Hypothesis test for a population variance
Hypothesis test for a population variance in the camshaft situation of Section 9.7 on pages 383 and 384:
• Enter a label (in this case Depth) into cell A1, the sample variance (here equal to .0885) into cell A2, and the sample size (here equal to 30) into cell A3.
• Select Add-Ins: MegaStat: Hypothesis Tests : Chi-square Variance Test
• Click on “summary input.”
• Enter the range A1.A3 into the Input Range window—that is, enter the range containing the data label, the sample variance, and the sample size.
• Enter the hypothesized value (here equal to 0.2209) into the “Hypothesized variance” window.
• Select the desired alternative (in this case “less than”) from the drop-down menu in the Alternative box.
• Check the “Display confidence interval” checkbox (if desired) and select or type the appropriate level of confidence.
• Click OK in the “Chi-square Variance Test” dialog box.
• A chi-square variance test may be carried out using data input by entering the observed sample values into a column in the Excel worksheet, and by then using the autoexpand feature to enter the range containing the label and sample values into the Input Range window.
1 This case is based on conversations by the authors with several employees working for a leading producer of trash bags. For purposes of confidentiality, we have agreed to withhold the company’s name.
2 Thanks to Krogers of Oxford, Ohio, for helpful discussions concerning this case.
3 Some statisticians suggest using the more conservative rule that both np0 and n(1 − p0) must be at least 10.
4 Source: “Driving Organic Growth at Bank of America” Quality Progress (February 2005), pp. 23–27.
5 Consumer Reports, January 2005, page 51.
(Bowerman 346)
Bowerman, Bruce L. Business Statistics in Practice, 5th Edition. McGraw-Hill Learning Solutions, 022008. <vbk:007-7376870#outline(9)>.
CHAPTER 19: Decision Theory
Chapter Outline
19.1 Bayes’ Theorem
19.2 Introduction to Decision Theory
19.3 Decision Making Using Posterior Probabilities
19.4 Introduction to Utility Theory
Every day businesses and the people who run them face a myriad of decisions. For instance, a manufacturer might need to decide where to locate a new factory and might also need to decide how large the new facility should be. Or, an investor might decide where to invest money from among several possible investment choices. In this chapter we study some probabilistic methods that can help a decision maker to make intelligent decisions. In Section 19.1 we present Bayes’ Theorem, which is useful for updating probabilities on the basis of newly obtained information that may help in making a decision. In Section 19.2 we formally introduce decision theory. We discuss the elements of a decision problem, and we present strategies for making decisions when we face various levels of uncertainty. We also show how to construct a decision tree, which is a diagram that can help us analyze a decision problem, and we show how the concept of expected value can help us make decisions. In Section 19.3 we show how to use sample information to help make decisions, and we demonstrate how to assess the worth of sample information in order to decide whether the sample information should be obtained. We conclude this chapter with Section 19.4, which introduces using utility theory to help make decisions.
Many of this chapter’s concepts are presented in the context of
The Oil Drilling Case: An oil company uses decision theory to help to decide whether to drill for oil on a particular site. The company can perform a seismic experiment at the site to obtain information about the site’s potential, and the company uses decision theory to decide whether to drill based on the various possible survey results. In addition, decision theory is employed to determine whether the seismic experiment should be carried out.
19.1: Bayes’ Theorem
Sometimes we have an initial or prior probability that an event will occur. Then, based on new information, we revise the prior probability to what is called a posterior probability. This revision can be done by using a theorem called Bayes’ Theorem.
EXAMPLE 19.1
To illustrate Bayes’ Theorem, consider the event that a randomly selected American has the deadly disease AIDS. We let AIDS represent this event, and we let represent the event that the randomly selected American does not have AIDS. Since it is estimated that .6 percent of the American population has AIDS,
A diagnostic test is used to attempt to detect whether a person has AIDS. According to historical data, 99.9 percent of people with AIDS receive a positive (POS) result when this test is administered, while 1 percent of people who do not have AIDS receive a positive result. That is,