Applied Reliability Applied Reliability

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Applied Reliability ------------------------------------------------------------------------------------------------------------ Applied Reliability"

Transcription

1 Applied Reliability Techniques for Reliability Analysis with Applied Reliability Tools (ART) (an EXCEL Add-In) and JMP Software AM216 Class 6 Notes Santa Clara University Copyright David C. Trindade, Ph. D. STAT-TECH Spring 2010

2 2 Reference Applied Reliability Quality Control Material Based on Chapter 9 in Text: Applied Reliability, 2 nd ed. by Paul A. Tobias and David C. Trindade Published 1995 by Chapman & Hall, New York ISBN No

3 3 Table of Contents Combinatorics Multiplication Rule Permutations (Spreadsheet Function) Combinations (Spreadsheet Function) Binomial Distribution Binomial Experiment Probability Calculations Parameter Estimation and Application Confidence Intervals Poisson Distribution Approximation to Binomial Probability Calculations Parameter Estimation and Application Confidence Intervals Hypergeometric Distribution Probability Calculations Application Acceptance Sampling Risks Operating Characteristic Curves Sampling Plan LTPD Plans Adjusting Sampling Plans Minimum Sample Size Plans AOQ Curve and AOQL Control Charts for Reliability Cumulative Count Control Charts

4 4 Multiplication Rule for Outcomes If the first experiment can result in n 1 possible outcomes, and for each outcome, the second experiment can result in n 2 possible outcomes, then there are a total of n 1 n 2 possible outcomes for the two experiments together. The multiplication rule is extendable to any number of experiments in a sequence. Example: On a restaurant menu, there are five appetizers, seven entrees, and six desserts. How many different three course meals are possible? Answer: 5x7x6 = 210.

5 5 Combinatorics Permutations The number of ways to arrange n objects in order: If all n objects are used: n(n-1)(n-2) 1 = n! If only r objects of n are used: 1 ( 2) 1 n n n n r r terms n! n r! Examples: Your child receives five birthday presents. How many different ways can your child open them? Answer: 5! = 5x4x3x2x1 = 120. Two musical notes are needed to complete the song. In one octave, thirteen possible notes exist. Without repeating the same note, how many different ways can the song be ended? Answer: 13x12 = 156.

6 6 Combinations The number of ways of combining r objects taken from n available in which order is not important: Note: 0! = 1 Combinatorics n n 1 n r 1 n! r! r! n r! n 0 1 n n 1 Examples: Eight players offer to play for the company golf team in a tournament. How many different four person teams can I make up? Answer: 8!/(4!4!) = (8x7x6x5)/(4x3x2x1) = 70. There are seven blade servers and five open positions in the rack. How many ways can any five be chosen for installing in any order on the five rack positions? (Two will be left out.) Answer: 7!/(5!2!) = (7x6x5x4x3)/(5x4x3x2x1) = 21. n r

7 7 Class Project #1 Combinatorics: Consider the three objects: A, B, C. 1. Taking all three objects, how many permutations are there? List them: 2. Taking two objects at a time, how many permutations are there? List them: 3. Taking two objects at a time, how many combinations are there? List them:

8 8 Class Project #1 Combinatorics: Consider the three objects: A, B, C. 1. Taking all three objects, how many permutations are there? 3! = 3x2x1 = 6. List them: ABC ACB BAC BCA CAB CBA 2. Taking two objects at a time, how many permutations are there? 3!/(3-2)! = (3x2x1)/(1) = 6. List them: AB AC BA BC CA CB 3. Taking two objects at a time, how many combinations are there? List them: 3!/(2!1!) = 3. AB AC BC

9 9 Combinatorics Class Project #2 1. How many different eleven letter words can be formed from the letters in the word MISSISSIPPI? 2. Let S denote a survival and F, a failure. Given two failures among five items on stress, how many different ways can the two failures occur on the five objects? (Hint: One way is SSSFF.) List all words. 3.How many different ways can r failures occur among n units on stress? This problem is analogous to asking how many different words of n letters can be formed from r F's and (n-r) S's.

10 10 Combinatorics Class Project #2 1. How many different eleven letter words can be formed from the letters in the word MISSISSIPPI? 11! 4!4!2! 34, Let S denote a survival and F, a failure. Given two failures among five items on stress, how many different ways can the two failures occur on the five objects? (Hint: One way is SSSFF.) List all words. 5! 5 2 2!3! 10 SSSFF SSFSF SSFFS SFFSS SFSFS SFSSF FFSSS FSFSS FSSFS FSSSF 3.How many different ways can r failures occur among n units on stress? r! n! n n r! r

11 11 Permutations and Combinations in EXCEL

12 12 Binomial Experiment Necessary Conditions 1. Only two possible outcomes (success, failure). 2. Fixed number of trials, n. 3. Constant probability of success, p, across trials. 4. Trial outcomes are independent.

13 13 Binomial Distribution Derivation Let p be the probability of failure, given by the CDF at time t; then 1 - p is the probability of survival. Consider n units on test for time t. Designate a failure by F and a survival by S. For the specific sequence with x failures and (n-x) survivals: P F F... F S S... S The probability of this sequence is: Since there are p x 1 n x n x n x x p 1 p x p nx ways of x failures occurring among n units, the probability of x failures among n units on stress is given by the binomial distribution probability mass function (PMF)

14 Probability of x Failures Applied Reliability 14 Binomial Distribution Probability Function 20 units are placed on stress for 100 hours. The CDF at 100 hours is 20%. The graph below shows the binomial distribution probabilities for x failures at 100 hours, x = 0,1,2, 25.0% n = 20, p = 0.2: 20.0% 15.0% 10.0% 5.0% 0.0% x = Number of Failures

15 15 Binomial Distribution Cumulative Function 0 1 leastone) (at 1 ) ( ) ( ) ( 1) ( ) ( Note ) ( ) ( (1) (0) ) ( 1 0 P P x X P x X P k P x X P x X P k P x P P P x X P n x k x k To get the cumulative probability of x or less events, sum the individual binomial probabilities.

