Department of Quantitative Methods & Information Systems. Business Statistics. Chapter 6 Normal Probability Distribution QMIS 120. Dr.

Transcription

1 Department of Quantitative Methods & Information Systems Business Statistics Chapter 6 Normal Probability Distribution QMIS 120 Dr. Mohammad Zainal

2 Chapter Goals After completing this chapter, you should be able to: Convert values from any normal distribution to a standardized z-score Find probabilities using a normal distribution table Apply the normal distribution to business problems QMIS 120, By Dr. M. Zainal Chap 6-2

3 Probability Distributions Probability Distributions Discrete Continuous Ch. 5 Probability Probability Ch. 6 Distributions Distributions Binomial Poisson Hypergeometric Normal Uniform Exponential QMIS 120, By Dr. M. Zainal Chap 6-3

4 Continuous Probability Distributions A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure accurately. QMIS 120, By Dr. M. Zainal Chap 6-4

5 The Normal Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Exponential QMIS 120, By Dr. M. Zainal Chap 6-5

6 The Normal Distribution Bell Shaped Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: + to f(x) μ σ Mean = Median = Mode x QMIS 120, By Dr. M. Zainal Chap 6-6

7 Many Normal Distributions By varying the parameters μ and σ, we obtain different normal distributions QMIS 120, By Dr. M. Zainal Chap 6-7

8 The Normal Distribution Shape f(x) Changing μ shifts the distribution left or right. σ Changing σ increases or decreases the spread. μ x QMIS 120, By Dr. M. Zainal Chap 6-8

9 The Normal Distribution Shape QMIS 120, By Dr. M. Zainal Chap 6-9

10 Finding Normal Probabilities Probability is measured by the area under the curve f(x) P ( a x b ) a b x QMIS 120, By Dr. M. Zainal Chap 6-10

11 Probability as Area Under the Curve The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below f(x) P( x μ) 0.5 P(μ x ) μ P( x ) 1.0 x QMIS 120, By Dr. M. Zainal Chap 6-11

12 Empirical Rules What can we say about the distribution of values around the mean? There are some general rules: f(x) μ ± 1σ encloses about 68% of x s σ σ μ 1σ μ μ+1σ 68.26% x QMIS 120, By Dr. M. Zainal Chap 6-12

13 The Empirical Rule μ ± 2σ covers about 95% of x s (continued) μ ± 3σ covers about 99.7% of x s 2σ μ 2σ x 3σ μ 3σ x 95.44% 99.72% QMIS 120, By Dr. M. Zainal Chap 6-13

14 Importance of the Rule If a value is about 3 or more standard deviations away from the mean in a normal distribution, then it is an outlier The chance that a value that far or farther away from the mean is highly unlikely, given that particular mean and standard deviation QMIS 120, By Dr. M. Zainal Chap 6-14

15 The Standard Normal Distribution Also known as the z distribution Mean is defined to be 0 Standard Deviation is 1 f(z) 1 0 z Values above the mean have positive z-values, values below the mean have negative z-values QMIS 120, By Dr. M. Zainal Chap 6-15

16 The Standard Normal Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (z) Need to transform x units into z units z x σ μ QMIS 120, By Dr. M. Zainal Chap 6-16

17 Example If x is distributed normally with mean of 100 and standard deviation of 50, the z value for x = 250 is z x σ μ This says that x = 250 is three standard deviations (3 increments of 50 units) above the mean of 100. QMIS 120, By Dr. M. Zainal Chap 6-17

18 Comparing x and z units μ = 100 σ = x 3.0 z Note that the distribution is the same, only the scale has changed. We can express the problem in original units (x) or in standardized units (z) QMIS 120, By Dr. M. Zainal Chap 6-18

19 The Standard Normal Table The Standard Normal table in the textbook gives the probability from the mean (zero) up to a desired value for z Example: P(0 < z < 2.00) = z QMIS 120, By Dr. M. Zainal Chap 6-19

20 The Standard Normal Table z (continued) The column gives the value of z to the second decimal point The row shows the value of z to the first decimal point P(0 < z < 2.00) 2.0 = The value within the table gives the probability from z = 0 up to the desired z value QMIS 120, By Dr. M. Zainal Chap 6-20

