Normal Distribution v The most commonly used distribution
Normal Distribution v The most commonly used distribution for continuous random variable f(x) μ x v “Bell Shaped” v Symmetrical v Mean, Median and Mode are Equal BUS 304 – Chapter 5 Normal Probability Theory 1
Examples of Normal Distribution v Examples: § Heights and weights of people § Students’ test scores § Amounts of rainfall in a year § Average temperature in a year v Characteristics: § Most of the time, the value is around the mean; § The higher (lower) a value is from the mean, the less likely it happens; § The probability a value is higher than the mean is the same as the probability that the value is lower than the mean; § It could be extremely higher or lower than the mean. BUS 304 – Chapter 5 Normal Probability Theory 2
Understanding the Curve v One striking assumption of continuous random variables is: § The curve doesn’t represent a probability. § The area below the curve does! v We don’t study the probability that the random variable takes one value. We care about the range. f(x) v E. g. we don’t care about the likelihood of a person’s income is $50, 102 per year. We care the percentage a person’ income falls in the range of $50, 000~$60, 000 x a BUS 304 – Chapter 5 Normal Probability Theory b 3
Characteristics of Normal Distribution f(x) σ μ μ v Location is determined by the mean (expected value), μ v Spread is determined by the standard deviation, σ v The shape of the normal distribution is decided based on μ and σ v In theory, the random variable has an infinite range: from - to + BUS 304 – Chapter 5 Normal Probability Theory 4
Rules to assign probability for normal distribution v Events: § Typical events we care about: § x<100 lbs, x>300 K, 50<x<60, etc. § Think: what are the complement events for the above events? v Basic Rules: 1. P(- < x < + ) = 1 ~ x is certainly between - and + f(x) 2. P(x = a) = 0 for any given a. (It is almost impossible to find that x equals to a specific value. E. g. it is impossible to find a person with a height exactly as 6. 1334 feet. ) 3. P(x > ) = P(x < ) = P(x ) = 0. 5 – symmetry. 0. μ 5 BUS 304 – Chapter 5 Normal Probability Theory x 5
Determine the probability for normal distribution (2) v More rules: 3. P(x< +b)= 0. 5 + P( <x< +b) f(x) P ( x +b ) 0. 5 +b x 4. P( - b<x< ) = P( <x< +b) BUS 304 – Chapter 5 Normal Probability Theory 6
Exercise v Assume that =3, P(3<x< 4)=0. 2, determine the following probabilities: f(x) 2 3 4 x § P(x< 4)? § P(2 <x< 3)? § P(2<x< 4)? § P(x<2)? BUS 304 – Chapter 5 Normal Probability Theory 7
Empirical Rules v The probability that x falls in to the range (μ-1σ, μ+1σ) is 0. 6826 (or 68. 26%) v That is, the area under the curve is 0. 6826 v Derivation: § A half of the area will be 0. 6826/2 = 0. 3413 f(x) 34. 13% f(x) σ μ-1σ σ μ μ+1σ x f(x) 68. 26% 34. 13% μ-1σ μ μ+1σ σ x μ-1σ μ μ+1σ BUS 304 – Chapter 5 Normal Probability Theory x 8
Empirical Rules v More Derivation: v Derivation: § The tailed area is 0. 5 -0. 3413 =0. 1587 f(x) § The complement of tailed area is 1 -0. 1587=0. 8413 f(x) 15. 87% 84. 13% σ μ-1σ μ σ μ+1σ x f(x) μ-1σ σ μ μ+1σ x f(x) μ-1σ μ 15. 87% 84. 13% σ σ μ+1σ x μ-1σ μ μ+1σ BUS 304 – Chapter 5 Normal Probability Theory x 9
Exercise A supermarket (Walmart) wants to reward its customers. Based on the past experience, the accumulative expenditure of each customer in a month follows a normal distribution with mean $700 and standard deviation $300. § The criterion to reward the customer is that if a customer spend more than $1000 will receive a reward of free digital camera. § If you randomly select a customer to check whether he/she receives a digital camera, what is the probability that you will get a confirmative answer? § If all the customers whose accumulative expenditure in Oct exceeds $400 will receive a free burger from Mc. Donalds, what is the probability that you meet a customer gets a burger? BUS 304 – Chapter 5 Normal Probability Theory 10
More Empirical Rule v μ ± 2σ covers about 95% of x’s 2σ 2σ μ v μ ± 3σ covers about 99. 7% of x’s x 95. 44% 3σ 3σ μ x 99. 72% BUS 304 – Chapter 5 Normal Probability Theory 11
Exercise v Problem 5. 40 (Page 210) BUS 304 – Chapter 5 Normal Probability Theory 12
Use the normal table to compute the probability for any range: v Concept 1: z score § The z score of x is computed based on 1. the value of x, 2. the mean of the normal distribution , and 3. the standard deviation of the normal distribution . § Z score represents how many standard deviation x is from the mean. • E. g. if x= , z =0. no deviation. if x = + , z = 1. one standard deviation above. if x = - , z = -1. one standard deviation below. BUS 304 – Chapter 5 Normal Probability Theory 13
Use the table to compute the probability v Standard normal table: (Page 595) § Use the z score to figure out the probability. § The z-score has to be positive. § The table shows the probability between the mean to the value. f(x) The area is the probability P( <x<a). a x BUS 304 – Chapter 5 Normal Probability Theory 14
How to use the table? v Steps: 1. 2. 3. 4. 5. f(x) Calculate the z-score. (z = (4 -3)/1. 5=0. 667) Round the z-score to two decimals (z =0. 67) Find the integer and first decimal part from the row Find the 2 nd decimal from the column Find the corresponding value the probability. =1. 5 3 4 x z … 0. 06 0. 07 0. 08 … … … … 0. 5 … 0. 2123 0. 2157 0. 2190 … 0. 6 … 0. 2454 0. 2486 0. 2517 … 0. 2764 0. 2794 0. 2823 … BUS 304 – Chapter 5 Normal Probability Theory … … … 15 …
Other cases: drawing helps! Case 1: 3<x<4 f(x) P=0. 2486 Case 3: x<4 f(x) x 3 4 4 Case 4: x>2 f(x) x P=0. 5+0. 2486=0. 7486 Case 2: 2<x<3 f(x) 3 Case 5: x>4 x 4 P=0. 5 -0. 2486=0. 2514 Case 6: x<2 f(x) P=0. 2486 x 2 3 x x P=0. 5+0. 2486=0. 7486 2 3 P=0. 5 -0. 2486=0. 2514 BUS 304 – Chapter 5 Normal Probability Theory 16
Advanced Cases f(x) x x 3 4 5 f(x) ? x 1 2 3 BUS 304 – Chapter 5 Normal Probability Theory 17
More advanced case f(x) x 2 5 Use the normal table, you should be able to figure out any probability! BUS 304 – Chapter 5 Normal Probability Theory 18
Exercise. v Page 210 § Problem 5. 45 § Problem 5. 52 BUS 304 – Chapter 5 Normal Probability Theory 19
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