Statistical Method of PROBIT · the value of the functional parameter for which the probability of success is equal to R. The PROBIT ( Prob ability Un it ) represents the value plus

STATISTICAL METHODS

PROBIT METHOD

GTPS DOCUMENT 11 A

Courtesy translation from MBDA.F

Version date: Octobre 2014

Document prepared by the « Reliability » committee

Statistical Methodof PROBIT

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Document GTPS N°11AOctober 2014 edition

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Recommandation for designing and mastering reliable pyrotechnic devices


Document prepared by the « Reliability » committee

Statistical Methodof PROBIT

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Document GTPS N°11AOctober 2014 edition

D dddddddddddddddddddddddddddddddddddddddddddddddd

Recommandation for designing and mastering reliable pyrotechnic devices


GTPS 11 A – October 2014 – Page 1

Table of Contents

A. GENERAL................................................................................................................................................................ 2

A.1 OBJECTIVE.......................................................................................................................................................... 2 A.2 BIBLIOGRAPHIC REFERENCES....................................................................................................................... 2

A.2.1 Reference documents .................................................................................................................................... 2 A.2.2 Other documents ........................................................................................................................................... 3

A.3 SPECIFIC TERMINOLOGY ................................................................................................................................ 4

B. PART ONE: RECOMMENDATION FOR DESIGNING AND MASTERIN G RELIABLE PYROTECHNIC DEVICES............................................................................................................................................ 5

B.1 SCOPE..................................................................................................................................................................... 5 B.2 METHODOLOGY BY PHASE ............................................................................................................................ 5

B.2.1 GENERAL RULES........................................................................................................................................ 5 B.2.2 FEASIBLITY ................................................................................................................................................. 6 B.2.3 PRE-PROJECT............................................................................................................................................. 7 B.2.4 DEVELOPMENT.......................................................................................................................................... 8

B.3 PRESENTATION OF THE STATISCAL METHODS USED.............................................................................. 9 B.3.1 PURPOSE OF METHODS ........................................................................................................................... 9 B.3.2 CONDITIONS FOR IMPLEMENTING METHODS..................................................................................... 9

B.4 COMPARISON OF METHODS FOR EVALUATING THE RELIABILITY OF PYROTECHNIC PRODUCTS12

C. PART TWO: USING THE PROBIT METHOD ................................................................................................ 13

C.1 NOTATION CONVENTIONS ........................................................................................................................... 13 C.2 GENERAL PRINCIPLE OF THE PROBIT METHOD....................................................................................... 14

C.2.1 Advantages and limitations of the PROBIT method.................................................................................... 14 C.2.2 SPECIAL CONDITIONS: SELECTING TEST PARAMETERS .................................................................. 14

C.3 USING TESTS ....................................................................................................................................................17 C.3.1 Purpose ....................................................................................................................................................... 17 C.3.2 VERIFICATION OF INITIAL HYPOTHESIS............................................................................................. 17 C.3.3 DetermiNATION OF the PROBIT line ....................................................................................................... 17 C.3.4 Confidence intervals on estimators............................................................................................................. 19 C.3.5 VERIFICATION OF distribution normality................................................................................................ 22 C.3.6 EvaluatiON OF Success (or failure) probability and the associated operating thresholds........................ 22 C.3.7 Spreadsheet................................................................................................................................................. 23

C.4 APPLICATION EXAMPLES ............................................................................................................................. 24 C.4.1 Reliability of an electro-pyrotechnic igniter ............................................................................................... 24 C.4.2 Acceptance procedure for a batch of cutting cords .................................................................................... 27 C.4.3 “True distribution” PROBIT of an electro-pyrotechnic igniter ................................................................. 32 C.4.4 Re-using results from an unusable Bruceton test........................................................................................ 37

D. CONCLUSION ...................................................................................................................................................... 40

E. APPENDICES (TABLES): ................................................................................................................................... 41

E.1 EMPIRICAL PROBIT Y I AS A FUNCTION OF OBSERVED SUCCESS RATE PI............................................................. 42 E.2 WEIGHTING COEFFICIENT WI AS A FUNCTION OF PROVISIONAL PROBIT VALUE YPI........................................... 44 E.3 WORKING PROBIT ZI = FN (OBSERVED SUCCESS RATE PI, PROVISIONAL PROBIT YPI) ...................................... 45 E.4 VALUES OF THE STANDARD NORMAL DISTRIBUTION FUNCTION............................................................................ 55