16 Probability of x or less Failures Applied Reliability 16 Binomial Distribution Cumulative Function 20 units are placed on stress for 100 hours. The CDF at 100 hours is 20%. The graph below shows the binomial distribution cumulative probabilities of x or less failures at 100 hours, x = 0,1,2, n = 20, p = 0.2: Binomial CDF x = Number of Failures

17 17 Binomial Distribution Properties Mean or expected number: Expected number of successes (or failures) in n trials with probability of success (or failure) p per trial equals np Variance: 2 np 1 p Standard Deviation: np 1p

18 18 Binomial Distribution Calculations in EXCEL =BINOMDIST(x, n, p, 0 or 1) Fourth argument is: 0 or false = individual probability 1 or true = cumulative probability

19 19 Class Project #3 Binomial Distribution: A fair coin is tossed one hundred times. 1. What is the expected number of heads? 2. What is the standard deviation? 3. What is the probability of exactly fifty heads? 4. What is the probability of at most fifty heads? 5. What is the probability of 60 or more heads? 6. What is the probability of 40 or less heads? 7. What is the probability of greater than 40 but less than 60 heads? 8. What is the probability of 100 heads?

20 20 Class Project #3 Binomial Distribution: A fair coin is tossed one hundred times. 1. What is the expected number of heads? np = 100(0.5) = What is the standard deviation? 3. What is the probability of exactly fifty heads? 4. What is the probability of at most fifty heads? 5. What is the probability of 60 or more heads? 6. What is the probability of 40 or less heads? 7. What is the probability of greater than 40 but less than 60 heads? P 8. What is the probability of 100 heads? np 1 p 100(0.5)(0.5) P X % P X % P X % P X % 41 X (2.844) 94.31% P X x

21 21 Binomial Distribution PMF n = 100, p = 0.5

22 22 Cumulative Binomial Distribution n = 100, p = 0.5

23 23 Binomial Distribution Parameter Estimation Estimation of p: In a binomial experiment, the probability of failure p at a given time is estimated by the proportion of failures observed out of n units. Thus, pˆ x / n where x is the number of failures: x = 0,1,2,...,n. Note this is the same statistic for estimating the CDF F(t) at time t, that is, pˆ Fˆ t Confidence Intervals on p: Instead of just a point estimate for p, we may specify a confidence interval that captures the true (population) p with a certain level of confidence.

24 24 Binomial Distribution 95% Confidence Intervals for Proportions Clopper and Pearson Chart

25 25 Binomial Distribution Exact Confidence Limits Using EXCEL: Exact confidence limits for the population parameter p may be found using the Inverse Beta function in EXCEL. Lower limit: 0 for x = 0; =BETAINV(a/2, x, n-x+1) for x between 1 and n. Upper limit: =BETAINV(1-a/2, x+1, n-x) for x between 1 and n-1; 1 for x = n. Example: n =100, x = 2, The estimate of p = 2/100 = 0.02 = 2%. Thus, a 90% 2-sided confidence interval on the population p is: LCL: = BETAINV(.05, 2, 99) = 0.36% UCL: = BETAINV(.95, 3, 98) = 6.16%

26 26 Class Project #4 Binomial Distribution: Fifty devices are stressed for 168 hrs. The probability of a device failing by 168 hrs is 0.05 or 5%. Determine: A. The probability all devices survive 168 hours. B. The expected number of failures. C. The probability of at least one failure. D. At the end of the test there are 10 failures. Provide a 95% confidence interval for the population failure proportion. Does the interval include the assumed 5% failure probability?

27 27 Class Project #4 Binomial Distribution: Fifty devices are stressed for 168 hrs. The probability of a device failing 168 hrs is 0.05 or 5%. Determine: A. The probability all devices survive 168 hours. PX ( 0) B. The expected number of failures. np = 50x0.05 = 2.5 C. The probability of at least one failure. P( X 1) 1 P( X 0) D. At the end of the test there are 10 failures. Provide a 95% confidence interval for the population failure proportion. Does the interval include the assumed 5% failure probability? From the Clopper Pearson chart, we get the interval (10% to 34%), which does not include the 5% failure probability.

28 28 The Poisson Distribution Approximation to Binomial Distribution For n large and p small, the Poisson distribution provides an excellent approximation to the binomial distribution. Let l = n x p, the expected number. The probability of exactly x occurrences is: P x l x e l x! The mean or expected number is l. The variance is also l. The standard deviation is l. Approximation is good for large n and small p, such that l= n x p is less than, roughly, 7.

29 Probability Function Applied Reliability 29 The Poisson Distribution Approximation to Binomial: There are ninety units placed on stress for 100 hours. If the probability of failure is 0.01 for an individual unit, what is the probability of 2 or less units failing? Comparison of Binomial and Poisson Probabilitiess n = 90, p = 0.01, l = np = Number of Failures Binomial Poisson We want P( X 2) P P Binomial Solution: P 0 P1 2 X Poisson Solution: P X

30 30 Poisson Distribution Applications of Poisson Distribution The Poisson is also a distribution on its own to model situations where the probability of a single event over a period of time or space is constant. For example, for calculations involving density, such as the number of defects per wafer, the Poisson distribution (with l = the mean number of defects for a given area) can be used to model the defect distribution. The Poisson has been applied to the number of phone calls in a given period, the number of repairs in time, the number of bugs in software, the number of raisins in a box of cereal, the number of people in a queue, the number of flaws per yard of insulated wire, the number of misprints per page, and so on.

31 31 Poisson Distribution Calculations in EXCEL =POISSON(x, l, 0 or 1)

32 32 Class Project #5 Poisson Distribution: The average defect density is 0.01 defects per square inch. The wafer area is 200 square inches. Ten wafers are randomly selected from a lot of fifty. What is: 1. The expected number of defects per wafer? 2. The expected number of defects for the ten wafers? 3. The probability any wafer is defect free? 4. The probability any wafer has at least one defect? 5. The probability any wafer has exactly five defects? 6. The probability all ten wafers are defect free? 7. The probability exactly two of the ten wafers are defect free?

