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# chapter7 - Chapter 7 The Normal Probability Distribution...

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362 Chapter 7 The Normal Probability Distribution 7.1 Properties of the Normal Distribution 1. For the graph to be that of a probability density function, (1) The area under the graph over all possible values of the random variable must equal 1; (2) The graph must be greater than or equal to 0 for all possible values of the random variable. That is, the graph of the equation must lie on or above the horizontal axis for all possible values of the random variable. 2. distribution; density 3. The area under the graph of a probability density function can be interpreted as either: (1) The proportion of the population with the characteristic described by the interval; or (2) The probability that a randomly selected individual from the population has the characteristic described by the interval. 4. We standardize the normal random variables so that one table can be used to find the area under the curve of any normal density function. 5. ; μ σμσ −+ 6. As σ increases, the height of the graph of the normal density function decreases. 7. No, the graph cannot represent a normal density function because it is not symmetric. 8. Yes, the graph can represent a normal density function. 9. No, the graph cannot represent a normal density function because it crosses below the horizontal axis. That is, it is not always greater than or equal to 0. 10. No, the graph cannot represent a normal density function because it does not approach the horizontal axis as X increases (and decreases) without bound. 11. Yes, the graph can represent a normal density function. 12. No, the graph cannot represent a normal density function because it is not bell-shaped. 13. The figure presents the graph of the density function with the area we wish to find shaded. The width of the rectangle is 10 5 5 −= and the height is 1 30 . Thus, the area between 5 and 10 is 5 11 30 6 ⎛⎞ = ⎜⎟ ⎝⎠ . The probability that the friend is between 5 and 10 minutes late is 1 6 . 03 0 510 1 30 __ X Random Variable (time) Density

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Section 7.1 Properties of the Normal Distribution 363 14. The figure presents the graph of the density function with the area we wish to find shaded. The width of the rectangle is 25 15 10 −= and the height is 1 30 . Thus, the area between 15 and 25 is 10 11 30 3 ⎛⎞ = ⎜⎟ ⎝⎠ . The probability that the friend is between 15 and 25 minutes late is 1 3 . 03 0 25 15 1 30 __ X Random Variable (time) Density 15. The figure presents the graph of the density function with the area we wish to find shaded. The width of the rectangle is 30 20 10 and the height is 1 30 . Thus, the area between 20 and 30 is 10 30 3 = . The probability that the friend is at least 20 minutes late is 1 3 . 0 20 1 30 __ X Random Variable (time) 16. The figure presents the graph of the density function with the area we wish to find shaded. The width of the rectangle is 505 and the height is 1 30 . Thus, the area between 0 and 5 is 5 30 6 = . The probability that the friend is no more than 5 minutes late is 1 6 . 0
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chapter7 - Chapter 7 The Normal Probability Distribution...

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