Introduction to Business Statistics QM 120 Chapter 6

Transcription

1 DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 6 Spring 2008 Chapter 6: Continuous Probability Distribution 2 When a RV x is discrete, we can assign a positive probability to each value that x can take and get the probability distribution for x. Thesumof all probabilities biliti of all values of x is 1. Not all experiments result in RVs that are discrete. Continuous RV, such as heights, weights, lifetime of a particular product, experimental laboratory error etc. can assume infinitely many values corresponding to points on line interval 1

2 Chapter 6: Continuous Probability Distribution 3 Suppose we have a set of measurements on a continuous RV, and we want to create a relative frequency histogram to describe their distribution. For a small number of measurements, we can use small number of classes Chapter 6: Continuous Probability Distribution 4 For a larger number of measurements, we must use a larger number of classes and reduce the class width 2

3 Chapter 6: Continuous Probability Distribution 5 As the number of measurements become very large, the class width become very narrow, the relative frequency histogram appears more and more like a smooth curve. This smooth curve describes the probability distribution of a continuous random variable. Chapter 6: Continuous Probability Distribution 6 A continuous RV can take on any of an infinite number of values on the real line The depth or density of the probability, which varies with x, maybe described by a mathematical formula f(x), called the probability density function for the RV x. 3

4 Chapter 6: Continuous Probability Distribution 7 The probability distribution f(x); P(a< x <b) is equal to the shaded area under the curve Several important properties of continuous probability distribution parallel their discrete part. Chapter 6: Continuous Probability Distribution 8 Just as the sum of discrete probabilities is equal to 1, i.e. ΣP(x) = 1, and the probability that x falls into a certain interval can be found by summing the probabilities in that interval, continuous probability distributions have the following two characteristics: The area under a continuous probability distribution is equal to 1. The probability that x will fall into a particular interval say, from a to b is equal to the area under the curve between the two points a and b 4

5 Chapter 6: Continuous Probability Distribution 9 There is a very important difference between the continuous case and the discrete one which is: P(x = a) = 0 P(a < x < b) = P(a x b) P( x < a) = P( x a) Chapter 6: The Normal Distribution 10 Among many continuous probability distributions, the normal distribution is the most important and widely used one. A large number of real world phenomena are either exactly or approximately normally distributed. The normal distribution or Gaussian distribution is given by abell shaped curve. A continuous RV x that has a normal distribution is said to be anormalrvwithameanμ and a standard deviation σ or simply x~n(μ,σ). 5

6 Chapter 6: The Normal Distribution 11 Note that not all the bell shaped curves represent a normal distribution Chapter 6: The Normal Distribution 12 The normal probability distribution, when plotted, gives a bell shaped curve such that The total area under the curve is 1.0. The curve is symmetric around the mean. The two tails of the curve extended indefinitely. The mean μ and the standard deviation σ are the parameters of the normal distribution. Given the values of these two parameters, we can find the area under the normal curve for any interval. 6

7 Chapter 6: The Normal Distribution 13 There is not just one normal distribution curve but rather a family of normal distribution. Each different set of values of μ and σ gives different normal curve. The value of μ determines the center of the curve on the Horizontal axis and the value of σ gives the spread of the normal curve Chapter 6: The Normal Distribution 14 Like all other distributions, the normal probability distribution can be expressed by a mathematical function f( x) = 1 e σ 2π 2 1 x μ 2 σ in which the probability that x falls between a and b is the integral of the above function from a to b, i.e. 2 b 1 x μ 1 2 σ P( a < x< b) = e dx σ 2π a 7

8 Chapter 6: The Normal Distribution 15 However we will not use the formula to find the probability. instead, we will use Table VII of Appendix C. Chapter 6: The Standard Normal Distribution 16 The standard normal distribution is a special case of the normal distribution where the μ is zero and the σ is 1. The RV that possesses the standard normal distribution is denoted by z and it is called z values or z scores. μ=0 σ= z 8

9 Chapter 6: The Standard Normal Distribution 17 Since μ is zero and the σ is 1 for the standard normal, a specific value of z gives the distance between the mean and the point represented by z in terms of the standard deviation. The z values to the right side of the mean are positive and thoseontheleftarenegativebuttheareaunderthecurveis always positive. For a value of z = 2, we are 2 standard deviations from the mean (to the right). Similarly, for z = 2, we are 2 standard deviations from the mean (to the left) Chapter 6: The Standard Normal Distribution 18 9