33 33 Poisson Distribution: Class Project #5 The average defect density is 0.01 defects per square inch. The wafer area is 200 square inches. Ten wafers are randomly selected from a lot of fifty. 1. The expected number of defects per wafer? np = 200x0.01 = The expected number of defects for the ten wafers? 10x2 = The probability any wafer is defect free? P 2 0 e The probability any wafer has at least one defect? 2 1 P(0) 1 e The probability any wafer has exactly five defects? e ! 6. The probability all ten wafers are defect free? e e 0 7. The probability exactly two of the ten wafers are defect free? 2 8 Binomial: P(2) 2 e 1 e Poisson: l = np = 10(0.135) = e P(2)

34 34 Poisson Distribution Estimation of l The estimate of the population parameter l is just the average observed number of occurrences over the collection of time periods, samples, objects, lines, etc. Confidence Intervals for l Tables for the 90%, 95%, and 99% confidence levels for various observed counts are shown next. Note the confidence limits depend only on the confidence level and the observed count. Conversion to a CI for l may be done using these tables. Exact confidence limits can also be calculated directly using functions in EXCEL.

35 35 Poisson Distribution Table of Confidence Limits Based on Observed Count Confidence Level alpha Observed Count 90% % 99% X lower limit upper limit lower limit upper limit lower limit upper limit

36 36 Poisson Distribution Exact Confidence Limits Using EXCEL: Exact confidence limits for the population parameter l may be found using the inverse chi-squared distribution (CHIINV(probability; degrees of freedom)) in EXCEL. Lower limit: 0 for x = 0; =0.5* CHIINV(1-a/2, 2x) for x > 0. Upper limit: =0.5* CHIINV(a/2, 2*(x+1)) for x > 0. Example: Observe 2 bugs in 100 lines of code, The estimate of l: x = 2 bugs/100 LOC Thus, a 90% 2-sided confidence interval on the population l = bugs/100 lines of code is: LCL: = 0.5* CHIINV(.95, 4) = 0.36 UCL: = 0.5* CHIINV(.05, 6) = 6.30

37 37 Poisson Confidence Limits Distinguish between Poisson confidence limits on: the total count (as in the previous table) the average count per unit of measure (as a % in ART). EXAMPLE: Suppose we open three boxes of cereal and find a total of 9 raisins. The average estimate lis 3 raisins per box. The 90% confidence interval on: the total count for three boxes is 4.70 to the average per box l is 1.57 to (CI width is 3.67.) Suppose we open ten boxes and find a total of 30 raisins, The average number of raisins per box is 3, as before. The 90% confidence interval on: the total count for ten boxes is to the average per box l is to (CI width for l is now 1.91 since the sample is larger.) The width of the confidence interval depends on the total count.

38 38 Class Project #6 Poisson Distribution: 1000 randomly selected lines of code are inspected. Twenty three bugs are found. Assuming a Poisson distribution: A. What's the estimate of the expected number of bugs per 1000 lines of code (KLOC) in the population? B. Provide a 90% confidence interval on the expected number of bugs per KLOC in the population.

39 39 Class Project #6 Poisson Distribution: 1000 randomly selected lines of code are inspected. Twenty three bugs are found. Assuming a Poisson distribution: A. What's the estimate of the expected number of bugs per 1000 lines of code (KLOC) in the population? Answer: 23 B. Provide a 90% confidence interval on the expected number of bugs per KLOC in the population. Answer: From Table of CL for Poisson, CI is to Using EXCEL: LCL: = 0.5* CHIINV(0.95, 46) = UCL: = 0.5* CHIINV(0.05, 48) = 32.59

40 40 Poisson Distribution No Limit on Defect Count The Poisson distribution has no limit (like n for the binomial distribution) on the number of defects per unit of measure that can be counted. Defective Unit versus Defects A unit is defective if it has one or more defects (a defect is defined as a nonconformance to specifications). Thus, defective units are treated separately from the number of defects per unit or area. A defective unit is handled by the binomial or hypergeometric distributions. For calculations involving density (e.g., defects per wafer, failures per period, etc.), the Poisson distribution is often used.

41 41 Hypergeometric Distribution Application: Fixed number of trials. Two possible outcomes: Success or failure. Sampling from finite population. Hence, probability of success with each trial changes and events are not independent. Example: What s the probability of drawing two aces in a row from a well shuffled deck of cards? Example: 20 lines of code in 100 are defective (contain a bug). Randomly select 10 lines. What s the probability of finding 0,1,2, defective lines?

42 42 Hypergeometric Distribution Formula: The probability of getting exactly x rejects in a sample of size n drawn from a finite lot of size N that contains a total of m rejects is P where x = 0,1,2,3,... m Nm x nx x N For n<0.1n, binomial distribution is a good approximation for P(X) where p = m/n. n

43 43 Class Project #7 Hypergeometric Distribution: Your dresser drawer contains twenty socks, half red and half black. It s early morning and still dark. You don t want to turn on the light and disturb your spouse. So in darkness you randomly select two socks. 1. What is the probability that you choose a matched pair of black socks? 2. What is the probability that you choose a matched pair of either color?

44 44 Class Project #7 Hypergeometric Distribution: 1. What is the probability that you choose a matched pair of black socks? P m Nm x nx N n x What is the probability that you choose a matched pair of either color? P(matched pair, either color) = 2xP(2) = 0.474

45 45 Hypergeometric Distribution Calculations in EXCEL = hypgeomdist( x; n; m; N)

46 46 Fisher s Exact Test In two groups we observe failure proportions 1/6 and 7/9. The small sample sizes are small, but we want to test whether Group 1 is significantly better than Group 2. The results are displayed in a 2x2 contingency table with margin totals: Group 1 Group 2 Totals Pass Fail Totals Is the population from which the first group was drawn significantly better than the second group population or is the difference likely to occur by chance alone? For small sample sizes, Fisher s Exact Test, based on the hypergeometric distribution, calculates the probability of observing differences equal to or greater than the results observed.

47 47 Fisher s Exact Test Group 1 Group 2 Totals Pass Fail Totals For both groups, note the total sample size is N = 15 and the total failures is m = 8. Assume a randomly drawn sample size of n = 6 for Group 1. The probability of getting exactly 1 failure out of the 8 possible is given by the hypergeometric distribution P

48 48 Fisher s Exact Test (Continued) An even more extreme event (zero failures in Group 1) with the same marginal totals is shown below. This probability is P Group 1 Group 2 Totals Pass Fail Totals So the one-sided probability of observing the results or stronger is the sum = % At ~96.5% confidence, the test results are significantly different

49 49 Class Project #8 Fisher s Exact Test In an experiment to compare different treatments, the old method produced four rejects out of twenty. In the second experiment, the new procedure resulted in zero rejects out of fifteen. How statistically significant is the improvement between the new and old?

