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Unformatted text preview: Chapter 6 Normal Probability Distributions 62 The Standard Normal Distribution 1. The word normal as used when referring to a normal distribution does carry with it some of the meaning the word has in ordinary language. Normal distributions occur in nature and describe the normal, or natural, state of many common phenomena. But in statistics the term normal has a specific and welldefined meaning in addition to its generic connotations of being typical it refers to a specific bellshaped distribution generated by a particular mathematical formula. 2. Bellshaped describes a smooth, symmetric distribution that has its highest point in the center and tapers off to zero in either direction. 3. A normal distribution can be centered about any value and have any level of spread. A standard normal distribution has a center (as measured by the mean) of 0 and has a spread (as measured by the standard deviation) of 1. 4. The notation z indicates the z score that has an area of to its right. When = 0.05, for example, z 0.05 indicates the z score with an area of 0.05 to its right. 5. The height of the rectangle is 0.5. Probability corresponds to area, and the area of a rectangle is (width)(height). P(x&gt;124.0) = (width)(height) = (125.0 124.0)(0.5) = (1.0)(0.5) = 0.50 6. The height of the rectangle is 0.5. Probability corresponds to area, and the area of a rectangle is (width)(height). P(x&lt;123.5) = (width)(height) = (123.5 123.0)(0.5) = (0.5)(0.5) = 0.25 x (volts) f(x) 125.0 124.5 124.0 123.5 123.0 0.5 0.4 0.3 0.2 0.1 0.0 x (volts) f(x) 125.0 124.5 124.0 123.5 123.0 0.5 0.4 0.3 0.2 0.1 0.0 The Standard Normal Distribution SECTION 62 137 7. The height of the rectangle is 0.5. Probability corresponds to area, and the area of a rectangle is (width)(height). P(123.2&lt;x&lt;124.7) = (width)(height) = (124.7 123.2)(0.5) = (1.7)(0.5) = 0.75 8. The height of the rectangle is 0.5. Probability corresponds to area, and the area of a rectangle is (width)(height). P(124.1&lt;x&lt;124.5) = (width)(height) = (124.5 124.1)(0.5) = (0.4)(0.5) = 0.20 NOTE: For problems 916, the answers are reexpressed (when necessary) in terms of items that can be read directly from Table A2. In general, this step is omitted in subsequent exercises and the reader is referred to the accompanying sketches to se how the indicated probabilities and z scores relate to Table A2. A is used to denote the tabled value of the area to the left of the given z score. As a crude check, always verify that A&gt;0.5000 corresponds to a positive z score and z&gt;0 corresponds to an A &gt;0.5000 A&lt;0.5000 corresponds to a negative z score and z&lt;0 corresponds to an A &lt; 0.5000 9. P(z&lt;0.75) = 0.7734 10. P(z &gt;0.75) = 1 P(z&lt;0.75) = 1 0.2266 = 0.7734 11. P(0.60&lt;z&lt;1.20) = P(z&lt;1.20) P(z&lt;0.60) = 0.8849 0.2743 = 0.6106 12. P(0.90&lt;z&lt;1.60) = P(z&lt;1.60) P(x&lt;0.90) = 0.9452 0.1841 = 0.7611 13. For A = 0.9798, z = 2.05....
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This note was uploaded on 03/07/2011 for the course STATISTICS 1022 taught by Professor Dr.kalluri during the Spring '11 term at University of South Florida  Tampa.
 Spring '11
 Dr.Kalluri
 Normal Distribution, Probability

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