Continuous Probability Distributions

Transcription

1 Continuous Probability Distributions Chapter 7 McGraw-Hill/Irwin Copyright 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

2 제 6 장연속확률분포와정규분포 7-2

3 제 6 장연속확률분포와정규분포 7-3

4 LEARNING OBJECTIVES LO1. Understand the difference between discrete and continuous distributions. LO2. Compute the mean and the standard deviation for a uniform distribution. LO3. Compute probabilities by using the uniform distribution. LO4. List the characteristics of the normal probability distribution. LO5. Define and calculate z values. LO6. Determine the probability an observation is between two points on a normal probability distribution. LO7. Determine the probability an observation is above (or below) a point on a normal probability distribution. 7-4

5 Types of Random Variables Learning Objective 1 Distinguish between discrete and continuous probability distributions. DISCRETE RANDOM VARIABLE A random variable that can assume only certain clearly separated values. It is usually the result of counting something. CONTINUOUS RANDOM VARIABLE can assume an infinite number of values within a given range. It is usually the result of some type of measurement 7-5

6 LO1 Discrete Random Variables DISCRETE RANDOM VARIABLE A random variable that can assume only certain clearly separated values. It is usually the result of counting something. EXAMPLES 1. The number of students in a class. 2. The number of children in a family. 3. The number of cars entering a carwash in a hour. 4. Number of home mortgages approved by Coastal Federal Bank last week. 7-6

7 LO1 Continuous Random Variables CONTINUOUS RANDOM VARIABLE can assume an infinite number of values within a given range. It is usually the result of some type of measurement EXAMPLES The length of each song on the latest Tim McGraw album. The weight of each student in this class. The temperature outside as you are reading this book. The amount of money earned by each of the more than 750 players currently on Major League Baseball team rosters. 7-7

8 LO1 The Uniform Distribution The uniform probability distribution is perhaps the simplest distribution for a continuous random variable. This distribution is rectangular in shape and is defined by minimum and maximum values. 7-8

9 The Uniform Distribution Mean and Standard Deviation Learning Objective 2 Compute the mean and the standard deviation for a uniform probability distribution. 7-9

10 The Uniform Distribution An Example Learning Objective 3 Compute probabilities by using the uniform probability distribution. Southwest Arizona State University provides bus service to students while they are on campus. A bus arrives at the North Main Street and College Drive stop every 30 minutes between 6 A.M. and 11 P.M. during weekdays. Students arrive at the bus stop at random times. The time that a student waits is uniformly distributed from 0 to 30 minutes. 1. Draw a graph of this distribution. 2. Show that the area of this uniform distribution is How long will a student typically have to wait for a bus? In other words what is the mean waiting time? What is the standard deviation of the waiting times? 4. What is the probability a student will wait more than 25 minutes 5. What is the probability a student will wait between 10 and 20 minutes? 7-10

11 LO3 The Uniform Distribution - Example 1. Draw a graph of this distribution. 7-11

12 LO3 The Uniform Distribution - Example 2. Show that the area of this distribution is

13 The Uniform Distribution - Example LO3 3. How long will a student typically have to wait for a bus? In other words what is the mean waiting time? What is the standard deviation of the waiting times? 7-13

14 The Uniform Distribution - Example LO3 4. What is the probability a student will wait more than 25 minutes? P(25 Wait Time 30) (height)(base) 1 (5) (30 0)

15 The Uniform Distribution - Example LO3 5. What is the P(10 Wait Time 20) probability a 1 student will wait between 10 and 20 (30 0) minutes? (height)(base) (10) 7-15

16 Characteristics of a Normal Probability Distribution Learning Objective 4 List the characteristics of the normal probability distribution. 1. It is bell-shaped and has a single peak at the center of the distribution. 2. It is symmetrical about the mean 3. It is asymptotic: The curve gets closer and closer to the X-axis but never actually touches it. 4. The location of a normal distribution is determined by the mean,, the dispersion or spread of the distribution is determined by the standard deviation,σ. 5. The arithmetic mean, median, and mode are equal 6. The total area under the curve is 1.00; half the area under the normal curve is to the right of this center point and the other half to the left of it 7-16

17 LO4 The Normal Distribution - Graphically 7-17

18 The Family of Normal Distribution LO4 Equal Means and Different Standard Deviations Different Means and Standard Deviations Different Means and Equal Standard Deviations 7-18

19 The Standard Normal Probability Distribution Learning Objective 5 Define and calculate z values. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is also called the z distribution. A z-value is the signed distance between a selected value, designated X, and the population mean, divided by the population standard deviation, σ. The formula is: 7-19

