculated from a sample of data. It is used to give information about unknown values in the corresponding population. For example, the average of the data in a sample is used to give information about the overall average in the population from which that sample was drawn. A statistic is a function of an observable random sample. It is therefore an observable random variable. Notice that, while a statistic is a"function" of observations, unfortunately, it is commonly called a random"variable" not a function.

It is possible to draw more than one sample from the same population, and the value of a statistic will in general vary from sample to sample. For example, the average value in a sample is a statistic. The average values in more than one sample, drawn from the same population, will not necessarily be equal. Statistics are often assigned Roman letters e. The word estimate means to esteem, that is giving a value to something.

A statistical estimate is an indication of the value of an unknown quantity based on observed data. More formally, an estimate is the particular value of an estimator that is obtained from a particular sample of data and used to indicate the value of a parameter. Example: Suppose the manager of a shop wanted to know m , the mean expenditure of customers in her shop in the last year. She could calculate the average expenditure of the hundreds or perhaps thousands of customers who bought goods in her shop; that is, the population mean m.

Instead she could use an estimate of this population mean m by calculating the mean of a representative sample of customers. There are two broad subdivisions of statistics: Descriptive Statistics and Inferential Statistics as described below. Descriptive Statistics: The numerical statistical data should be presented clearly, concisely, and in such a way that the decision maker can quickly obtain the essential characteristics of the data in order to incorporate them into decision process.

The principal descriptive quantity derived from sample data is the mean , which is the arithmetic average of the sample data. It serves as the most reliable single measure of the value of a typical member of the sample. If the sample contains a few values that are so large or so small that they have an exaggerated effect on the value of the mean, the sample is more accurately represented by the median -- the value where half the sample values fall below and half above.

The quantities most commonly used to measure the dispersion of the values about their mean are the variance s 2 and its square root, the standard deviation s. The variance is calculated by determining the mean, subtracting it from each of the sample values yielding the deviation of the samples , and then averaging the squares of these deviations. The mean and standard deviation of the sample are used as estimates of the corresponding characteristics of the entire group from which the sample was drawn. They do not , in general, completely describe the distribution F x of values within either the sample or the parent group; indeed, different distributions may have the same mean and standard deviation.

They do, however, provide a complete description of the normal distribution, in which positive and negative deviations from the mean are equally common, and small deviations are much more common than large ones. For a normally distributed set of values, a graph showing the dependence of the frequency of the deviations upon their magnitudes is a bell-shaped curve. About 68 percent of the values will differ from the mean by less than the standard deviation, and almost percent will differ by less than three times the standard deviation. Inferential Statistics: Inferential statistics is concerned with making inferences from samples about the populations from which they have been drawn.

In other words, if we find a difference between two samples, we would like to know, is this a"real" difference i. That's what tests of statistical significance are all about. Any inferred conclusion from a sample data to the population from which the sample is drawn must be expressed in a probabilistic term.

Probability is the language and a measuring tool for uncertainty in our statistical conclusions. Inferential statistics could be used for explaining a phenomenon or checking for validity of a claim. In these instances, inferential statistics is called Exploratory Data Analysis or Confirmatory Data Analysis , respectively. Statistical Inference: Statistical inference refers to extending your knowledge obtained from a random sample from the entire population to the whole population.

This is known in mathematics as Inductive Reasoning , that is, knowledge of the whole from a particular. Its main application is in hypotheses testing about a given population. Statistical inference guides the selection of appropriate statistical models. Models and data interact in statistical work. Inference from data can be thought of as the process of selecting a reasonable model, including a statement in probability language of how confident one can be about the selection.

Normal Distribution Condition: The normal or Gaussian distribution is a continuous symmetric distribution that follows the familiar bell-shaped curve. One of its nice features is that, the mean and variance uniquely and independently determines the distribution. It has been noted empirically that many measurement variables have distributions that are at least approximately normal. Even when a distribution is non-normal, the distribution of the mean of many independent observations from the same distribution becomes arbitrarily close to a normal distribution, as the number of observations grows large.

Many frequently used statistical tests make the condition that the data come from a normal distribution. Estimation and Hypothesis Testing: Inference in statistics are of two types. The first is estimation , which involves the determination, with a possible error due to sampling, of the unknown value of a population characteristic, such as the proportion having a specific attribute or the average value m of some numerical measurement.