21 General Procedure for Finding Probabilities To find P(a < x < b) when x is distributed normally: Draw the normal curve for the problem in terms of x Translate x-values to z-values Use the Standard Normal Table QMIS 120, By Dr. M. Zainal Chap 6-21

22 Z Table example Suppose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) Calculate z-values: x QMIS 120, By Dr. M. Zainal Chap 6-22

23 Z Table example Suppose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) (continued) = 8 = x z P(8 < x < 8.6) QMIS 120, By Dr. M. Zainal Chap 6-23

24 Solution: Finding P(0 < z < 0.12) Standard Normal Probability Table (Portion) z Z QMIS 120, By Dr. M. Zainal Chap 6-24

25 Finding Normal Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x < 8.6) Z QMIS 120, By Dr. M. Zainal Chap 6-25

26 Finding Normal Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x < 8.6) (continued) P(x < 8.6) 0.00 Z QMIS 120, By Dr. M. Zainal Chap 6-26

27 Upper Tail Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x > 8.6) Z QMIS 120, By Dr. M. Zainal Chap 6-27

28 Upper Tail Probabilities Now Find P(x > 8.6) (continued) 0 Z 0 Z QMIS 120, By Dr. M. Zainal Chap 6-28

29 Lower Tail Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(7.4 < x < 8) Z QMIS 120, By Dr. M. Zainal Chap 6-29

30 Lower Tail Probabilities Now Find P(7.4 < x < 8) (continued) The Normal distribution is symmetric, so we use the same table even if z-values are negative: P(7.4 < x < 8) Z QMIS 120, By Dr. M. Zainal Chap 6-30

31 Z Table example Example: Find the area under the standard normal curve between z = 0 to z = 1.95 Example: Find the area under the standard normal curve between z = to z = 0 QMIS 120, By Dr. M. Zainal Chap 6-31

32 Z Table example 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 = QMIS 120, By Dr. M. Zainal Chap 6-32

33 Z Table example Example: Find the following areas under the standard normal curve. a) P(1.19 < z < 2.12) b) P(-1.56 < z < 2.31) c) P(z > -.75) QMIS 120, By Dr. M. Zainal Chap 6-33

34 Z Table example Example: Find the following areas under the standard normal curve. a) P(0 < z < 5.65) b) P( z < - 5.3) QMIS 120, By Dr. M. Zainal Chap 6-34

35 Z Table example 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) QMIS 120, By Dr. M. Zainal Chap 6-35

36 Z Table example Example: The assembly time for a racing car toy follows a normal distribution with a mean of 55 minutes and a standard deviation of 4 minutes. The factory closes at 5 PM every day. If one worker starts assembling that car at 4 PM, what is the probability that she will finish this job before the company closes for the day? QMIS 120, By Dr. M. Zainal Chap 6-36

37 Z Table example Example: The lifetime of a calculator manufactured by a company 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. What percentage of such calculators are expected to be replaced? QMIS 120, By Dr. M. Zainal Chap 6-37

38 Determining the z and x values 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. z QMIS 120, By Dr. M. Zainal Chap 6-38

39 Determining the z and x values 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 QMIS 120, By Dr. M. Zainal Chap 6-39

40 Finding x for a normal dist. To find an x value when an area under a normal distribution curve is given, we do the following Find the z value corresponding to that x value from the standard normal curve. Transform the z value to x by substituting the values of,, and z in the following formula x + z QMIS 120, By Dr. M. Zainal Chap 6-40

41 Finding x for a normal dist. 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 of 550 and a standard deviation of 90. You are planning to take this test. What should your score be in this test so that only 10% of all the examinees score higher than he does? QMIS 120, By Dr. M. Zainal Chap 6-41

42 Finding x for a normal dist. Example: Recall the calculators example, it is known that the life of a calculator manufactured by a factory has a normal distribution with a mean of 54 months and a standard deviation of 8 months. What should the warranty period be to replace a malfunctioning calculator if the company does not want to replace more than 0.5 % of all the calculators sold? QMIS 120, By Dr. M. Zainal Chap 6-42

43 Copyright The materials of this presentation were mostly taken from the PowerPoint files accompanied Business Statistics: A Decision-Making Approach, 7e 2008 Prentice-Hall, Inc. Chap 6-43 QMIS 120, By Dr. M. Zainal