50 50 Class Project #8 Fisher s Exact Test In an experiment to compare different treatments, the old method produced four rejects out of twenty. In the second experiment, the new procedure resulted in zero rejects out of fifteen. How statistically significant is the improvement between the new and old? New Old Totals Pass Fail Totals P =HYPGEOMDIST(0,4,35,15)=

51 51 Acceptance Sampling Risks Accept or reject a lot based on the results of looking at a sample of size n randomly drawn from a lot of size N. Several possible risks involved in deciding whether or not a lot (the population) has a maximum allowed percent defective level, say Y%: -Reject a good lot with percent defective below Y% and commit a Type 1, a, or producer's error. -Accept a bad lot with percent defective above Y% and commit a Type 2, b, or consumer's error. Matrix of Possible Choices Population Value (% Defective) < Y % > Y % Decision on Lot Accept Correct Type II error Reject Type I error Correct

52 Probability of Acceptance Applied Reliability 52 Operating Characteristic (O.C.) Curve Plot of PA versus p: For a given sampling plan (sample size n and acceptance number c), the probability of acceptance will depend on the incoming lot percent defective p. A plot of the probability of acceptance P A versus the lot percent defective p is called an O.C. curve. OC Curve 1.00 n = 50 c = % 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% Lot Percent Defective

53 53 Calculations for an O.C. Curve Method: Use binomial distribution with n, varying p, to calculate probability of acceptance, that is, P( X < c ): Example: Sketch the O.C. curve for a sample size n = 100 and acceptance number c = 2. The probability of acceptance is: P X P P P P 0 1p 1 1 nn1 n1 2 P 2 p 1p 2 With n = 100, now try different p values, using ART. For p = 0.01, the probability of acceptance P is = For p = 0.05, the probability of acceptance P is = For p = 0.10, the probability of acceptance P is = n P np p n2

54 Probability of Acceptance Probability of Acceptance Applied Reliability 54 OC Curves Fixed c, Varying Sample Sizes n: c = n = 100 n = n = n = % 5% 10% 15% 20% 25% Lot Percent Defective Fixed n, Varying Acceptance Numbers c: c = 0 c = 1 c = 2 c = 3 n = % 5% 10% 15% 20% 25% Lot Percent Defective

55 55 O.C. Curve Calculations Using ART

56 56 Sampling Plan A sampling plan is a specified sample size n and acceptance number c. A sampling plan generates a specific O.C. curve. The sampling plan is determined by specifying four values, equivalent to two points on the O.C. curve: 1. The p value for an incoming lot for which the probability of acceptance is very high (e.g., 95% or producer s risk a = 5%). 2. The p value for an incoming lot for which the probability of rejection is very high ( e.g., 90% or consumer s risk b = 10%). These four numbers (two p values and associated risks) uniquely determine the sampling plan. We assume the lot size N is large relative to n, or equivalently, that we are sampling from a process, rather than from lots. This assumption allows using simpler binomial distribution for calculations rather than the more complex hypergeometric distribution. Several procedures exist to generate sampling plans: published tables, nomograph, computer programs, iterative calculations, etc.

57 Sampling Plan Nomograph 57

58 58 Generating a Sampling Plan Using ART AQL 2%, a 5%; RQL 8%, b 10%

59 59 LTPD Sampling Plans What if our primary concern was viewing sampling plans that would assure rejection with high probability, say 90%, of a defect level that was the maximum value a consumer could tolerate. Such plans are called LTPD (lot tolerance percent defective) plans. The next table is based on a 10% consumer s risk at various rejectable quality levels.

60 60 LTPD Table Consumer s Risk b = 10%

61 61 Adjusting Sampling Plans Reducing the Sample Size: Suppose the qualification sampling plan is to accept on 3 or less rejects out of 300 units. Because of the cost or lack of availability of 300 units, we wish to reduce the sample size while holding the consumer s risk constant. What is the sample size if 2 or less rejects are allowed, or 1 or less, or even zero?

62 Adjusting Sampling Plans 62 Procedure: Refer to figures following. 1. Find the graph corresponding to the present acceptance number, say Find the intersection of the sample size on the horizontal x-axis with the diagonal line in the graph labeled with the probability of acceptance value 10, corresponding to a 10% consumer s risk. 3. Read, on the vertical y-axis, the percent defective value associated with the 10% consumer s risk. 4. Find the graph with a lower acceptance number and reverse the above procedure. 5. Using the percent defective value previously found for the consumer s risk on the y-axis of the new graph, find its intersection with the labeled 10 line. 6. Dropping vertically at this intersection point to the x - axis, read off the new sample size for this lower acceptance number.

63 Graphs for Adjusting Sampling Plans 63

64 Graphs for Adjusting Sampling Plans 64

65 65 Class Project #9 Adjusting a Qualification Sampling Plan: The qualification plan calls for allowing two rejects out of two hundred units. The engineer needs to reduce the sample size of the study. If he wants to keep the same defect level at which the consumer s risk is 10% for lot acceptance, what is the necessary sample size, allowing only one reject. What is the LTPD?

66 66 Class Project #9 Adjusting a Qualification Sampling Plan: The qualification plan calls for allowing two rejects out of two hundred units. The engineer needs to reduce the sample size of the study. If he wants to keep the same defect level at which the consumer s risk is 10% for lot acceptance, what is the necessary sample size, allowing only one reject. What is the LTPD? From Figure 9.15 (c = 3), 10% consumer s risk is at 2.2% defective. From Figure 9.13 (c = 1), at 2.2% defective, the 10% consumer s risk corresponds to sample size 175 units.

67 67 Minimum Sample Size Plans The smallest sample size for a plan having a specified rejectable quality level occurs when the acceptance number is zero.