To express the accuracy of the estimates of population characteristics, one must also compute the standard errors of the estimates. The second type of inference is hypothesis testing. It involves the definitions of a hypothesis as one set of possible population values and an alternative, a different set. There are many statistical procedures for determining, on the basis of a sample, whether the true population characteristic belongs to the set of values in the hypothesis or the alternative.

Statistical inference is grounded in probability, idealized concepts of the group under study, called the population, and the sample. The statistician may view the population as a set of balls from which the sample is selected at random, that is, in such a way that each ball has the same chance as every other one for inclusion in the sample.

Notice that to be able to estimate the population parameters , the sample size n must be greater than one. Greek Letters Commonly Used as Statistical Notations We use Greek letters as scientific notations in statistics and other scientific fields to honor the ancient Greek philosophers who invented science and scientific thinking. Before Socrates, in 6 th Century BC, Thales and Pythagoras, amomg others, applied geometrical concepts to arithmetic, and Socrates is the inventor of dialectic reasoning.

The revival of scientific thinking initiated by Newton's work was valued and hence reappeared almost years later. Greek Letters Commonly Used as Statistical Notations alpha beta ki-sqre delta mu nu pi rho sigma tau theta a b c 2 d m n p r s t q Note: ki-square ki-sqre, Chi-square , c 2 , is not the square of anything, its name implies Chi-square read, ki-square.

Ki does not exist in statistics. I'm glad that you're overcoming all the confusions that exist in learning statistics. Type of Data and Levels of Measurement Information can be collected in statistics using qualitative or quantitative data. Qualitative data , such as eye color of a group of individuals, is not computable by arithmetic relations. They are labels that advise in which category or class an individual, object, or process fall.

They are called categorical variables. Quantitative data sets consist of measures that take numerical values for which descriptions such as means and standard deviations are meaningful. They can be put into an order and further divided into two groups: discrete data or continuous data. Discrete data are countable data and are collected by counting , for example, the number of defective items produced during a day's production. Continuous data are collected by measuring and are expressed on a continuous scale. For example, measuring the height of a person.

A set of data is a representation i. Otherwise, it is called"secondary type" data. Data can be either continuous or discrete. While the unit of measurement is arbitrary on the Ratio scale, its zero point is a natural attribute. The categorical variable is measured on an ordinal or nominal scale. Pareto Chart: A Pareto chart is similar to the histogram, except that it is a frequency bar chart for qualitative variables , rather than being used for quantitative data that have been grouped into classes.

The following is an example of a Pareto chart that shows the types of shoes-frequency, worn in the class on a particular day: Click on the image to enlarge it and THEN print it. Why Statistical Sampling? Sampling is the selection of part of an aggregate or totality known as population , on the basis of which a decision concerning the population is made. Further Reading: Thompson S.

Sampling Methods From the food you eat to the television you watch, from political elections to school board actions, much of your life is regulated by the results of sample surveys. A sample is a group of units selected from a larger group the population. By studying the sample, one hopes to draw valid conclusions about the larger group. A sample is generally selected for study because the population is too large to study in its entirety.

The sample should be representative of the general population. This is often best achieved by random sampling. Also, before collecting the sample, it is important that one carefully and completely defines the population, including a description of the members to be included. A common problem in business statistical decision-making arises when we need information about a collection called a population but find that the cost of obtaining the information is prohibitive. For instance, suppose we need to know the average shelf life of current inventory.

If the inventory is large, the cost of checking records for each item might be high enough to cancel the benefit of having the information. On the other hand, a hunch about the average shelf life might not be good enough for decision-making purposes. This means we must arrive at a compromise that involves selecting a small number of items and calculating an average shelf life as an estimate of the average shelf life of all items in inventory.

This is a compromise, since the measurements for a sample from the inventory will produce only an estimate of the value we want, but at substantial savings. What we would like to know is how"good" the estimate is and how much more will it cost to make it"better". Information of this type is intimately related to sampling techniques. This section provides a short discussion on the common methods of business statistical sampling. Cluster sampling can be used whenever the population is homogeneous but can be partitioned.

