Subjects: Business mathematics

Testing ideas lies at the heart of statistics. Researchers must have unifying criteria to determine whether the results they see are evidence of real effects or difference, or whether they are simply due to chance. You have learned in previous lessons about probability and how to calculate it. But how do you know when chance accounts for sample results? What is the cutoff? To answer that question, statisticians have developed tests to determine whether findings are likely to be real or the result of chance. This lesson explains the reasoning behind such tests.

Statistics is a powerful tool for discovering how the world works, but we must be cautious when interpreting what the numbers mean when we calculate a confidence interval or perform a test of significance. Results are not always as clear-cut as they seem. This lesson explores some of the errors that can occur when interpreting the results of these two statistical procedures. The sampling errors you learned about in early lessons are just one pitfall that awaits the careless statistician. In this lesson, you will encounter two scenarios in which hypotheses are evaluated incorrectly even though the numbers seem to tell a different story. These scenarios reveal two types of errors that can occur when interpreting results using a test of significance. The cautions learned in this lesson will continue to apply in future lessons when more complex tests of significance are introduced.

This lesson pursues a more realistic test of significance than the z test discussed in Lesson 15. In that lesson, you learned to use the z test to determine whether or not a sample population mean is significantly different from the claimed population mean. The z-test assumes that we know the population standard deviation ?. In reality, we rarely know ?. In this lesson, you will learn how researchers get around that problem by using a different test with more realistic parameters.

Experimental studies are powerful tools that researchers use to determine cause and effect relationships. These studies are picked up in the media and disseminated to people all over the world. The results of many of these studies provide information critical to decisions we make. Whether it’s deciding on what car to buy, which candidate to vote for, or which medical treatment regimen to follow, we all rely on studies to give us the information we need to make decisions and to accurately understand our world.

Inferences are something we make every day. Every time we draw a conclusion based on evidence, we are making an inference. For example, if we meet someone new, we draw conclusions about that person based on what we see and hear. If we see a long line outside of a local dance club, we might assume that it is an exclusive club and probably full of VIPs. When we hear a couple shouting at each other, we may conclude that they are arguing. When statistician gathers information, the analysis includes making a graphic display and correlate the data. All of these examples involve the practice of inference.

What are the chances?” We think this many times a day. Why is it raining when the weatherman said there was only a five percent chance of rain? Why do people play the lottery if the chance of winning is so low? In mathematics, the science of probability describes such chance behavior. As statisticians strive to make sense out of data, a major part of their endeavor is the use of probability in making predictions and explaining results.

We have all had the experience watching our friends and family play sports or games. Based on prior performance, we predict in our heads how they will do in the next game, and we are often right. By keeping track of variables like bowling scores or chips in a poker game, one can predict how well someone will do in the next event.

Researchers are often interested in variables that are categorical in nature. These often involve people who fall into certain groups, such as age, gender, political affiliation, or religion. These groups can also be defined as yes or no categories, especially in medicine. Patients get well or remain ill, survive cancer or don’t, improve on mobility tests or decline.

Polls are something we encounter everyday in the media. We are constantly bombarded with numbers of how many people approve of the President’s performance or how many people prefer one brand of soda over another. Politicians and corporations want to know what we are thinking and doing. But they can’t ask all of us all the time. Instead they pick a few people, a sample, that can be easily surveyed. Just as one can taste a small sample of ice cream flavors at an ice cream counter, so can we sample a few people to find out what the public at large is thinking.

Every day, the media bombard us with the results of recent statistical studies. Statements like, “Studies show our products are better than ever…” appear frequently in news releases and advertisements. Many of these studies claim results based on experiments. But what constitutes a proper experiment? How do we know the results are reliable and have real meaning? How does one design an experiment to ensure that the research objective is properly addressed? Resolving these issues are part of the statistician’s role in research.