E.5 VALUES OF ),2/(2

ναχ = FN (NUMBER OF DEGREES OF FREEDOM ν, CONFIDENCE LEVEL 1-α) ............................ 56


A. GENERAL

A.1 OBJECTIVE The first part of this recommendation is intended for designers to enable them to create and master the manufacturing of reliable pyrotechnic products. It should provide a basis for discussions between customers and suppliers whenever a contract binding them requires reliability specifications. It covers:

• the type of the design phases; • the procedures for ensuring reliability for each of these phases, • a presentation of various statistical methods available to designers including:

∗ suitable methods for each of the design phases, ∗ the advantages and drawbacks of each of the methods explained.

The second part of this recommendation presents the procedure for using the statistical technique known as the “PROBIT” method. This is a way to evaluate the success or failure probability of a one-shot device (pyrotechnic product), based on a limited number of tests. The PROBIT method, originally developed by biologists, is a non-sequential testing method. There is no guarantee, despite implementing the method perfectly in tests, that it will yield results. This method is therefore recommended only when (for practical testing reasons) it is the only one applicable method.

A.2 BIBLIOGRAPHIC REFERENCES A.2.1 REFERENCE DOCUMENTS

GTPS (Space Pyrotechnic Working Group)

1. Dictionnaire de pyrotechnie / Pyrotechnic Dictionary (ed. 6, 2008)

2. N° 11B: Méthode statistique One-Shot

3. N° 11C: Méthode statistique de Bruceton

4. N° 11F: Méthode statistique des essais durcis

AFNOR 5. Recueil de normes françaises AFNOR – Statistique Tome 1, éd. 7, 2008: Vocabulaire,

estimation et tests statistiques

6. Groupe fiabilité (éd. 1, 1981): Guide d’évaluation de fiabilité en mécanique par l’A.F.C.I.Q

7. NF X06-021 (01/10/1991): Application of statistics. Principles of the statistical control of batches

8. NF X06-050 (01/12/1995): Application of statistics. Study of the normality of a distribution

9. NF X07-009, NF EN ISO 10012 (01/09/2003): Measurement management systems – Requirements for measurement processes and measuring equipment

10. NF X50-130, NF EN ISO 9000 (01/10/2005): Quality management systems – Fundamentals and vocabulary

11. FD X50-127 (01/04/2002): Management tools – Design and development control

12. NF X60-500 (October 1988): Terminology relating to reliability, maintainability and availability


A.2.2 OTHER DOCUMENTS

Standards documents

13. BNAe - RE Aéro 703.05, March 2000: Guide pour la maîtrise de la fiabilité

14. ARMP-1, 08/2008: NATO Requirements for Reliability and Maintainability

15. IEC 60812, CEI 60812 (2006-01-01): Procedure for Failure Mode and Effects Analysis (FMEA) - Analysis techniques for system reliability

16. IEC 61025, CEI 61025 (2006-12-01): Fault Tree Analysis (FTA)

17. Technical Note A5-NT-1-X-542-ASAI of 28/08/1990: “Systèmes pyrotechniques Marges de dimensionnement et de performances” (Pyrotechnic systems: Sizing margins and performance)

PROBIT bibliography

18. Caussinus H., Mathieu J.R., Meste M., Milhaud X. (ATS GIAT, Université P.Sabatier, 06/85): Etude de méthodes statistiques de contrôle d'un composant pyrotechnique

19. Caussinus H., Mathieu J.R., Meste M., Milhaud X. (Laboratoire de statistique et probabilités CNRS Université P.Sabatier, 11/85): Evaluation de la distribution de sensibilité d'amorces pour les très petites valeurs du stimulus

20. Caussinus H., Mathieu J.R., Meste M., Milhaud X., Refouvelet J. (ATS GIAT, Université P.Sabatier, 03/87, 3rd GTPS convention, Juan-Les-Pins): Analyse statistique de la sensibilité d'un composant pyrotechnique

21. Finney D.J. (Cambridge University Press, 1947, 3rd Ed. 1971): PROBIT analysis: A statistical treatment of the sigmoid response curve