68 68 Minimum Sample Size Plans Zero Failures Protection Against Consumer's Risk: For zero failures and rejection probability (1 - b), the minimum sample size to reject a lot with fraction defective value p is given by: n ln b ln 1 p Plot of Minimum Sample Size Versus Fraction Defective

69 69 Using ART Minimum Size Sampling Plans

70 70 Class Project #10 Sampling Plans: We're willing to accept at most a 10% chance of permitting a failure fallout as high as 1.0%. A. If we allow up to three rejects, what sample size do we need? See Table 9.3 or ART Sampling Plans. B. It turns out, we have only 500 pieces available. What should our acceptance number be now? C. What is the minimum sample size we need, that is, allowing zero failures?

71 71 Class Project #10 Sampling Plans: We're willing to accept at most a 10% chance of permitting a failure fallout as high as 1.0%. A. Allowing up to three rejects, what sample size do we need? Table 9.3: 668 B. It turns out, we have only 500 pieces available. What should our acceptance number be now? Table 9.3: c = 1 for 390 units c = 2 for 533 units C. What is the minimum sample size we need, that is, allowing zero failures?. Table 9.3: 231 ART (Min. SS): 230 Formula: n ln0.1 ln(1.01) 229

72 72 Sampling Plans The Meaning of AQL and LTPD AQL: Acceptable quality level The incoming lot percent defective with typically a 95% chance of acceptance. If a lot is rejected (that is, the number of rejects is above the acceptance number), we state with 95% confidence that the lot defect level is above the AQL. LTPD: Lot tolerance percent defective The incoming lot percent defective with a 10% chance of acceptance. If a lot is accepted (that is, the number of rejects is equal to or below the acceptance number), we can state with 90% confidence that the lot defect level is below the LTPD.

73 73 Sampling Plans Outgoing Quality AOQ: Average outgoing quality Average level of defects shipped. Found by multiplying lot percent defective by probability of acceptance, that, is pxp. Generates AOQ curve. Assumes rejected lots are screened to make lot perfect. AOQL: Average outgoing quality limit Maximum value of AOQ curve. With 100% inspection of rejected lots, AOQ to customer is never any worse than AOQL.

74 74 Sampling Plans AOQ Curve and AOQL EXCEL Worksheet

75 75 AOQ Curves Using ART

76 76 Statistical Process Control Control Charts for Reliability If p is the historical process average, the upper control limit is given by UCL pz 1a p 1 n p Three-Sigma Control Chart for Binomial Proportions

77 77 Cumulative Count Control Chart Control Based on Good Units CCC chart concentrates on number of good units produced instead of number of defective items. Accumulated number of good units is plotted in time and compared to control limits such that a cumulative count of good units will fall inside the limits when a process is in control.

78 78 Cumulative Count Control Chart Example

79 79 Cumulative Count Control Chart Points Outside Control Limits Interpretation If the first failure occurs before the cumulative count exceeds the lower limit, the process is not capable of meeting the PPM level specified for the control limits. If the first failure occurs after the cumulative count exceeds the upper limit, the process is demonstrating even better performance than the PPM level specified for the control limits.

80 80 Cumulative Count Control Charts for Low PPM Centerline Set at Median: Median n LCL Lower Control Limit: ln ln p 1 a 1 / 2 p Upper Control Limit: n UCL ln ln a / 1 2 p Centerline and 90% LCL and UCL can be obtained from the following table by reading in the row for a given PPM under the columns labeled 0.5, 0.95, and 0.05, respectively.

81 81 Minimum Sample Sizes for Zero Rejects at Various Probabilities Probability of Zero Rejects PPM sigma: sigma: sigma: sigma:

82 82 Class Project #11 Cumulative Count Control Chart: A process is assumed to run at 500 PPM. A cumulative count control chart is desired for monitoring purposes. Find the centerline and upper and lower non-defective cumulative count values for a 95% control limit band.

83 83 Class Project #11 Cumulative Count Control Chart: A process is assumed to run at 500 PPM. A cumulative count control chart is desired for monitoring purposes. Find the centerline and upper and lower nondefective cumulative count values for a 95% control limit band. From Table: CL = 1386, LCL = 51, UCL = 7376 Formulas: CL ln( / 2) LCL 51 ln( ) UCL ln(0.05 / 2) ln( ) 7376 Compare to the tabulated values for Minimum Sample Sizes for Zero Rejects at Various Probabilities. Matches!

84 84 Using ART for Cumulative Count Control Charts

Acceptance Sampling OPRE

Acceptance Sampling OPRE Acceptance Sampling OPRE 6364 1 Acceptance Sampling Accept/reject entire lot based on sample results Created by Dodge and Romig during WWII Not consistent with TQM of Zero Defects Does not estimate the

More information

Acceptance Sampling OPRE

Acceptance Sampling OPRE Acceptance Sampling OPRE 6364 1 Acceptance Sampling Accept/reject entire lot based on sample results Created by Dodge and Romig during WWII Not consistent with TQM of Zero Defects Does not estimate the

More information

Confidence Intervals for Cp

Confidence Intervals for Cp Chapter 296 Confidence Intervals for Cp Introduction This routine calculates the sample size needed to obtain a specified width of a Cp confidence interval at a stated confidence level. Cp is a process

More information

WHERE DOES THE 10% CONDITION COME FROM?

WHERE DOES THE 10% CONDITION COME FROM? 1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

More information

Attributes Acceptance Sampling Understanding How it Works

Attributes Acceptance Sampling Understanding How it Works Attributes Acceptance Sampling Understanding How it Works Dan O Leary CBE, CQE, CRE, CSSBB, CIRM, LLC 603-209-0600 OmbuEnterprises@msn.com Copyright 2008, 2009 by, LLC Acceptance Sampling 1 Instructor

More information

in a Production Process

in a Production Process White Paper WP-2 Statistical Sampling in a Production Process A Practical Guide to Statistical Sampling A White Paper by David C. Wilson, MSEE December 23, 2004 WP-2: DCW_Statistical Sampling in a Production

More information

Minitab Guide. This packet contains: A Friendly Guide to Minitab. Minitab Step-By-Step

Minitab Guide. This packet contains: A Friendly Guide to Minitab. Minitab Step-By-Step Minitab Guide This packet contains: A Friendly Guide to Minitab An introduction to Minitab; including basic Minitab functions, how to create sets of data, and how to create and edit graphs of different

More information

Applied Reliability Page 1 APPLIED RELIABILITY. Techniques for Reliability Analysis

Applied Reliability Page 1 APPLIED RELIABILITY. Techniques for Reliability Analysis Applied Reliability Page 1 APPLIED RELIABILITY Techniques for Reliability Analysis with Applied Reliability Tools (ART) (an EXCEL Add-In) and JMP Software AM216 Class 1 Notes Santa Clara University Copyright

More information

Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

More information

Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...

Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test... Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................

More information

Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4) Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

More information

1 Introduction. 2 Basic Principles. 2.1 Multiplication Rule. [Ch 9] Counting Methods. 400 lecture note #9

1 Introduction. 2 Basic Principles. 2.1 Multiplication Rule. [Ch 9] Counting Methods. 400 lecture note #9 400 lecture note #9 [Ch 9] Counting Methods 1 Introduction In many discrete problems, we are confronted with the problem of counting. Here we develop tools which help us counting. Examples: o [9.1.2 (p.

More information

Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition

Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition Online Learning Centre Technology Step-by-Step - Excel Microsoft Excel is a spreadsheet software application

More information

Random variables, probability distributions, binomial random variable

Random variables, probability distributions, binomial random variable Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

More information

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution

More information

Discrete probability and the laws of chance

Discrete probability and the laws of chance Chapter 8 Discrete probability and the laws of chance 8.1 Introduction In this chapter we lay the groundwork for calculations and rules governing simple discrete probabilities. These steps will be essential

More information

4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions 4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

More information

Toss a coin twice. Let Y denote the number of heads.

Toss a coin twice. Let Y denote the number of heads. ! Let S be a discrete sample space with the set of elementary events denoted by E = {e i, i = 1, 2, 3 }. A random variable is a function Y(e i ) that assigns a real value to each elementary event, e i.

More information

Stats on the TI 83 and TI 84 Calculator

Stats on the TI 83 and TI 84 Calculator Stats on the TI 83 and TI 84 Calculator Entering the sample values STAT button Left bracket { Right bracket } Store (STO) List L1 Comma Enter Example: Sample data are {5, 10, 15, 20} 1. Press 2 ND and

More information

Discrete Random Variables and their Probability Distributions

Discrete Random Variables and their Probability Distributions CHAPTER 5 Discrete Random Variables and their Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Discrete Random Variable

More information

STATISTICAL QUALITY CONTROL (SQC)

STATISTICAL QUALITY CONTROL (SQC) Statistical Quality Control 1 SQC consists of two major areas: STATISTICAL QUALITY CONTOL (SQC) - Acceptance Sampling - Process Control or Control Charts Both of these statistical techniques may be applied

More information

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions... MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

More information

Random Variable: A function that assigns numerical values to all the outcomes in the sample space.

Random Variable: A function that assigns numerical values to all the outcomes in the sample space. STAT 509 Section 3.2: Discrete Random Variables Random Variable: A function that assigns numerical values to all the outcomes in the sample space. Notation: Capital letters (like Y) denote a random variable.

More information

4. Joint Distributions

4. Joint Distributions Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose

More information

Tests for Two Proportions

Tests for Two Proportions Chapter 200 Tests for Two Proportions Introduction This module computes power and sample size for hypothesis tests of the difference, ratio, or odds ratio of two independent proportions. The test statistics

More information

Learning Objectives. Understand how to select the correct control chart for an application. Know how to fill out and maintain a control chart.

Learning Objectives. Understand how to select the correct control chart for an application. Know how to fill out and maintain a control chart. CONTROL CHARTS Learning Objectives Understand how to select the correct control chart for an application. Know how to fill out and maintain a control chart. Know how to interpret a control chart to determine

More information

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution Sections 3-7, 3-8 The remaining discrete random

More information

STAT 35A HW2 Solutions

STAT 35A HW2 Solutions STAT 35A HW2 Solutions http://www.stat.ucla.edu/~dinov/courses_students.dir/09/spring/stat35.dir 1. A computer consulting firm presently has bids out on three projects. Let A i = { awarded project i },

More information

Gamma Distribution Fitting

Gamma Distribution Fitting Chapter 552 Gamma Distribution Fitting Introduction This module fits the gamma probability distributions to a complete or censored set of individual or grouped data values. It outputs various statistics

More information

Confidence Intervals for the Difference Between Two Means

Confidence Intervals for the Difference Between Two Means Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

More information

Math 202-0 Quizzes Winter 2009

Math 202-0 Quizzes Winter 2009 Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile

More information

Some special discrete probability distributions

Some special discrete probability distributions University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Some special discrete probability distributions Bernoulli random variable: It is a variable that

More information

7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

More information

Analysis of Variance. MINITAB User s Guide 2 3-1

Analysis of Variance. MINITAB User s Guide 2 3-1 3 Analysis of Variance Analysis of Variance Overview, 3-2 One-Way Analysis of Variance, 3-5 Two-Way Analysis of Variance, 3-11 Analysis of Means, 3-13 Overview of Balanced ANOVA and GLM, 3-18 Balanced

More information

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

More information

Online Materials: Counting Rules The Poisson Distribution

Online Materials: Counting Rules The Poisson Distribution Chapter 6 The Poisson Distribution Counting Rules When the outcomes of a chance experiment are equally likely, one way to determine the probability of some event E is to calculate number of outcomes for

More information

PROBLEM SET 1. For the first three answer true or false and explain your answer. A picture is often helpful.

PROBLEM SET 1. For the first three answer true or false and explain your answer. A picture is often helpful. PROBLEM SET 1 For the first three answer true or false and explain your answer. A picture is often helpful. 1. Suppose the significance level of a hypothesis test is α=0.05. If the p-value of the test

More information

Confidence Intervals for Cpk

Confidence Intervals for Cpk Chapter 297 Confidence Intervals for Cpk Introduction This routine calculates the sample size needed to obtain a specified width of a Cpk confidence interval at a stated confidence level. Cpk is a process

More information

Two-Sample T-Tests Assuming Equal Variance (Enter Means)

Two-Sample T-Tests Assuming Equal Variance (Enter Means) Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of

More information

5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.

