Lecture Business statistics in practice (7/e): Chapter 6 - Bowerman, O'Connell, Murphree

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Chapter 6 - Continuous random variables. After mastering the material in this chapter, you will be able to: Define a continuous probability distribution and explain how it is used, use the uniform distribution to compute probabilities, describe the properties of the normal distribution and use a cumulative normal table, use the normal distribution to compute probabilities,...

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Nội dung Text: Lecture Business statistics in practice (7/e): Chapter 6 - Bowerman, O'Connell, Murphree

Chapter 6 Continuous Random Variables McGrawHill/Irwin Copyright © 2014 by The McGrawHill Companies, Inc. All rights reserved.
Continuous Random Variables 6.1 Continuous Probability Distributions 6.2 The Uniform Distribution 6.3 The Normal Probability Distribution 6.4 Approximating the Binomial Distribution by Using the Normal Distribution (Optional) 6.5 The Exponential Distribution (Optional) 6.6 The Normal Probability Plot (Optional) 62
LO6-1: Define a continuous probability distribution and explain how it is used. 6.1 Continuous Probability Distributions A continuous random variable may assume any numerical value in one or more intervals ◦For example, time spent waiting in line Use a continuous probability distribution to assign probabilities to intervals of values The curve f(x) is the continuous probability distribution of the random variable x if the probability that x will be in a specified interval of numbers is the area under the curve f(x) corresponding to the interval 63
LO6-1 Properties of Continuous Probability Distributions  Properties of f(x): f(x) is a continuous function such that 1. f(x) ≥ 0 for all x 2. The total area under the f(x) curve is equal to 1  Essential point: An area under a continuous probability distribution is a probability 64
LO6-2: Use the uniform distribution to compute probabilities. 6.2 The Uniform Distribution  If c and d are numbers on the real line (c
LO6-3: Describe the properties of the normal distribution and use a cumulative normal table. 6.3 The Normal Probability Distribution The normal probability distribution is defined by the equation 2 1 x 1 2 f( x) = e σ 2π for all values x on the real number line ◦ σ is the mean and σ is the standard deviation ◦ π = 3.14159… and e = 2.71828 is the base of natural logarithms 66
LO6-3 The Standard Normal Table The standard normal table is a table that lists the area under the standard normal curve to the right of negative infinity up to the z value of interest ◦Table 6.1 ◦Other standard normal tables will display the area between the mean of zero and the z value of interest Always look at the accompanying figure for guidance on how to use the table 67
LO6-4: Use the normal distribution to compute probabilities. Find P(0 ≤ z ≤ 1)  Find the area listed in the table corresponding to a z value of 1.00  Starting from the top of the far left column, go down to “1.0”  Read across the row z = 1.0 until under the column headed by “.00”  The area is in the cell that is the intersection of this row with this column  As listed in the table, the area is 0.8413, so  P(– ≤ z ≤ 1) = 0.8413  P(0 ≤ z ≤ 1) = P(– ≤ z ≤ 1) – 0.5000 = 0.3413 68
LO6-5: Find population values that correspond to specified normal Finding Normal Probabilities distribution probabilities. 1. Formulate the problem in terms of x values 2. Calculate the corresponding z values, and restate the problem in terms of these z values 3. Find the required areas under the standard normal curve by using the table Note: It is always useful to draw a picture showing the required areas before using the normal table 69
LO6-5 Some Areas under the Standard Normal Curve 610 Figure 6.15
LO6-5 Finding a Tolerance Interval Finding a tolerance interval [ k ] that contains 99% of the measurements in a normal population 611 Figure 6.23
LO6-6: Use the normal distribution to 6.4 Approximating the Binomial Distribution approximate binomial probabilities (Optional). by Using the Normal Distribution (Optional) The figure below shows several binomial distributions Can see that as n gets larger and as p gets closer to 0.5, the graph of the binomial distribution tends to have the symmetrical, bellshaped, form of the normal curve 612 Figure 6.24
LO6-6 Normal Approximation to the Binomial Continued Generalize observation from last slide for large p Suppose x is a binomial random variable, where n is the number of trials, each having a probability of success p ◦Then the probability of failure is 1 – p If n and p are such that np 5 and n(1–p) 5, then x is approximately normal with np and np 1 p 613
LO6-7: Use the exponential distribution to compute probabilities (Optional). 6.5 The Exponential Distribution (Optional) Suppose that some event occurs as a Poisson process ◦That is, the number of times an event occurs is a Poisson random variable Let x be the random variable of the interval between successive occurrences of the event ◦The interval can be some unit of time or space Then x is described by the exponential distribution ◦With parameter , which is the mean number of events that can occur per given interval 614
LO6-8: Use a normal probability plot to help decide whether data 6.6 The Normal Probability Plot come from a normal distribution (Optional). A graphic used to visually check to see if sample data comes from a normal distribution A straight line indicates a normal distribution The more curved the line, the less normal the data is 615