22. Malabiau R. (Direction des Constructions et Armes Navales de Toulon, 1979): Méthodes statistiques utilisables pour déterminer la sensibilité des initiateurs électropyrotechniques à diverses excitations d'origine électrique ou électromagnétique

23. NOL (US NOL, White Oak. NOLM 9910, AD 106-866, 11/48): Comparison of the PROBIT method and the Bruceton up and down method as applied to sensitivity data

24. U.S. Naval Ordnance Laboratory (NAVORD report 2101, 20/09/54): Statistical method appropriate for evaluation of fuze explosive – Train safety and reliability

25. Raspaud L. (Interface SA 843/91/CNES/6310, 02/92): Essais de sensibilité par les méthodes Bruceton et PROBIT

26. Raspaud L., Auger P. (Interface SA 840/93/CNES/0132, 06/94): Essais de sensibilité par les méthodes One-Shot, Bruceton et PROBIT

27. Thomson Brandt (Technical Note 23.110: Etablissement de la fiabilité expérimentale d’éléments et systèmes monocoups): Proposition technique concernant la comparaison des efficacités des tests de Bruceton, One-Shot et PROBIT

28. PAQTE2004-DLA-05 Project, CNES document ref. DLA-NT-0-988-AP, 24/01/2005: “Comparaison expérimentale des méthodes One Shot, Bruceton et PROBIT”.

29. Gond D, ref. 2476484 du 28/05/2013 : GTPS Exploitation des essais pyrotechniques : méthode des probits, Etablissement des expressions utilisées et application.


A.3 SPECIFIC TERMINOLOGY

In order to avoid any misunderstanding in this recommendation, it was decided that the following concepts should be clearly defined:

� Design: Based on expressed needs and existing knowledge, it is a creative activity that leads to product definition compliant with these needs and that is processable,

� Product: a term covering any item resulting from a production operation or any service provided such as production of components (raw materials, semi-finished or finished products, ingredients, parts, components, hardware, systems, etc.),

� A functional parameter is a quantifiable physical magnitude, associated with the product, whose value affects the success-failure criteria during their implementation,

� The success or failure criterion is a way of characterising the response of the product to stress,

� The operating threshold of a product for a given reliability R is defined as being the value of the functional parameter for which the probability of success is equal to R.

� The PROBIT (Probability Unit ) represents the value plus 5 of the nth quantile of the standard normal distribution (the value 5 was originally introduced into the method in order to obtain only positive PROBIT).


B. PART ONE: RECOMMENDATION FOR DESIGNING AND MASTERING RELIABLE PYROTECHNIC DEVICES

B.1 SCOPE

This document is intended for all industrial designers who need to respond to a formal quantitative reliability requirement of a pyrotechnic device. It covers:

� In conformity with [10], design activities, including feasibility, pre-project and development phases, during which reliability must be taken into account to define a product which can be manufactured at optimised cost.

� Continuous design activities to improve the reliability of a given product.

It applies to products using pyrotechnic devices defined in the document cited at paragraph 1 at section A.2.1 (one-shot devices).

B.2 METHODOLOGY BY PHASE

B.2.1 GENERAL RULES

1. Determine the objectives to be achieved in terms of performance, characteristics, costs and timeframe,

2. Integrate and manage reliability during the project design phases,

3. Have a systematic dialogue structure between the parties concerned,

4. Ensure consistency of the objectives with:

� Actions planned,

� Results obtained,

5. Ensure that the technical and human resources used correspond to the product being designed.

Associated with these rules are certain tasks such as management, calculation, analysis or testing. In particular, they are due to the necessary iteration between the dimensioning of the product and its reliability expressed in terms of margins and design factors.


B.2.2 FEASIBLITY B.2.2.1 Purpose

The purpose of this phase is to show if the stated requirements can be met, by detailing the possible concepts, technological routes and architectures. Such requirements are generally expressed in terms of the mission objectives, information concerning the operational environment (life cycle with associated environmental conditions) and reliability objectives.

It has to work towards establishing the reliability requirements to be included in the functional performance specifications, and possible reliability management requirements.