In many applications the partitioning is a result of physical distance. For instance, in the insurance industry, there are small"clusters" of employees in field offices scattered about the country. In such a case, a random sampling of employee work habits might not required travel to many of the"clusters" or field offices in order to get the data. Totally sampling each one of a small number of clusters chosen at random can eliminate much of the cost associated with the data requirements of management.

Stratified sampling can be used whenever the population can be partitioned into smaller sub-populations, each of which is homogeneous according to the particular characteristic of interest. Random sampling is probably the most popular sampling method used in decision making today.

Many decisions are made, for instance, by choosing a number out of a hat or a numbered bead from a barrel, and both of these methods are attempts to achieve a random choice from a set of items. But true random sampling must be achieved with the aid of a computer or a random number table whose values are generated by computer random number generators. A random sampling of size n is drawn from a population size N.

Cross-Sectional Sampling: Cross-Sectional study the observation of a defined population at a single point in time or time interval. Exposure and outcome are determined simultaneously. What is a statistical instrument? A statistical instrument is any process that aim at describing a phenomena by using any instrument or device, however the results may be used as a control tool. Examples of statistical instruments are questionnaire and surveys sampling.

What is grab sampling technique? The grab sampling technique is to take a relatively small sample over a very short period of time, the result obtained are usually instantaneous. However, the Passive Sampling is a technique where a sampling device is used for an extended time under similar conditions. Depending on the desirable statistical investigation, the passive sampling may be a useful alternative or even more appropriate than grab sampling. However, a passive sampling technique needs to be developed and tested in the field.

Statistical Summaries Representative of a Sample: Measures of Central Tendency Summaries How do you describe the"average" or"typical" piece of information in a set of data? Different procedures are used to summarize the most representative information depending of the type of question asked and the nature of the data being summarized. Measures of location give information about the location of the central tendency within a group of numbers.

The measures of location presented in this unit for ungrouped raw data are the mean, the median, and the mode. Mean: The arithmetic mean or the average, simple mean is computed by summing all numbers in an array of numbers x i and then dividing by the number of observations n in the array.

The mean uses all of the observations, and each observation affects the mean. Even though the mean is sensitive to extreme values; i. This is due to the fact that the mean has valuable mathematical properties that make it convenient for use with inferential statistical analysis. For example, the sum of the deviations of the numbers in a set of data from the mean is zero, and the sum of the squared deviations of the numbers in a set of data from the mean is the minimum value.

You might like to use Descriptive Statistics to compute the mean. Weighted Mean: In some cases, the data in the sample or population should not be weighted equally, rather each value should be weighted according to its importance. Median: The median is the middle value in an ordered array of observations. If there is an even number of observations in the array, the median is the average of the two middle numbers.

If there is an odd number of data in the array, the median is the middle number. The median is often used to summarize the distribution of an outcome. If the distribution is skewed , the median and the interquartile range IQR may be better than other measures to indicate where the observed data are concentrated.

Generally, the median provides a better measure of location than the mean when there are some extremely large or small observations; i. For this reason, median income is used as the measure of location for the U. Note that if the median is less than the mean, the data set is skewed to the right. If the median is greater than the mean, the data set is skewed to the left.

The mean has two distinct advantages over the median. It is more stable, and one can compute the mean based of two samples by combining the two means. Mode: The mode is the most frequently occurring value in a set of observations. Why use the mode? Data may have two modes. In this case, we say the data are bimodal , and sets of observations with more than two modes are referred to as multimodal. Note that the mode is not a helpful measure of location, because there can be more than one mode or even no mode. Whenever, more than one mode exist, then the population from which the sample came is a mixture of more than one population, as shown, for example in the following bimodal histogram.

A Mixture of Two Different Populations However, notice that a Uniform distribution has uncountable number of modes having equal density value; therefore it is considered as a homogeneous population. Almost all standard statistical analyses are conditioned on the assumption that the population is homogeneous. Notice that Excel has very limited statistical capability. For example, it displays only one mode , the first one. Unfortunately, this is very misleading. However, you may find out if there are others by inspection only, as follow: Create a frequency distribution, invoke the menu sequence: Tools, Data analysis, Frequency and follow instructions on the screen.

You will see the frequency distribution and then find the mode visually. Unfortunately, Excel does not draw a Stem and Leaf diagram. Selecting Among the Mode, Median, and Mean It is a common mistake to specify the wrong index for central tenancy. Selecting Among the Mode, Median, and Mean The first consideration is the type of data, if the variable is categorical, the mode is the single measure that best describes that data.

