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The Normal Distribution DeMoivre-Laplace Theorem: Let X be a binomial random variable defined on n independent trials each having success probability p. For any number c and d: X np 1 lim P c d n npq 2 d e 2 x2 dx c If this integral is a continuous probability density function, then: 1 2 e 2 x2 dx 1 Standard Normal Distribution: A Random Variable Z is said to have a standard normal distribution given by the following continuous pdf: 1 12 z 2 f ( z) e 2 A Standardized Normal Distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Cumulative Distribution Function for a Standard Normal Distribution There is a special notation used for a cumulative distribution of a standard normal distribution or P ( X x ). P( X x) or P( X x) P( X x) ( z ) Let the random variable X have a normal distribution. Then: E( X ) Var ( X ) 2 Find the probability of getting a z-value less than 0.43 in a standard normal distribution. Answer: 0.6664 Find the area between z = 0 and z = 1 in a standard normal curve. Answer: 0.3413 Find the probability of getting a z-value in a standard normal distribution that is greater than 1.82. Answer: 0.0344 If the probability of getting less than a certain z-value is 0.1190, what is the z-value? Answer: -1.18 Find the probability of getting a z-value in a standard normal distribution that is between 0.46 and 1.75. Answer: 0.2827 Find the z-value for which the area under the standard normal curve to the right of that value is 0.025. Answer: 1.96 Properties of the Standard Normal Distribution: 1. The probability that a z-score falls within 1 standard deviation of the mean on either side, that is, between z = -1 and z = 1, is approximately 68%. 2. The probability that a z-score falls within 2 standard deviations of the mean on either side, that is, between z = -2 and z = 2, is approximately 95%. 3. The probability that a z-score falls within 3 standard deviations of the mean on either side, that is, between z = -3 and z = 3, is approximately 99.7%. A Random Variable X is said to have a normal distribution with standard mean and variance is given by: X N ( , 2 ) or 12 x 1 f ( x) e 2 This function can be used for normal distributions with a mean different from 0 and a standard deviation different from 1. We can convert this into a standardized normal variable by converting each of its scores into standard scores, given by: z x 2 In a Normal Distribution, the mean and standard deviation are given below. What is the probability of obtaining a value greater than 30? 25 5 Answer: 0.1587 In a Normal Distribution, the mean and standard deviation are given below. What is the probability of obtaining a value less than 20? 15 2.3 Answer: 0.9850 In a Normal Distribution, the mean and standard deviation are given below. What is the probability of obtaining a value greater than 120? 100 16 Answer: 0.1056 In a Normal Distribution, the mean and standard deviation are given below. What is the probability of obtaining a value between -6 and 9? 3 4 Answer: 0.9210 In a Normal Distribution, the mean and standard deviation are given below. Find the x-value for which the area to the left of that value is 0.04. 100 16 Answer: 72 In a Normal Distribution, the mean and standard deviation are given below. Find the x-value for which the area to the right of that value is 0.6. 3 4 Answer: 1.99 In a 1905 study, R. Pearl determined that the brain weights of Swedish men are approximately normally distributed with a mean and standard deviation given below. Determine the percentage of Swedish men with brain weights between 1.50 and 1.70 kg. 1.4 0.11 Answer: 0.178 or 17.8% The annual wages, excluding board, of US farm laborers in 1926 are normally distributed with a mean and standard deviation given below. In 1926, what is the probability that a US farm laborer will have an annual wage of at least $400? $586 $97 Answer: 0.972 The masses of a certain species of bird have a normal distribution with mean 50 grams. If 10 percent of the birds weigh more than 60 grams, find the variance of their masses. Answer: 61 grams The annual rainfall in a town has a normal distribution with standard deviation of 5 inches. If the rainfall is over 20 inches for a third of the years, find the mean rainfall. Answer: 17.8 inches The weights of a certain species of bird are normally distributed with mean 0.8 kg and standard deviation 0.12 kg. Find the probability that the weight of a randomly chosen bird of the species lies between 0.74 kg and 0.95 kg. Answer: 0.586 M02/HL1/11 The random variable X is normally distributed and P( X 10) 0.670 P( X 12) 0.937. Answer: 9.19 Find E(X). M03/HL1/14 Z is the standard normal variable with mean 0 and variance 1. Find the value of a such that: P( Z a) 0.75 Answer: a = 1.15 M01/HL1/13 The speed of cars at a certain point on a straight road are normally distributed with mean and standard deviation. 15% of the cars travelled at speeds greater than 90 km/hr and 12% of them at speeds less than 40 km/hr. Find the mean and standard deviation. Answer: Mean = 66.6 Standard Deviation = 22.6 SPEC06/HL1/8 A certain type of vegetable has a weight which follows a normal distribution with mean 450 grams and standard deviation 50 grams. a) In a load of 2000 of these vegetables, calculate the expected number with a weight greater than 525 grams. b) Find the upper quartile of the distribution. Answers: 134, 484 N06/HL1/9 The weights in grams of bread loaves sold at a supermarket are normally distributed with mean 200 grams. The weights of 88% of the loaves are less than 220 grams. Find the standard deviation. Answer: Standard Deviation = 17.0 M06/HL1/8