10 Chapter 6: The Standard Normal Distribution 19 Table VII of Appendix C, list the area under the standard normal curve between z = 0 to the values of z to To read the standard normal table, we always start at z = 0, which represents the mean. Always remember that the normal curve is symmetric, that is the area from 0 to any positive z value is equal to the area from 0 to that value in the negative side. Example: Find the area under the standard normal curve betweenz=0toz=1.95 Chapter 6: The Standard Normal Distribution 20 Example: Find the area under the standard normal curve between z = 2.17 to z = 0 10

11 Chapter 6: The Standard Normal Distribution 21 Example: Find the following areas under the standard normal curve. a) Area to the right of z = 2.32 b) Area to the left of z = 1.54 Chapter 6: The Standard Normal Distribution 22 Example: Find the following probabilities for the standard normal curve. a) P(1.19 < z < 2.12) b) P( 1.56 < z < 2.31) c) P(z >.75) 11

12 Chapter 6: The Standard Normal Distribution 23 Remember: When two points are on the same side of the mean, first find the areas between the mean and each of the two points. Then, subtract the smaller area from the larger area When two points are on different side of the mean, first find the areas between the mean and each of the two points. Then, add these two area Chapter 6: The Standard Normal Distribution 24 Empirical rule 12

13 Chapter 6: The Standard Normal Distribution 25 Example: Find the following probabilities for the standard normal curve. a) P(0 < z < 5.65) b) P( z < 5.3) Chapter 6: Standardizing a Normal Distribution 26 As it was shown in the previous section, Table VII of Appendix C can be used only to find the areas under the standard normal curve. However, in real world applications, most of continuous RVs that are normally distributed come with mean and standard deviation different from 0 and 1, respectively. What shall we do? Is there any way to bring μ to zero and σ to 1. Yes,itcanbedonebysubtractingμ from x and dividing the result by σ (standardizing) 13

14 Chapter 6: Standardizing a Normal Distribution 27 Standardizing x: For a normal RV x with mean μ and standard deviation σ. The standardized RV z can be found using the following formula z = x μ σ Chapter 6: Standardizing a Normal Distribution 28 Example: For the following data [3.2, 1.9, 2.7, 4.3, 3.5, 2.8, 3.9, 4.4, 1.5, 1.8], a) find the mean and standard deviation b) standardize d each value in thesample x n 4.3 X i 3.5 i= μ = = n σ = Transformation z 14

15 Chapter 6: Standardizing a Normal Distribution 29 Example: Let x be a continuous RV that has a normal distribution with a mean 50 and a standard deviation of 10. Convert the following x values to z values a) x = 55 b) x = 35 c) x = 50 Chapter 6: Standardizing a Normal Distribution 30 To find the area between two values of x for a normal distribution Convert both values of x to their respective z values Find the area under the standard normal curve between those two values Example: Let x be a continuous RV that is normally distributed with a mean of 25 and a standard deviation of 4. Find the area a) betweenx=25 ee and x=32 b) betweenx=18 ee and x=34 15

16 Chapter 6: Standardizing a Normal Distribution 31 Example: Let x be a continuous RV that has a normal distribution with a mean 40 and a standard deviation of 5. Find the following probabilities a) P(x > 55) b) P(x < 49) Chapter 6: Standardizing a Normal Distribution 32 Example: Let x be a continuous RV that has a normal distribution with a mean 50 and a standard deviation of 8. Find the probability P(30 < x < 39) 16

17 Chapter 6: Standardizing a Normal Distribution 33 Example: Let x be a continuous RV that has a normal distribution with a mean 80 and a standard deviation of 12. Find the following probabilities a) P(70 <x < 135) b) P(x < 27) Chapter 6: Application of the Normal Distribution 34 Example: According to the U.S. Bureau of Census, the mean income of all U.S. families was $43,237 in Assume that the 1991 incomes of all U.S. families have a normal distribution with a mean of $43,237 and a standard deviation of $10,500. Find the probability that the 1991 income of a randomly selected U.S. family was between $30,000 and $50,

