## Intermediate Applied Statistics

Although descriptive statistics are invaluable for organizing and describing data, frequently one needs to be able to draw inferences from the data in order to make decisions and develop plans of action to help the organization reach its goals and objectives. This often involves the use of inferential statistics, a subset of mathematical statistics used in the analysis and interpretation of data. Most intermediate statistics courses discuss a number of statistical procedures that can be used for these purposes including analysis of variance, a family of statistical techniques that analyze the joint and separate effects of multiple independent variables on a single dependent variable; the Pearson product moment coefficient of correlation which estimates the degree to which two events or variables are consistently related; and linear regression, a statistical technique in which a line of best fit is extrapolated through a set of data points to analyze the effect of an independent variable on a dependent variable.

Statistics -- either in the form of numerical data or in their analysis and interpretation -- seem to be everywhere one looks. The newspaper tells us the statistics of Senator Harvey's voting record: the percentage of votes in which he participated, how many of his votes were in support of environmental issues, and whether or not his actions were in favor of or opposed to higher taxes. Television advertisements tell us that the Woody has been found to be 59 percent safer than the Speed Racer. The latest diet book promises to help us lose 14 percent more weight on average. Professors are rated by students on a five-point scale and students' grades are judged based on the normal curve. In the business world, descriptive statistics, a subset of mathematical statistics that describes and summaries data, is useful, too. Marketers find it helpful to know that on average, people rate the new widget design as an 8.5 on a 10 point scale. Quality control engineers graph the number of defects found in a series of random samples to determine whether processes are in or out of control. Employees are given job performance ratings that determine whether they will be given a raise or put on probation. In these ways and more, statistics are part of our lives.

However, one often wants to be able to do more than merely describe data. Although descriptive statistics are invaluable for organizing and describing data through various graphing techniques, measures of central tendency, and measures of variability, one frequently needs to be able to draw inferences from the data in order to make decisions and develop plans of action to help the organization reach its goals and objectives. Good business strategy is based on the rigorous analysis of empirical data, including market needs and trends, competitor capabilities and offerings, and the organization's resources and abilities. Developing a good business strategy often involves the use of inferential statistics, a subset of mathematical statistics used in the analysis and interpretation of data. Inferential statistics are used to make inferences such as drawing conclusions about a population from a sample and for making many business decisions.

In addition to descriptive statistics, most beginning statistics courses also teach basic inferential statistical techniques, including z-tests to estimate the mean of a population from the mean of a sample. In addition, the t-tests taught in basic statistics courses can be used for hypothesis testing to determine the probability that various theories about business phenomena are true. However, these statistical techniques can only test simple hypotheses comparing two samples to determine if they come from the same population. Real world business problems, however, are often more complex and there is frequently a need to compare more than two conditions at a time. Inferential statistics offers other techniques that can help managers and other business decision makers to answer more complex questions. Some of these techniques include analysis of variance (ANOVA), correlation, and linear regression.

### Interpreting Statistical Results

These statistical techniques are powerful tools that can be invaluable in assisting managers and other organizational decision makers in their tasks. However, it needs to be borne in mind that the results of statistical techniques do not yield black-and-white answers, but probabilities. Inferential statistics are used to test the probability that the null hypothesis (H0) is true. This is the statement that there is no statistical difference between the status quo and the experimental condition. If the null hypothesis is true, then the treatment or characteristic being studied made no difference on the end result. For example, a null hypothesis might state that there is no difference in preference for Super Crunchies cereal than for Nutty Flakies cereal in children versus adults. The alternative hypothesis (H1), on the other hand, would state that there is a relationship between the two variables.

A lack of understanding of the way that probability works can result in poor experimental design that yields spurious results. It is important to remember that the results of a statistical data analysis do not prove whether or not the hypothesis is true, but what the probability is of the hypothesis being true at a given confidence level. So, for example, if a t-test or analysis of variance results in a value that is significant at the a = .05 level, this means not that the hypothesis is true, but that the analyst is willing to run the risk of being wrong five times out of 100. This means that there is a possibility of error when interpreting statistics and either accepting or rejecting the null hypothesis. Type I errors occur when one incorrectly rejects the null hypothesis and accepts the alternate hypothesis. An example of a Type I error would be if an analyst concluded that adults enjoy Super Crunchies while children do not enjoy them when, in fact, there is no difference. Type II errors, on the other hand, occur when one incorrectly accepts the null hypothesis. For example, if the analyst interpreted the results to mean that both children and adults equally enjoy Super Crunchies when in actuality adults prefer it more than children do, then a Type II error would have occurred.

### Applications

The techniques taught in intermediate statistics courses vary from course to course. However, most intermediate statistics courses discuss a powerful family of statistical techniques called analysis of variance; a family of statistical techniques that analyze the joint and separate effects of multiple independent variables on a single dependent variable and determine the statistical significance of the effect. Another statistical technique that is often taught in intermediate statistics courses is the Pearson product moment coefficient of correlation which estimates the degree to which two events or variables are consistently related. In addition, most intermediate statistics courses introduce students to the concept of linear regression, a statistical technique in which a line of best fit is extrapolated through a...

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