371chapter1 - 0 Chapter 1 Probability 1.1 The Standard...

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Chapter 1 Probability 1.1 The Standard Normal Curve. In this section we will learn how to use an approximating device that we will need throughout the semester. The standard normal curve , henceforth abbreviated the snc , is pictured on page 98 of the text. Below are the essential features of this curve: It is symmetric about the number 0. This means that the curve to the right of 0 is the mir- ror image of the curve to the left of 0. In Statistics, symmetry always means left-to-right symmetry, like the letter ‘U,’ not top-to-bottom symmetry, like the letter ‘E.’ The total area under the curve equals 1. (Enrichment, for students of calculus: The snc has a point of inflection at +1 and, of course b/c of symmetry, another at - 1 .) The snc will be used often in this course to obtain approximate answers to computational problems. The approximations will be obtained by calculating an area under a portion of the snc. Enrichment, for students of calculus: For many functions it is easy to obtain the area under (a portion of) a curve. It is not, however, easy for the snc; trust me on this. Sophisticated ‘numerical methods’ are needed and they are beyond the scope of this course. There are many resources that can be used to obtain the area under the snc. These include: 1. Table A.2 of the text. Table A.2 appears twice in the text: on pages 650–651 and inside its front cover. 2. A fancy calculator. 3. A statistical software package on a computer. In this course you are responsible for the first of the methods listed above. I will now explain how to use Table A.2 and then give you some problems so that you can practice your new skill. 1
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Table A.2 is specifically designed to find ‘areas to the right,’ and, naturally, we will begin with one of those problems. Alan wants to find the area under the snc to the right of 0.72. He should proceed as follows. The number of interest, in this case 0.72, is denoted by z . Thus, for Alan, z = 0 . 72 . We then proceed as follows: 1. Break z into two pieces: 0.7 and 0.02. 2. Locate the row of table A.2 that corresponds to 0.7. 3. Locate the column of table A.2 that corresponds to 0.02. 4. Run one finger across the row and another down the column; at the point of intersection of your fingers is the answer. For Alan’s problem, the entry is 0.2358. Thus, we have determined that the area under the snc to the right of z = 0 . 72 is 0.2358. Let’s do another example. Betsy wants to find the area under the snc to the right of z = - 1 . 47 . Proceeding as above, 1. Break z into two pieces: - 1 . 4 and 0.07. 2. Locate the row of table A.2 that corresponds to - 1 . 4 . 3. Locate the column of table A.2 that corresponds to 0.07. 4. Run one finger across the row and another down the column; at the point of intersection
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371chapter1 - 0 Chapter 1 Probability 1.1 The Standard...

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