18 Chapter 6: Application of the Normal Distribution 35 Example: A racing car is one of the many toys manufactured by Mack Corporation. The assembly time for this toy follows a normal distribution with a mean of 55 minutes and a standard deviation of 4 minutes. The company closes at 5 P.M. every day. If one worker starts assembling a racing car at 4 P.M., what is the probability that she will finish this job before the company closes for the day? Chapter 6: Application of the Normal Distribution 36 Example: Hupper Corporation produces many types of soft drinks, including Orange Cola. The filling machines are adjusted to pour 12 ounces of soda in each 12 ounce can of Orange Cola. However, the actual amount of soda poured into each can is not exactly 12 ounces; it varies from can to can. It is found that the net amount of soda in such a can has a normal distribution with a mean of 12 ounces and a standard deviation of.015 ounces. (a) What is the probability that a randomly selected can of Orange Cola contains to ounces of soda? (b) What percentage of the Orange Cola cans contain to ounces of soda? 18

19 Chapter 6: Application of the Normal Distribution 37 Example: The life span of a calculator manufactured by lntal Corporation has a normal distribution with a mean of 54 months and a standard deviation of 8 months. The company guarantees that any calculator that starts malfunctioning within 36 months of the purchase will be replaced by a new one. About what percentage of such calculators made by this company are expected to be replaced? Chapter 6: Determining the z and x values 38 Determining the z and x values when the area under the normal distribution is known We reverse the procedure of finding the area under the normal curve for a specific value of z or x to finding a specific value of z or x for a known area under the normal curve. Example: Find a point z such that the area under the standard normal curve between 0 and z is.4251 and the value of z is positive 19

20 Chapter 6: Determining the z and x values 39 Chapter 6: Determining the z and x values 40 Example: Find the value of z such that the area under the standard normal curve in the right tail is

21 Chapter 6: Determining the z and x values 41 Example: Find the value of z such that the area under the standard normal curve in the left tail is.050. Chapter 6: Determining the z and x values 42 Finding an x value for a normal distribution: To find an x value when an area under a normal distribution curve is given, we do the following 1. Find the z value corresponding to that x value from the standard normal curve. 2. Transform the z value to x by substituting the values of μ, σ, and z in the following formula x = μ + σ z 21

22 Chapter 6: Determining the z and x values 43 Example: Recall the calculators example, it is known that the life of a calculator manufactured by Intal Corporation has a normal distributionwithameanof54monthsandastandarddeviation of 8 months. What should the warranty period be to replace a malfunctioning calculator if the company does not want to replace more than 1 % of all the calculators sold Chapter 6: Determining the z and x values 44 Example: Most business schools require that every applicant for admission to a graduate degree program take the GMAT. Suppose the GMAT scores of all students have a normal distribution with a mean of550 and a standard deviation of 90. Matt Sanger is planning to take this test soon. What should his score be on this test so that only 10% of all the examinees score higher than he does? 22

23 Chapter 6: Determining the z and x values 45 Example: Find the value of z such that 0.95 of the area is within ±z standard deviations of the mean Chapter 6: The Uniform Distribution 46 In many of the discrete cases, the sample points were assigned equal probabilities (equally likely). Tossing a die Tossing a coin For a continuous RV, there is an infinite number of values in the sample space but with equally likely outcomes too (sometimes). If a short exists in a 5 m of an electrical wire. If a safety inspector wants to inspect a plant over a time interval 23

24 47 Chapter 6: The Uniform Distribution Definition: Let the RV x be defined over the interval c to d probability function given by: E( x) = μ = Var( x) = σ x f ( x) dx 2 = x 2 f ( x) dx ) [ E( x ] 2 Then x is said to be uniform random variable. Theprobability b can be found with 1 f ( x) =, c - x d d c (c < d) with 48 Chapter 6: The Uniform Distribution Definition: Let the RV x be defined over the interval c to d (c < d) with probability function given by: 1 f ( x) = d c 0 if c x d otherwise Then x is said to be uniform random variable. Theprobability bili can be found with ih b P( a < x < b) = f ( x) dx, with c a < b d a b - a = d - c 24

25 Chapter 6: The Uniform Distribution 49 Example: A special case of the continuous uniform distribution occurs whenever c = 0 and d = 1. The distribution is called the standardized uniform distribution. Determine the probability function, f(x), the mean, E(x), the variance, Var(x), of the standard uniform. Chapter 6: The Uniform Distribution 50 Example: A uniform random variable is defined over the interval 1 to 6. a) What is the probability b that the RV assumes a value between 2 and 4? b) What is the mean and the variance of that RV? 25