20 LO5 Areas Under the Normal Curve 7-20

21 The Normal Distribution Example Learning Objective 6 Determine the probability an observation is between two points on a normal probability distribution. The weekly incomes of shift foremen in the glass industry follow the normal probability distribution with a mean of $1,000 and a standard deviation of $100. What is the z value for the income, let s call it X, of a foreman who earns $1,100 per week? For a foreman who earns $900 per week? 7-21

22 LO6 The Empirical Rule About 68 percent of the area under the normal curve is within one standard deviation of the mean. About 95 percent is within two standard deviations of the mean. Practically all is within three standard deviations of the mean. 7-22

23 LO6 The Empirical Rule - Example As part of its quality assurance program, the Autolite Battery Company conducts tests on battery life. For a particular D-cell alkaline battery, the mean life is 19 hours. The useful life of the battery follows a normal distribution with a standard deviation of 1.2 hours. Answer the following questions. 1. About 68 percent of the batteries failed between what two values? 2. About 95 percent of the batteries failed between what two values? 3. Virtually all of the batteries failed between what two values? 7-23

24 Normal Distribution Finding Probabilities LO6 In an earlier example we reported that the mean weekly income of a shift foreman in the glass industry is normally distributed with a mean of $1,000 and a standard deviation of $100. What is the likelihood of selecting a foreman whose weekly income is between $1,000 and $1,100? 7-24

25 LO6 Normal Distribution Finding Probabilities 7-25

26 LO6 Finding Areas for Z Using Excel The Excel function =NORMDIST(x,Mean,Standard_dev,Cumu) =NORMDIST(1100,1000,100,true) generates area (probability) from Z=1 and below 7-26

27 Normal Distribution Finding Probabilities (Example 2) LO6 Refer to the information regarding the weekly income of shift foremen in the glass industry. The distribution of weekly incomes follows the normal probability distribution with a mean of $1,000 and a standard deviation of $100. What is the probability of selecting a shift foreman in the glass industry whose income is: Between $790 and $1,000? 7-27

28 Normal Distribution Finding Probabilities (Example 3) LO6 Refer to the information regarding the weekly income of shift foremen in the glass industry. The distribution of weekly incomes follows the normal probability distribution with a mean of $1,000 and a standard deviation of $100. What is the probability of selecting a shift foreman in the glass industry whose income is: Less than $790? 7-28

29 Normal Distribution Finding Probabilities (Example 4) LO6 Refer to the information regarding the weekly income of shift foremen in the glass industry. The distribution of weekly incomes follows the normal probability distribution with a mean of $1,000 and a standard deviation of $100. What is the probability of selecting a shift foreman in the glass industry whose income is: Between $840 and $1,200? 7-29

30 Normal Distribution Finding LO6 Probabilities (Example 5) Refer to the information regarding the weekly income of shift foremen in the glass industry. The distribution of weekly incomes follows the normal probability distribution with a mean of $1,000 and a standard deviation of $100. What is the probability of selecting a shift foreman in the glass industry whose income is: Between $1,150 and $1,

31 Using Z in Finding X Given Area Example Learning Objective 7 Determine the probability an observation is above (or below) a normal probability distribution. Layton Tire and Rubber Company wishes to set a minimum mileage guarantee on its new MX100 tire. Tests reveal the mean mileage is 67,900 with a standard deviation of 2,050 miles and that the distribution of miles follows the normal probability distribution. Layton wants to set the minimum guaranteed mileage so that no more than 4 percent of the tires will have to be replaced. What minimum guaranteed mileage should Layton announce? 7-31

32 Using Z in Finding X Given Area - Example LO7 Solve X using the formula: x - z x 67,900 2,050 The valueof The area between 67,900 and x is , found by Using Appendix B.1, the area closest to is , which gives a z alue of z is found using the 4% information Then substituting into the equation : x - 67,900, then solving for x 2, (2,050) x - 67,900 x 67, (2,050) x 64,

33 Using Z in Finding X Given Area - Excel LO6 7-33

34 제 6 장연속확률분포와정규분포 7-34

35 제 6 장연속확률분포와정규분포 7-35

36 제 6 장연속확률분포와정규분포 7-36

37 제 6 장연속확률분포와정규분포 7-37

38 제 6 장연속확률분포와정규분포 7-38

39 제 6 장연속확률분포와정규분포 7-39

40 제 6 장연속확률분포와정규분포 7-40

41 제 6 장연속확률분포와정규분포 7-41

42 제 6 장연속확률분포와정규분포 7-42