If the occurrence of major earthquakes can be modeled by the Poisson distribution, none of the 12 turtles. b. at least 2 of the 12 turtles. What is the probability of no eruptions or instances of major seismic activities What is the probability of at leas

4.111 A wildlife biologist is studying turtles that have been exposed to oil spills in the Gulf of Mexico. Previous studies have determined that a particular blood disorder occurs in turtles ex- posed for a length of time to oil at a rate of 1 in every 8 exposed turtles. The biologist examines 12 turtles exposed for a considerable period of time to oil. If the rate of occurrence of the blood disorder has not changed, what is the probability of each of the following events? She finds the disorder in:

a. none of the 12 turtles. b. at least 2 of the 12 turtles.

c. no more than 4 turtles. 4.112 Airlines overbook sell more tickets than there are seats flights, based on past records that indicate that approximately 5 of all passengers fail to arrive on time for their flight. Sup- pose a plane will hold 250 passengers, but the airline books 260 seats. What is the probability that at least one passenger will be bumped from the flight? 4.113 For the last 300 years, extensive records have been kept on volcanic activity in Japan. In 2002, there were five eruptions or instances of major seismic activity. From historical records, the mean number of eruptions or instances of major seismic activity is 2.4 per year. A researcher is interested in modeling the number of eruptions or major seismic activities over the 5-year period, 2005 –2010. a. What probability model might be appropriate? b. What is the expected number of eruptions or instances of major seismic activity during 2005 –2010?

c. What is the probability of no eruptions or instances of major seismic activities

during 2005 –2010?

d. What is the probability of at least two eruptions or instances of major seismic

activity? Text not available due to copyright restrictions This page intentionally left blank 5 Inferences about Population Central Values 6 Inferences Comparing Two Population Central Values 7 Inferences about Population Variances 8 Inferences about More Than Two Population Central Values 9 Multiple Comparisons 10 Categorical Data 11 Linear Regression and Correlation 12 Multiple Regression and the General Linear Model 13 Further Regression Topics 14 Analysis of Variance for Completely Randomized Designs 15 Analysis of Variance for Blocked Designs 16 The Analysis of Covariance 17 Analysis of Variance for Some Fixed-, Random-, and Mixed-Effects Models 18 Split-Plot, Repeated Measures, and Crossover Designs 19 Analysis of Variance for Some Unbalanced Designs P A R T 4 Analyzing Data, Interpreting the Analyses, and Communicating Results 222 CHAPTER 5 Inferences about Population Central Values 5.1 Introduction and Abstract of Research Study

5.2 Estimation of M

5.3 Choosing the Sample Size for Estimating M 5.4 A Statistical Test for M 5.5 Choosing the Sample Size for Testing M 5.6 The Level of Significance of a Statistical Test 5.7 Inferences about M for a Normal Population, S Unknown 5.8 Inferences about M When Population Is Nonnormal and n Is Small: Bootstrap Methods 5.9 Inferences about the Median

5.10 Research Study: Percent Calories from Fat

5.11 Summary and Key Formulas

5.12 Exercises

5.1 Introduction and Abstract of Research Study

Inference, specifically decision making and prediction, is centuries old and plays a very important role in our lives. Each of us faces daily personal decisions and situa- tions that require predictions concerning the future. The U.S. government is concerned with the balance of trade with countries in Europe and Asia. An invest- ment advisor wants to know whether inflation will be rising in the next 6 months. A metallurgist would like to use the results of an experiment to determine whether a new light-weight alloy possesses the strength characteristics necessary for use in au- tomobile manufacturing. A veterinarian investigates the effectiveness of a new chemical for treating heartworm in dogs. The inferences that these individuals make should be based on relevant facts, which we call observations, or data. In many practical situations, the relevant facts are abundant, seemingly incon- sistent, and, in many respects, overwhelming. As a result, a careful decision or prediction is often little better than an outright guess. You need only refer to the “Market Views’’ section of the Wall Street Journal or one of the financial news shows on cable TV to observe the diversity of expert opinion concerning future stock market behavior. Similarly, a visual analysis of data by scientists and engineers often yields conflicting opinions regarding conclusions to be drawn from an experiment. Many individuals tend to feel that their own built-in inference-making equip- ment is quite good. However, experience suggests that most people are incapable of utilizing large amounts of data, mentally weighing each bit of relevant information, and arriving at a good inference. You may test your own inference-making ability by using the exercises in Chapters 5 through 10. Scan the data and make an inference before you use the appropriate statistical procedure. Then compare the results. The statistician, rather than relying upon his or her own intuition, uses statistical results to aid in making inferences. Although we touched on some of the notions involved in statistical inference in preceding chapters, we will now collect our ideas in a pres- entation of some of the basic ideas involved in statistical inference. The objective of statistics is to make inferences about a population based on information contained in a sample. Populations are characterized by numerical de- scriptive measures called parameters. Typical population parameters are the mean m , the median M, the standard deviation s, and a proportion p. Most inferential problems can be formulated as an inference about one or more parameters of a population. For example, a study is conducted by the Wisconsin Education De- partment to assess the reading ability of children in the primary grades. The popu- lation consists of the scores on a standard reading test of all children in the primary grades in Wisconsin. We are interested in estimating the value of the population mean score m and the proportion p of scores below a standard, which designates that a student needs remedial assistance. Methods for making inferences about parameters fall into one of two cate- gories. Either we will estimate the value of the population parameter of interest or we will test a hypothesis about the value of the parameter. These two methods of statistical inference—estimation and hypothesis testing—involve different proce- dures, and, more important, they answer two different questions about the param- eter. In estimating a population parameter, we are answering the question, “What is the value of the population parameter?” In testing a hypothesis, we are seeking an answer to the question, “Does the population parameter satisfy a specified con- dition?” For example, “m ⬎ 20” or “p ⬍ .3.” Consider a study in which an investigator wishes to examine the effectiveness of a drug product in reducing anxiety levels of anxious patients. The investigator uses a screening procedure to identify a group of anxious patients. After the pa- tients are admitted into the study, each one’s anxiety level is measured on a rating scale immediately before he or she receives the first dose of the drug and then at the end of 1 week of drug therapy. These sample data can be used to make infer- ences about the population from which the sample was drawn either by estimation or by a statistical test: Estimation: Information from the sample can be used to estimate the mean decrease in anxiety ratings for the set of all anxious patients who may conceivably be treated with the drug. Statistical test: Information from the sample can be used to determine whether the population mean decrease in anxiety ratings is greater than zero. Notice that the inference related to estimation is aimed at answering the question, “What is the mean decrease in anxiety ratings for the population?” In contrast, the statistical test attempts to answer the question, “Is the mean drop in anxiety ratings greater than zero?” estimation hypothesis testing