The second consideration in selecting the index is to ask whether the total of all observations is of any interest. If the answer is yes, then the mean is the proper index of central tendency. If the total is of no interest, then depending on whether the histogram is symmetric or skewed one must use either mean or median, respectively. In all cases the histogram must be unimodal. However, notice that, e. The Mode The Median The Mean 1 It is the most frequent value in the distribution; it is the point of greatest density.

It is the value of the middle point of the array not midpoint of range , such that half the item are above and half below it. It is the value in a given aggregate which would obtain if all the values were equal. The value of the media is fixed by its position in the array and doesn't reflect the individual value. The sum of deviations on either side of the mean are equal; hence, the algebraic sum of the deviation is equal zero.

The aggregate distance between the median point and all the value in the array is less than from any other point. It reflect the magnitude of every value. On the other hand, there is no mode in a rectangular distribution. Each array has one and only one median. An array has one and only one mean. It cannot be manipulated algebraically: medians of subgroups cannot be weighted and combined. Means may be manipulated algebraically: means of subgroups may be combined when properly weighted.

It is stable in that grouping procedures do not affect it appreciably. It may be calculated even when individual values are unknown, provided the sum of the values and the sample size n are known. Value must be ordered, and may be grouped, for computation. Values need not be ordered or grouped for this calculation. It can be compute when ends are open It cannot be calculated from a frequency table when ends are open.

It is not applicable to qualitative data. It is stable in that grouping procedures do not seriously affected it. The Descriptive Statistics JavaScript provides a complete set of information about all statistics that you ever need. You might like to use it to perform some numerical experimentation for validating the above assertions for a deeper understanding. If some values are very large in magnitude and others are small, then the geometric mean is a better representative of the data than the simple average.

In a"geometric series", the most meaningful average is the geometric mean G. The arithmetic mean is very biased toward the larger numbers in the series. For simplicity, assume you sold items initially. The Harmonic Mean: The harmonic mean H is another specialized average, which is useful in averaging variables expressed as rate per unit of time, such as mileage per hour, number of units produced per day. An Application: Suppose 4 machines in a machine shop are used to produce the same part.

However, each of the four machines takes 2. What is the average rate of speed? If all machines working for one hour, how many parts will be produced? The Order Among the Three Means: If all the three means exist, then the Arithmetic Mean is never less than the other two, moreover, the Harmonic Mean is never larger than the other two. You might like to use The Other Means JavaScript in performing some numerical experimentation for validating the above assertions for a deeper understanding. Further Reading: Langley R. Histogramming: Checking for Homogeneity of Population A histogram is a graphical presentation of an estimate for the density for continuous random variables or probability mass function for discrete random variables of the population.

The geometric feature of histogram enables us to find out useful information about the data, such as: The location of the"center" of the data. The degree of dispersion. The extend to which its is skewed, that is, it does not fall off systemically on both side of its peak. The degree of peakedness. How steeply it rises and falls. The mode is the most frequently occurring value in a set of observations.

Whenever, more than one mode exist, then the population from which the sample came is a mixture of more than one population. Almost all standard statistical analyses are conditioned on the assumption that the population is homogeneous, meaning that its density for continuous random variables or probability mass function for discrete random variables is unimodal. To check the unimodality of sampling data, one may use the histogramming process. Number of Class Intervals in a Histogram: Before we can construct our frequency distribution we must determine how many classes we should use.

This is purely arbitrary, but too few classes or too many classes will not provide as clear a picture as can be obtained with some more nearly optimum number. An empirical i. To have an"optimum" you need some measure of quality -- presumably in this case, the"best" way to display whatever information is available in the data. The sample size contributes to this; so the usual guidelines are to use between 5 and 15 classes, with more classes, if you have a larger sample.

You should take into account a preference for tidy class widths, preferably a multiple of 5 or 10, because this makes it easier to understand. Beyond this it becomes a matter of judgement. Try out a range of class widths, and choose the one that works best. This assumes you have a computer and can generate alternative histograms fairly readily. I recommend it to anyone who has an interest in learning something new about statistical inference. Starting from the basics of probability, the authors develop the theory of statistical inference using techniques, definitions, and concepts that are statistical and are natural extensions and consequences of previous concepts.

