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# post15 - th percentile of the standard normal distribution...

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1 STAT 511-2 Spring 2012 Lecture 15 Feb 13, 2012 Jun Xie 4.3 The normal distribution The standard normal Definition The normal distribution with parameter values = 0 and = 1 is called the standard normal distribution. A random variable having a standard normal distribution is called a standard normal random variable and will be denoted by Z . The pdf of Z is . The cdf P ( Z z ) is denoted as Φ (z). Appendix Table A.3 gives Φ (z), the area under the standard normal density curve to the left of z . From this table, various other probabilities involving Z can be calculated. Example 13 Let’s determine the following standard normal probabilities: (a) P ( Z 1.25), (b) P ( Z > 1.25), (c) P ( Z 1.25), and (d) P ( .38 Z 1.25). a b. c. P ( Z 1.25) = Φ ( 1.25), a lower-tail area. Directly from Appendix Table A.3, ( 1.25) = .1056. By symmetry of the z curve, this is the same answer as in part (b).

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2 d. Percentiles of the standard normal distribution Example 14 Find the 99 th percentile of the standard normal distribution.
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Unformatted text preview: th percentile of the standard normal distribution. Appendix Table A.3 gives for fixed z the area under the standard normal curve to the left of z , whereas here we have the area and want the value of z . This is the “inverse” problem to P ( Z z ) = ? Here .9901 lies at the intersection of the row marked 2.3 and column marked .03, so the 99th percentile is (approximately) z = 2.33. z Notation for z Critical Values Notation z will denote the value on the z axis for which of the area under the z curve lies to the right of z . For example, z .10 captures upper-tail area .10, and z .01 captures upper-tail area .01. Nonstandard normal distributions Proposition If X has a normal distribution with mean and standard deviation , then The key idea of the proposition is that by standardizing, any probability involving X can be expressed as a probability involving a standard normal rv Z , so that Appendix Table A.3 can be used....
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