The Standard Normal Distribution: Properties and Applications - Prof. J. R. Buchanan, Assignments of Statistics

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The Standard Normal Distribution

MATH 130, Elements of Statistics I

J. Robert Buchanan

Department of Mathematics

Spring 2008

Properties of the Standard Normal Curve

(^1) It is symmetric about its mean μ = 0 and has standard deviation σ = 1. (^2) The mean = mode = median, and thus the peak of the curve occurs at μ = 0. (^3) It has inflection points at μ − σ = −1 and μ + σ = 1. (^4) The area under the curve is 1. (^5) The area under the curve to the left of μ = 0 is 1/2 which is also the area under the curve to the right of μ = 0. (^6) The PDF approaches 0 but never reaches 0. (^7) The Empirical Rule applies.

Finding the Area to the Left of Z

We will use Table IV (pages A–10 and A–11) to look up areas associated with different values of the standard normal random variable Z.

1.65 Z

Examples

Example Find the areas under the standard normal curve associated with the following Z -scores. (^1) To the left of Z = 1 .43. (^2) To the left of Z = − 1 .34. (^3) To the right of Z = 1 .17. (^4) To the right of Z = − 0 .66. (^5) Between Z = − 0 .49 and Z = 1 .08. (^6) Between Z = 1 .13 and Z = 2 .05.

Finding Z -scores Given the Area

If we are given the area under the standard normal curve, we can search for the closest area found in Table IV and look up the Z -score corresponding to this area. Example Find the Z -scores associated with the following areas under the standard normal curve. (^1) Area to the left is 0.68. (^2) Area to the left is 0. 25 (^3) Area to the right is 0.72. (^4) Area to the right is 0.85.

Area in the Middle

Example Find the Z -scores associated with the following areas under the standard normal curve. (^1) Middle 80%. (^2) Middle 90%. (^3) Middle 95%. (^4) Middle 99%.

Examples

Example Find the Z -scores associated with the following z α’s. (^1) z 0. 20 (^2) z 0. 15 (^3) z 0. 10 (^4) z 0. 05

Area and Probability

The area under the standard normal curve is the probability that Z lies in a particular interval. P ( a < Z < b ) represents the probability that a standard normal random variable is between a and b. P ( Z > a ) represents the probability that a standard normal random variable is greater than a. P ( Z < a ) represents the probability that a standard normal random variable is less than a. We will not distinguish between strict (<, >) and non-strict (≤, ≥) inequality.

Homework

Read Section 7.2. Pages 341-343: 5–49 odd