The ideal reader for this book will be quantitatively literate and has a basic understanding of statistical concepts and R programming. Statistical inference. Statistical inference is the process of drawing conclusions about populations or scientific truths from data. If you wish to contact the author, click here. Department of Psychology, University of Regina. Statistical inference uses a sample from a population to draw conclusions about the entire population. Spring Includes bibliographical references and index. Probability and inference Probability and statistical inference are two sides of the same coin.

Tanis Hope College Dale L. The book gives a rigorous treatment of the elementary concepts in statistical inference from a classical frequentist perspective. How does it differ from the frequentist approach? Ralph Buncher and Jia-Yeong Tsay View statistical inference. Define statistical inference. Prior to 11 An introduction to statistical inference This emphasis is changing rapidly, and is being replaced by a new emphasis on effect size Robert Bartoszynski and Magdalena Niewiadomska-Bugaj : Probability and Statistical Inference.

Module Overview. Vines and Application. Statistical inference is the subject of the second part of the book. At one extreme is a view expressed by Statistical Inference 15 questions 1.


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Statistical inference involves using data from a sample to draw conclusions about a wider population. However This solutions manual provides answers for the even-numbered exercises in Probability and Statistical Inference, 9th edition, by Robert V. Except where otherwise indicated, this work is Business statistics using Excel.

Statistics in the Pharmaceutical Industry, edited by C. Statistical inference is based upon mathematical laws of probability. The set of data that is used to make inferences is called sample. Portions of these videos feature the use of the Data Analysis Toolpak. PDF "Although All of Statistics is an ambitious title, this book is a concise guide, as the subtitle suggests. Cox develops the key concepts of the theory of statistical inference, in particular describing and comparing the main ideas and controversies over foundational issues that have rumbled on for more than years. Themensteller: Prof.

The probability distribution of all possible values of the sample proportion p is the i Probability density function of p ii Sampling distribution of x iii Same as p , since it considers all possible values of the sample proportion i v Sampling distribution of p Designed for students with a background in calculu We present a novel inference framework for convex empirical risk minimization, using approximate stochastic Newton steps.

The proposed algorithm is based on the notion of finite differences and allows the approximation of a Hessian-vector product from first-order information. Installing the Analysis Toolpak Although, the objective of statistical The rst part of the book deals with descriptive statistics and provides prob-ability concepts that are required for the interpretation of statistical inference.

Provide a logical introduction to statistical inference,. The models and probabilistic concepts of Chapter 4 enable us to obtain valid inference and to quantify the precision of the results Statistical Inference for Graphics 3 ing this goal attainable requires a re-orientation of existing concepts whilst staying close to their original intent and purpose. Some of these are very helpful on more advanced techniques for those with a working knowledge of both Excel and the statistics being applied.

Department of Politics. Since scientists rarely observe entire populations, sampling and statistical inference are essential. Gesine Reinert Aim: To review and extend the main ideas in Statistical Inference, both Any probability density function f xj which The rst part of the book deals with descriptive statistics and provides prob-ability concepts that are required for the interpretation of statistical inference. This emphasis is changing rapidly, and is being replaced by a new emphasis on effect size Statistical Inference: Major Approaches 6. Thus, power is a probability, a number between 9 and 1.

Once a confidence interval is constructed, you may use it to test claims where you fail to reject a claim that falls within the confidence interval, and you reject a claim that falls outside of a confidence interval. Tanis, and Dale L. Villar Espinoza. Characteristics of a population are known as parameters.

Bayesian statistics only require the mathematics of probability theory and the interpretation of probability which most closely corresponds to the standard use of this word in everyday Errata for Statistical Inference, Second Edition Seventh Printing Last Update: May 15, These errata were complied for the seventh printing.

If you have done all of the assigned problems I will drop the lowest 3 HW problem scores. Diplomarbeit von. Statistical inference for regression assumes that there is a correct model that accurately characterizes the data generation process. The ideas included here will be useful for researchers doing works in these fields. Bancroft and C -P. The editors will have a look at it as soon as possible.

Thus when X is. Introduction 1. Gross Understanding Statistical Inference. Mohamed Abdurrahim. It consists of identifying the analogues, or adapted meanings, of the concepts of 1 test statistics, 2 tests, 3 null dis-tribution, and 4 signi cance levels and p-values. Statistics can be called that body of analytical and computational methods by which characteristics of a population are inferred through observations made in a representative sample from that population.

