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Unformatted text preview: density curves that can be used as models. In our next Section 8.6 we focus on an important family of densities called the NORMAL DISTRIBUTIONS. 53 8.6 Normal Random Variables We had our first introduction to normal random variables back in Chapter 2 as a special case of bell‐shaped distributions. The family of normal distributions is very important because many variables have this shape and form approximately and many statistics that we use in our inference methods are based on sums or averages which generally have (approximately) a normal distribution. A Normal Curve: Symmetric, bell‐shaped, centered at the mean and its spread is determined by the standard deviation . In fact, the points of inflection on each side of the mean mark the values which are one standard deviation away from the mean. Density (this axis and label often omitted) Notation: If a population of measurements follows a normal curve, and if X is the measurement for a randomly selected individual from the population, then X is said to be a normal random variable. X is also said to have a normal distribution. Any normal random variable can be completely characterized by its mean and its standard deviation. The variable X is normally distributed with mean and standard deviation is denoted by: _____ X is N()_____ A N(50,10) curve is sketched below. Add a N(80,5) curve to this picture. Keep in mind the features of the empirical rule (68‐95‐99.7) which applies to a normal curve. Ask: why
higher? 10 20 30 40 50 60 70 54 80 90 100 110 Standardized Scores: A normal distribution is indexed by its population mean , and its population standard deviation , and denoted by N ( , ) . Recall that the standard deviation is a useful “yardstick” for measuring how far an individual value falls from the mean. The standardized score or z‐score is the distance between the observed value and the mean, measured in terms of number of standard deviations. Values that are above the mean have positive z‐scores, and values that are below the mean have negative z‐scores. value mean
Standardized score or z‐score: z Standard deviation Finding Probabilities for z‐Scores: Standard scores play a role in how we will find areas (and thus probabilities) under a normal curve. We simply have to convert the endpoints of the interval of interest to the corresponding standardized scores and then use a table (or computer/calculator) to find probabilities associated with these standardized scores. When we convert to standardized scores, the random variable X is converted to what is called the Standard Normal Random Variable, denoted by Z, and it has the N (0,1) distribution. Table A.1 provides the areas to the left for various values of Z 55 56 Try It! Finding Probabilities for Z 1. Find P(Z ≤ 1.22). 0.8888 Think about it: What is P(Z < 1.22)? It is still 0.8888 as P(Z = 1.22) is 0. 2. Find P(Z > 1.22). 1 – 0.8888 = 0.1112 (or from P(Z < ‐1.22)) 3. Find P(‐1.58 < Z < 2.24) 0.9875 – 0.05...
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This document was uploaded on 02/25/2014 for the course STATS 250 at University of Michigan.
- Summer '10