CHAPTER TOPICS STATISTIK & PROBABILITAS. Copyright 2017 By. Ir. Arthur Daniel Limantara, MM, MT.

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2 CHAPTER TOPICS The Normal Distribution The Standardized Normal Distribution Evaluating the Normality Assumption The Uniform Distribution The Exponential Distribution 2

3 CONTINUOUS PROBABILITY DISTRIBUTIONS Continuous Random Variable Values from interval of numbers Absence of gaps Continuous Probability Distribution Distribution of continuous random variable Most Important Continuous Probability Distribution The normal distribution 3

4 THE NORMAL DISTRIBUTION Bell Shaped Symmetrical Mean, Median and Mode are Equal Interquartile Range Equals 1.33 σ Random Variable Has Infinite Range f(x) Mean Median Mode X 4

5 THE MATHEMATICAL MODEL 5 1 f X e 2 (1/ 2) X / f X : density of random variable X ; e : population mean : population standard deviation X : value of random variable 2 X

6 MANY NORMAL DISTRIBUTIONS There are an Infinite Number of Normal Distributions 6 Varying the Parameters and, We Obtain Different Normal Distributions

7 THE STANDARDIED NORMAL DISTRIBUTION When X is normally distributed with a mean and a standard deviation, follows a standardized (normalized) normal distribution with a mean 0 and a standard deviation 1. X f() f(x) X

8 FINDING PROBABILITIES Probability is the area under the curve! Pc X d? f(x) 8 c d X

9 WHICH TABLE TO USE? Infinitely Many Normal Distributions Means Infinitely Many Tables to Look Up! 9

10 SOLUTION: THE CUMULATIVE STANDARDIED NORMAL DISTRIBUTION Cumulative Standardized Normal Distribution Table (Portion) Probabilities 0 = Only One Table is Needed

11 STANDARDIING EXAMPLE Normal Distribution 10 X Standardized Normal Distribution X

12 EXAMPLE: X X Normal Distribution 10 P 2.9 X Standardized Normal Distribution X

13 EXAMPLE: Cumulative Standardized Normal Distribution Table (Portion) P 2.9 X = (continued)

14 14 P EXAMPLE: Cumulative Standardized Normal 2.9 X Distribution Table (Portion) = (continued)

15 NORMAL DISTRIBUTION IN PHSTAT PHStat Probability & Prob. Distributions Normal Example in Excel Spreadsheet Microsoft Excel Worksheet 15

16 EXAMPLE: P X X Normal Distribution 10 Standardized Normal Distribution X

17 EXAMPLE: P X (continued) 17 Cumulative Standardized Normal Distribution Table (Portion) =

18 FINDING VALUES FOR KNOWN PROBABILITIES What is Given Probability = ? Cumulative Standardized Normal Distribution Table (Portion)

19 RECOVERING X VALUES FOR KNOWN PROBABILITIES Normal Distribution 10 Standardized Normal Distribution ? X X

20 MORE EXAMPLES OF NORMAL DISTRIBUTION USING PHSTAT A set of final exam grades was found to be normally distributed with a mean of 73 and a standard deviation of 8. What is the probability of getting a grade no higher than 91 on this exam? 2 X N 73,8 P X 91? Mean 73 Standard Deviation 8 8 Probability for X <= X Value 91 Value 2.25 P(X<=91) X

21 MORE EXAMPLES OF NORMAL DISTRIBUTION USING PHSTAT (continued) What percentage of students scored between 65 and 89? 2 X N 73,8 P 65 X 89? Probability for a Range From X Value 65 To X Value 89 Value for 65-1 Value for 89 2 P(X<=65) P(X<=89) P(65<=X<=89) X

22 MORE EXAMPLES OF NORMAL DISTRIBUTION USING PHSTAT Only 5% of the students taking the test scored higher than what grade? 2 X N 73,8 P? X.05 (continued) Find X and Given Cum. Pctage. Cumulative Percentage 95.00% Value X Value ? = X

23 ASSESSING NORMALITY Not All Continuous Random Variables are Normally Distributed It is Important to Evaluate How Well the Data Set Seems to Be Adequately Approximated by a Normal Distribution 23

24 ASSESSING NORMALITY Construct Charts For small- or moderate-sized data sets, do the stem-andleaf display and box-and-whisker plot look symmetric? For large data sets, does the histogram or polygon appear bell-shaped? Compute Descriptive Summary Measures Do the mean, median and mode have similar values? Is the interquartile range approximately 1.33 σ? Is the range approximately 6 σ? (continued) 24

25 ASSESSING NORMALITY Observe the Distribution of the Data Set Do approximately 2/3 of the observations lie between mean 1 standard deviation? Do approximately 4/5 of the observations lie between mean 1.28 standard deviations? Do approximately 19/20 of the observations lie between mean 2 standard deviations? Evaluate Normal Probability Plot Do the points lie on or close to a straight line with positive slope? (continued) 25

26 ASSESSING NORMALITY Normal Probability Plot Arrange Data into Ordered Array Find Corresponding Standardized Normal Quantile Values Plot the Pairs of Points with Observed Data Values on the Vertical Axis and the Standardized Normal Quantile Values on the Horizontal Axis Evaluate the Plot for Evidence of Linearity (continued) 26

27 ASSESSING NORMALITY (continued) Normal Probability Plot for Normal Distribution X Look for Straight Line!

28 NORMAL PROBABILITY PLOT Left-Skewed Right-Skewed X 60 X Rectangular U-Shaped X 60 X

29 OBTAINING NORMAL PROBABILITY PLOT IN PHSTAT PHStat Probability & Prob. Distributions Normal Probability Plot Enter the range of the cells that contain the data in the Variable Cell Range window 29

30 THE UNIFORM DISTRIBUTION Properties: The probability of occurrence of a value is equally likely to occur anywhere in the range between the smallest value a and the largest value b Also called the rectangular distribution 30 2 a 2 b b a 2 12

31 THE UNIFORM 31 DISTRIBUTION The Probability Density Function 1 f X if a X b b a Application: Selection of random numbers E.g., A wooden wheel is spun on a horizontal surface and allowed to come to rest. What is the probability that a mark on the wheel will point to somewhere between the North and the East? P 90 0 X (continued)

32 EXPONENTIAL DISTRIBUTIONS P arrival tim e X 1 e X : any value of continuous random variable : the population average num ber of arrivals per unit of tim e 1/ : average tim e betw een arrivals e E.g., Drivers arriving at a toll bridge; customers arriving at an ATM machine X 32

33 EXPONENTIAL DISTRIBUTIONS Describes Time or Distance between Events Used for queues Density Function f(x) = 0.5 Parameters x 1 f x e (continued) = 2.0 X 33

34 EXAMPLE E.g., Customers arrive at the checkout line of a supermarket at the rate of 30 per hour. What is the probability that the arrival time between consecutive customers will be greater than 5 minutes? 3 0 X 5 / 6 0 h o u rs P arrival tim e > X 1 P arrival tim e X 1 1 e /

35 EXPONENTIAL DISTRIBUTION IN PHSTAT PHStat Probability & Prob. Distributions Exponential Example in Excel Spreadsheet Microsoft Excel Worksheet 35

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