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Unformatted text preview: CHAPTER 6 Ch 5 discusses discrete distributions. Ch 6 discusses a popular continuous distribution called the Normal distribution. Normal Distribution is a symmetric continuous distribution about the mean, μ . It is not the only symmetric continuous distribution. It is also known as the Gaussian distribution. It is often what is meant by the term Bell- shaped curve. To determine probabilities, we must use the formula 2 2 2 ) ( 2 2 1 σ μ σ − − = x e y ; where x can be any real number. We will not use the formula, but charts instead. Normal Distribution-15-10-5 5 10 15 The distribution curve looks like this: FACTS ABOUT THE NORMAL DISTRIBUTION 1.) bell-shaped 2.) continuous distribution 3.) symmetric about the mean We can fold a piece of paper over on 0 and the distribution matches. Mean=Median=Mode because the distribution is symmetric and not skewed. 4.) Unimodal (1 mode, which is the mean) 5.) Highest point on the curve is the mean. 6.) The area under the curve gives the probability 7.) Total area under the curve is 1 because total probability is 1. 8.) Empirical Rule applies a. 1 σ Rule: Area under the curve within 1 standard deviation from the mean is about 0.68 b. 2 σ Rule: Area under the curve within 2 standard deviation from the mean is about 0.95 c. 3 σ Rule: Area under the curve within 3 standard deviations from the mean is about 0.997 9.) The curve never touches 0 10.) The further the curve is spread out from the mean, the bigger the standard deviation. 11.) The probability of a single value is 0. We will find the probability of a range of values. Why we cannot get the probability of a single value? Think about the probability in terms of chapter 4. The probability is the number of ways to get that single 1 value over the number of possible outcomes. There is an infinite number of values with that 1 value on top. This would lead to a probability of 0. Try it, put this in your calculator: 99999999 1 . You should get something close to 0 and this is not infinity you are dividing by. This is a property of all continuous distributions. CHAPTER 6 STANDARD NORMAL DISTRIBUTION The normal distribution formula can be complicated for a number of people. So naturally, we want to use a chart to look up the values. However, the distribution curve of the normal distribution changes every time we change the mean and/or standard deviation. A particular form of the normal distribution is used instead. We use the standard normal distribution to determine the probability. Standard Normal Distribution is the normal distribution with a mean of 0 & a standard deviation of 1. Its formula for determining probability is 2 2 2 1 μ − = e y . How do we take a normal distribution and get a standard normal distribution?...
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This document was uploaded on 02/23/2010.
- Spring '09
- Normal Distribution