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Chapter 8 – Normal Probability Distribution • A probability distribution in which the random variable is continuous is a continuous probability distribution. • The normal probability distribution is the most common continuous probability distribution. Nature of the normal distribution • Characteristics of normal distribution 1. Bell-shaped with a single peak 2. Symmetrical so two halves are mirror images •Look at figure 8-3 on page 164 •There are numerous normal distributions that have the same mean, but different standard deviations. •Look at figure 8-4 on page 165. Importance of the normal distribution. • The normal distribution is very important for two good reasons. • 1. It can be used as an approximation for many other distributions. • 2. Many random variables in the real world follow a normal distribution. The standard normal distribution • Any normal distribution with a mean and a standard deviation can be converted to a standard normal distribution. The standard normal distribution has a mean of zero and a standard deviation of one. • So the standard normal distribution looks like the one shown below. Standard Distribution (con’t) • Once converted to the standard normal distribution, the random variable is denoted by Z. The conversion is done by using the following formula: • Z=(X-)/. Formula 8-1 p. 167 • Where X is the original random variable with a mean of and a standard deviation of . Standard Distribution (con’t) • Probability of X being greater than 700 is the same as the probability of Z being greater than 2. • P(X>700) = P(Z>2) Standard Distribution (con’t) • Example Problems 8-1, p. 167 • Z = (700-500) / 100 = 2 300 -2 400 -1 500 0 600 1 700 2 X-Scale Z-Scale Areas under the normal curve • Areas under the normal curve can be found by using appendix D, p. 478. • Let’s remember a few things 1. The area under the normal curve totals 100%. 2. Since the normal curve is symmetrical, 50% of the area is to the right of the mean and the other 50% to the left. Finding the area • Example problem 8-3, p. 168, • P(Z >1.64)=1- P(Z < 1.64) =1- 0.9495=0.0505 • Example Problem 8-4, page 169 • P(Z > -1.65) = P(Z < 1.65) = 0.9505 • Example Problem 8-5, p. 169-170 • Example Problem 8-6, page 170 • Problem #4, page 173 • Problem #7, page 174 • Problem #10, page 174 • Problem #11, page 174 Applications of Z-score • By now , we know how to use appendix D for finding probabilities. Let’s solve some real-life problems using appendix D. • Example problem 8-10, p. 175 • Example Problem 8-11, p. 176 • Problem#6, p. 180 • Problem #10, p. 181 • Problem #14, p. 181 Sampling Distribution of Mean • If several samples of size n are taken from a population (whose mean is and standard deviation is ) and their means are computed, these means are normally distributed with a mean of and a standard deviation of /n. Central Limit Theorem • Calculation of probabilities for sample _ mean, X: _ X- Z = ------- / n Formula 8-6, Page 189 •Example Problem 8-16 (Page 190) Problem #5 (Page 191-192), Problem #11 (Page 192-193)