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32 30.012 0.0052 4
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Determine E(Y) Show that 1X 2 2 Y c2 2Y 2 2 1X 22 4 Using part (b), show that E3 1X c2 2 E 3 Y4 Using part (c), complete the derivation of Chebyshev s inequality.
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CHAPTER OUTLINE
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Random Sampling and Data Description
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6-1 DATA SUMMARY AND DISPLAY 6-2 RANDOM SAMPLING 6-3 STEM-AND-LEAF DIAGRAMS 6-4 FREQUENCY DISTRIBUTIONS AND HISTOGRAMS
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6-5 BOX PLOTS 6-6 TIME SEQUENCE PLOTS 6-7 PROBABILITY PLOTS 6-8 MORE ABOUT PROBABILITY PLOTTING (CD ONLY)
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LEARNING OBJECTIVES
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After careful study of this chapter you should be able to do the following: 1. Compute and interpret the sample mean, sample variance, sample standard deviation, sample median, and sample range 2. Explain the concepts of sample mean, sample variance, population mean, and population variance 3. Construct and interpret visual data displays, including the stem-and-leaf display, the histogram, and the box plot 4. Explain the concept of random sampling 5. Construct and interpret normal probability plots 6. Explain how to use box plots and other data displays to visually compare two or more samples of data 7. Know how to use simple time series plots to visually display the important features of timeoriented data. CD MATERIAL 8. Interpret probability plots for distributions other than normal. Answers for most odd numbered exercises are at the end of the book. Answers to exercises whose numbers are surrounded by a box can be accessed in the e-Text by clicking on the box. Complete worked solutions to certain exercises are also available in the e-Text. These are indicated in the Answers to Selected Exercises section by a box around the exercise number. Exercises are also available for some of the text sections that appear on CD only. These exercises may be found within the e-Text immediately following the section they accompany.
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190 6-1
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CHAPTER 6 RANDOM SAMPLING AND DATA DESCRIPTION
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DATA SUMMARY AND DISPLAY
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Well-constructed data summaries and displays are essential to good statistical thinking, because they can focus the engineer on important features of the data or provide insight about the type of model that should be used in solving the problem. The computer has become an important tool in the presentation and analysis of data. While many statistical techniques require only a hand-held calculator, much time and effort may be required by this approach, and a computer will perform the tasks much more ef ciently. Most statistical analysis is done using a prewritten library of statistical programs. The user enters the data and then selects the types of analysis and output displays that are of interest. Statistical software packages are available for both mainframe machines and personal computers. We will present examples of output from Minitab (one of the most widely-used PC packages), throughout the book. We will not discuss the hands-on use of Minitab for entering and editing data or using commands. This information is found in the software documentation. We often nd it useful to describe data features numerically. For example, we can characterize the location or central tendency in the data by the ordinary arithmetic average or mean. Because we almost always think of our data as a sample, we will refer to the arithmetic mean as the sample mean.
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De nition If the n observations in a sample are denoted by x1, x2, p , xn, the sample mean is
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(6-1)
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EXAMPLE 6-1
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Let s consider the eight observations collected from the prototype engine connectors from 1. The eight observations are x1 12.6, x2 12.9, x3 13.4, x4 12.3, x5 13.6, x6 13.5, x7 12.6, and x8 13.1. The sample mean is
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x2 n
a xi
12.9 8
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A physical interpretation of the sample mean as a measure of location is shown in the dot diagram of the pull-off force data. See Figure 6-1. Notice that the sample mean x 13.0 can be thought of as a balance point. That is, if each observation represents 1 pound of mass placed at the point on the x-axis, a fulcrum located at x would exactly balance this system of weights. The sample mean is the average value of all the observations in the data set. Usually, these data are a sample of observations that have been selected from some larger population of observations. Here the population might consist of all the connectors that will be manufactured and sold to customers. Recall that this type of population is called a conceptual or