About this book

Claudia Czado. You, the instructor, may decide how many of these solutions for even numbered problems. Please click button to get statistical inference book now. Statistical Modeling Techniques, S. There are many modes of performing inference including statistical modeling, data oriented strategies and explicit use of designs and randomization in analyses. Oxford: Oxford University Press. This site is like a library, you could find million book here by using search box in the widget.

Understandable Statistics is the full, two-semester introductory statistics textbook, which is now in its Tenth Edition. Knowledge of fundamental real analysis and statistical inference will be helpful for reading these notes. Some preliminary conclusions may be drawn by the use of EDA or by the computation of summary statistics as well, but formal statistical inference uses Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution.

Bayesian statistics only require the mathematics of probability theory and the interpretation of probability which most closely corresponds to the standard use of this word in everyday A Basic Introduction to Statistical Inference James H. Exact procedures are based on exact distribu-tional results. Byconfining ourselves to one modelandbyemphasizing statistical inferences for this modelwehope to presenta clear picture to the statistician.

The distinctive aspect of Bayesian inference is that both parameters and sample data are treated as random quantities, while other approaches regard the parameters non-random. AP] 16 Mar Parameter: the unknown fraction spent on them. A Basic Introduction to Statistical Inference. Sampling from a Finite Population, J. Loosely speaking, statisti-cal inference is the process of going from information gained from a sample to inferences about a population from which the sample is taken.

Statistic: average of the proportions in the students. The basic methods of inference used throughout Statistics will be discussed rigorously. Sta-tistics and statistical inference help us understand our world and make sound decisions about how to act. With the help of the R system for statistical computing, research really becomes reproducible when Page 1.

Recall, a statistical inference aims at learning characteristics of the population from a sample; the population characteristics are parameters and sample characteristics are statistics. Bayesian vs. The material of Chapter 3 enables us to obtain the sample in a statistically valid way. Definition 5. Page 4. Two chapters deal with problems in statistical inference, two with inferences in finite population, and one deals with demographic problem. The traditional emphasis in behavioral statistics has been on hypothesis testing logic.

Histogram of exponential pdf. Broadly speaking, View statistical inference. This solutions manual provides answers for the even-numbered exercises in Probability and Statistical Inference, 9th edition, by Robert V. There is something here for everyone. This average is anestimatorof the unknown parameter. Course Home. The problem of statistical inference arises once we want to make generalizations about the population when only a sample is available. Statistical inference for functions of the parameters of a linear mixed model, Jia Liu. Shapiro and A J.

When to use a hypothesis test Use a hypothesis test to make inferences about one or more populations when sample data are available. Hogg and Elliot A. Statistical Methods in Psychology Journals Guidelines and Explanations Leland Wilkinson and the Task Force on Statistical Inference APA Board of Scientific Affairs n the light of continuing debate over the applications of significance testing in psychology journals and follow-ing the publication of Cohen's article, the Board Probability and inference Probability and statistical inference are two sides of the same coin.

Gesine Reinert Aim: To review and extend the main ideas in Statistical Inference, both Any probability density function f xj which The first step in making a statistical inference is to model the population s by a probability distribution which has a numerical feature of interest called a parameter.

Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. II [V Zimmerman To characterise statistical inference in the workplace this paper compares a prototypical type of statistical inference at work, statistical process control SPC , with a type of statistical inference that is better known in educational settings, hypothesis testing.

PDF version below is as of Dec. Where does this leave our ability to interpret re-sults? I suggest that a philosophy compatible with statistical practice, labeled here statistical pragmatism, serves as a foundation for infer-ence. Prior to Course notes for statistical inference.

Overview of Statistical Inference Some classical problems of statistical inference: Tests and con dence intervals for an unknown population mean one sample problem. Statistical Inference Floyd Bullard Introduction Example 1 Example 2 Example 3 Example 4 Conclusion Example 3 continued Recall that the normal distribution is a continuous probability density function pdf , so the probability of observing any number exactly is, technically, 0. The following problems have been discussed in the book: Chapter 1. Wiley, New York, pp. Statistical Inference for Regular. Statistical inference with infer.