5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives. The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution

More information

Confidence Intervals for the Area Under an ROC Curve

Confidence Intervals for the Area Under an ROC Curve Chapter 261 Confidence Intervals for the Area Under an ROC Curve Introduction Receiver operating characteristic (ROC) curves are used to assess the accuracy of a diagnostic test. The technique is used

More information

Normality Testing in Excel

Normality Testing in Excel Normality Testing in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com

More information

Statistical Functions in Excel

Statistical Functions in Excel Statistical Functions in Excel There are many statistical functions in Excel. Moreover, there are other functions that are not specified as statistical functions that are helpful in some statistical analyses.

More information

Probability Calculator

Probability Calculator Chapter 95 Introduction Most statisticians have a set of probability tables that they refer to in doing their statistical wor. This procedure provides you with a set of electronic statistical tables that

More information

Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

More information

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

More information

CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.

CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is. Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,

More information

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS UNIT I: RANDOM VARIABLES PART- A -TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1-x) 0

More information

Unit 29 Chi-Square Goodness-of-Fit Test

Unit 29 Chi-Square Goodness-of-Fit Test Unit 29 Chi-Square Goodness-of-Fit Test Objectives: To perform the chi-square hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni

More information

Data Analysis Tools. Tools for Summarizing Data

Data Analysis Tools. Tools for Summarizing Data Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool

More information

Statistical Intervals. Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage

Statistical Intervals. Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage 7 Statistical Intervals Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Confidence Intervals The CLT tells us that as the sample size n increases, the sample mean X is close to

More information

Chi-Square Test. Contingency Tables. Contingency Tables. Chi-Square Test for Independence. Chi-Square Tests for Goodnessof-Fit

Chi-Square Test. Contingency Tables. Contingency Tables. Chi-Square Test for Independence. Chi-Square Tests for Goodnessof-Fit Chi-Square Tests 15 Chapter Chi-Square Test for Independence Chi-Square Tests for Goodness Uniform Goodness- Poisson Goodness- Goodness Test ECDF Tests (Optional) McGraw-Hill/Irwin Copyright 2009 by The

More information

Matrices Worksheet. Adding the results together, using the matrices, gives

Matrices Worksheet. Adding the results together, using the matrices, gives Matrices Worksheet This worksheet is designed to help you increase your confidence in handling MATRICES. This worksheet contains both theory and exercises which cover. Introduction. Order, Addition and

More information

2. DATA AND EXERCISES (Geos2911 students please read page 8)

2. DATA AND EXERCISES (Geos2911 students please read page 8) 2. DATA AND EXERCISES (Geos2911 students please read page 8) 2.1 Data set The data set available to you is an Excel spreadsheet file called cyclones.xls. The file consists of 3 sheets. Only the third is

More information

Chi Square for Contingency Tables

Chi Square for Contingency Tables 2 x 2 Case Chi Square for Contingency Tables A test for p 1 = p 2 We have learned a confidence interval for p 1 p 2, the difference in the population proportions. We want a hypothesis testing procedure

More information

DISCRETE RANDOM VARIABLES

DISCRETE RANDOM VARIABLES DISCRETE RANDOM VARIABLES DISCRETE RANDOM VARIABLES Documents prepared for use in course B01.1305, New York University, Stern School of Business Definitions page 3 Discrete random variables are introduced

More information

HYPOTHESIS TESTING: POWER OF THE TEST

HYPOTHESIS TESTING: POWER OF THE TEST HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,

More information

Normal distribution. ) 2 /2σ. 2π σ

Normal distribution. ) 2 /2σ. 2π σ Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

More information

Common Tools for Displaying and Communicating Data for Process Improvement

Common Tools for Displaying and Communicating Data for Process Improvement Common Tools for Displaying and Communicating Data for Process Improvement Packet includes: Tool Use Page # Box and Whisker Plot Check Sheet Control Chart Histogram Pareto Diagram Run Chart Scatter Plot

More information

Point Biserial Correlation Tests

Point Biserial Correlation Tests Chapter 807 Point Biserial Correlation Tests Introduction The point biserial correlation coefficient (ρ in this chapter) is the product-moment correlation calculated between a continuous random variable

More information

Chapter 5. Random variables

Chapter 5. Random variables Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like

More information

Measuring the Power of a Test

Measuring the Power of a Test Textbook Reference: Chapter 9.5 Measuring the Power of a Test An economic problem motivates the statement of a null and alternative hypothesis. For a numeric data set, a decision rule can lead to the rejection

More information

4. How many integers between 2004 and 4002 are perfect squares?

4. How many integers between 2004 and 4002 are perfect squares? 5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started

More information

The normal approximation to the binomial

The normal approximation to the binomial The normal approximation to the binomial The binomial probability function is not useful for calculating probabilities when the number of trials n is large, as it involves multiplying a potentially very

More information

Practice Problems #4

Practice Problems #4 Practice Problems #4 PRACTICE PROBLEMS FOR HOMEWORK 4 (1) Read section 2.5 of the text. (2) Solve the practice problems below. (3) Open Homework Assignment #4, solve the problems, and submit multiple-choice

More information

The concept of probability is fundamental in statistical analysis. Theory of probability underpins most of the methods used in statistics.

The concept of probability is fundamental in statistical analysis. Theory of probability underpins most of the methods used in statistics. Elementary probability theory The concept of probability is fundamental in statistical analysis. Theory of probability underpins most of the methods used in statistics. 1.1 Experiments, outcomes and sample

More information

Probability Distributions

Probability Distributions CHAPTER 5 Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Probability Distribution 5.3 The Binomial Distribution

More information

Improving Validation Through the Use of Confidence Statements Based on Attributes Acceptance Sampling

Improving Validation Through the Use of Confidence Statements Based on Attributes Acceptance Sampling Improving Validation Through the Use of Confidence Statements Based on Attributes Acceptance Sampling John Haury INTRODUCTION Incorporating acceptance sampling into validation studies enhances their credibility

More information

MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

More information

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as... HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

More information

Chapter 7 Part 2. Hypothesis testing Power

Chapter 7 Part 2. Hypothesis testing Power Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship

More information

Notes on Continuous Random Variables

Notes on Continuous Random Variables Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

More information

Technology Step-by-Step Using StatCrunch

Technology Step-by-Step Using StatCrunch Technology Step-by-Step Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate

More information

Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs

Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs Types of Variables Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs Quantitative (numerical)variables: take numerical values for which arithmetic operations make sense (addition/averaging)

More information

Using Excel for inferential statistics

Using Excel for inferential statistics FACT SHEET Using Excel for inferential statistics Introduction When you collect data, you expect a certain amount of variation, just caused by chance. A wide variety of statistical tests can be applied

More information

Binomial Random Variables

Binomial Random Variables Binomial Random Variables Dr Tom Ilvento Department of Food and Resource Economics Overview A special case of a Discrete Random Variable is the Binomial This happens when the result of the eperiment is

More information

Hypothesis Testing. Learning Objectives. After completing this module, the student will be able to

Hypothesis Testing. Learning Objectives. After completing this module, the student will be able to Hypothesis Testing Learning Objectives After completing this module, the student will be able to carry out a statistical test of significance calculate the acceptance and rejection region calculate and

More information

Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test

Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely

More information

CHAPTER 5 THE BINOMIAL DISTRIBUTION AND RELATED TOPICS

CHAPTER 5 THE BINOMIAL DISTRIBUTION AND RELATED TOPICS CHAPTER 5 THE BINOMIAL DISTRIBUTION AND RELATED TOPICS THE BINOMIAL PROBABILITY DISTRIBUTION (SECTIONS 5.1, 5.2 OF UNDERSTANDABLE STATISTICS) The binomial probability distribution is discussed in Chapter

More information

MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 140 Lab 4: Probability and the Standard Normal Distribution MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes

More information

Module 5 Hypotheses Tests: Comparing Two Groups

Module 5 Hypotheses Tests: Comparing Two Groups Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this

More information

Engineering Problem Solving and Excel. EGN 1006 Introduction to Engineering

Engineering Problem Solving and Excel. EGN 1006 Introduction to Engineering Engineering Problem Solving and Excel EGN 1006 Introduction to Engineering Mathematical Solution Procedures Commonly Used in Engineering Analysis Data Analysis Techniques (Statistics) Curve Fitting techniques

More information

Two Correlated Proportions (McNemar Test)

Two Correlated Proportions (McNemar Test) Chapter 50 Two Correlated Proportions (Mcemar Test) Introduction This procedure computes confidence intervals and hypothesis tests for the comparison of the marginal frequencies of two factors (each with

More information

Two-Sample T-Tests Allowing Unequal Variance (Enter Difference)

Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Chapter 45 Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when no assumption

More information

Cork Institute of Technology. CIT Mathematics Examination, Paper 1 Sample Paper A

Cork Institute of Technology. CIT Mathematics Examination, Paper 1 Sample Paper A Cork Institute of Technology CIT Mathematics Examination, 2015 Paper 1 Sample Paper A Answer ALL FIVE questions. Each question is worth 20 marks. Total marks available: 100 marks. The standard Formulae

More information

Important Probability Distributions OPRE 6301

Important Probability Distributions OPRE 6301 Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in real-life applications that they have been given their own names.

More information

MATHEMATICAL THEORY FOR SOCIAL SCIENTISTS THE BINOMIAL THEOREM. (p + q) 0 =1,

MATHEMATICAL THEORY FOR SOCIAL SCIENTISTS THE BINOMIAL THEOREM. (p + q) 0 =1, THE BINOMIAL THEOREM Pascal s Triangle and the Binomial Expansion Consider the following binomial expansions: (p + q) 0 1, (p+q) 1 p+q, (p + q) p +pq + q, (p + q) 3 p 3 +3p q+3pq + q 3, (p + q) 4 p 4 +4p

More information

Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

More information

Instruction Manual for SPC for MS Excel V3.0

Instruction Manual for SPC for MS Excel V3.0 Frequency Business Process Improvement 281-304-9504 20314 Lakeland Falls www.spcforexcel.com Cypress, TX 77433 Instruction Manual for SPC for MS Excel V3.0 35 30 25 LSL=60 Nominal=70 Capability Analysis

More information

EE 302 Division 1. Homework 5 Solutions.

EE 302 Division 1. Homework 5 Solutions. EE 32 Division. Homework 5 Solutions. Problem. A fair four-sided die (with faces labeled,, 2, 3) is thrown once to determine how many times a fair coin is to be flipped: if N is the number that results

More information

9-3.4 Likelihood ratio test. Neyman-Pearson lemma

9-3.4 Likelihood ratio test. Neyman-Pearson lemma 9-3.4 Likelihood ratio test Neyman-Pearson lemma 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental

More information

Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

More information

STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables

STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random

More information

MBA 611 STATISTICS AND QUANTITATIVE METHODS

MBA 611 STATISTICS AND QUANTITATIVE METHODS MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain

More information

Solutions for the exam for Matematisk statistik och diskret matematik (MVE050/MSG810). Statistik för fysiker (MSG820). December 15, 2012.

Solutions for the exam for Matematisk statistik och diskret matematik (MVE050/MSG810). Statistik för fysiker (MSG820). December 15, 2012. Solutions for the exam for Matematisk statistik och diskret matematik (MVE050/MSG810). Statistik för fysiker (MSG8). December 15, 12. 1. (3p) The joint distribution of the discrete random variables X and

More information

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover

More information

EXCEL EXERCISE AND ACCELERATION DUE TO GRAVITY

EXCEL EXERCISE AND ACCELERATION DUE TO GRAVITY EXCEL EXERCISE AND ACCELERATION DUE TO GRAVITY Objective: To learn how to use the Excel spreadsheet to record your data, calculate values and make graphs. To analyze the data from the Acceleration Due

More information

Lecture 7: Binomial Test, Chisquare

Lecture 7: Binomial Test, Chisquare Lecture 7: Binomial Test, Chisquare Test, and ANOVA May, 01 GENOME 560, Spring 01 Goals ANOVA Binomial test Chi square test Fisher s exact test Su In Lee, CSE & GS suinlee@uw.edu 1 Whirlwind Tour of One/Two

More information

6.4 Normal Distribution

6.4 Normal Distribution Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

More information

E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

More information