B.2.2.2 Tasks

For each proposed technological solution, the tasks to be accomplished are:

� Preliminary risk/hazard analysis,

� Risk assessment by: o literature survey and/or experience acquired with similar products,

especially regarding anomalies or incidents encountered; reliability database research,

o computed simulation to gain quantitative and qualitative understanding of the phenomena involved and to highlight certain critical design features,

o use of an experimental design to establish the predominant parameters, their sensitivity on performance and their interactions,

o implementation of one of the methods recommended in the table section B.4 to estimate the mean for certain specific parameters,

� Appraisal of critical points highlighted for each solution, and comparison of solutions with respect to the stated requirements.

By the end of this phase, qualitative assessment criteria should comprise the input data necessary to start the next phase. Accordingly, they should be set down in the functional performance specifications in the chapter on reliability requirements.


B.2.3 PRE-PROJECT B.2.3.1 Purpose

The purpose of this phase is to investigate the possible approaches at the end of the feasibility study so as to suggest what can be developed.

It enables the preliminary product definition file to be prepared in accordance with the reliability requirements of the functional performance specifications established during the previous phase.

B.2.3.2 Tasks

For each solution considered feasible:

� Modelling: draw up a reliability block diagram in order to establish product architecture and identify the interfaces concerned by the reliability study. This approach is used to define the "product" tree diagram whose level of breakdown stops at the basic components with measurable characteristics,

� Allocation: distribute the overall reliability objective among items on the tree, allocating a predicted reliability objective to each of the itemised components and interfaces to indicate the probability of the function being fulfilled for each component, allowing for life cycle and/or its life time,

� Analysis: for each component listed, perform a Failure Mode, Effects, and Criticality Analysis (FMECA) to highlight the points considered critical based on: o existing databases and/or feedback on similar components, o possibly, and depending on the products developed, a specific

experiment using the method(s) specified in the table section B.4 to first confirm the initial assessment of the average m (see section B.2.2.2), and secondly to provide an initial estimate of the standard deviation σ of the dispersion around the mean value.

� Forecast: using the reliability block diagram model, piece together partial assessments in accordance with the product tree to assess how the proposed solution matches the requirement.

� Validation plan: draw up a pre-project development - reliability product plan to estimate what technical work is necessary to develop the product satisfactorily in terms of cost and timeframe.

� Trade-off: considering all the solutions, choose the one which best meets the stated requirement, and which will be developed in the subsequent phase, while justifying why the other solutions are rejected.


B.2.4 DEVELOPMENT B.2.4.1 Purpose

The purpose of this phase is:

� to draw up the product definition file to meet the reliability requirements as expressed in the Technical Specifications,

� to validate the design using the results of theoretical studies, tests, and exploitation of technical fact,

� to prepare production and operational phases, specifying which procedures will be necessary for ensuring reliability during these two phases.

B.2.4.2 Tasks

For the adopted solution:

� Conduct a reliability predicted study in order to: o rework and refine the previous reliability block diagram, o optimise the environmental constraints applied to each component, o possibly, update the reliability allocations and negotiate reliability

requirement,

� Analyse hazardous events by a fault tree analysis. Deductive analysis is a statistical analysis which does not take the sequential aspect of events into account. Limits inherent in implementing fault tree analysis are:

o To correctly define the feared event (origin of the tree) o To define elementary events, o To ensure the independence of the elementary events listed,

� Carry out FMECA for each elementary event listed,

� Define all the solutions needed to meet the required levels of reliability, by means of:

o studies and tests up to the product qualifying phase (design reliability ), by implementing the methods recommended in the table section B.4,

o manufacturing and acceptance procedures (manufacturing reliability),

� A posteriori, check and assess the independence of the events,

� Undertake long-term actions to ensure reliability throughout the life time of the product. In particular, define the ageing programme to be conducted in order to :

o ensure that the assumed level of reliability has been attained, o assess what advance warning is required to prevent or overcome a

possible long-term failure, o upgrade the databases, especially those used for reliability analysis

during development.

The associated sampling policy should be consistent with the operational needs.

At the end of the development phase, the product design shall meet reliability objectives.

The development phase is finalised by the approval dossier, qualification and/or certification report (definition file and supporting evidence, industrial file).


B.3 PRESENTATION OF THE STATISCAL METHODS USED

B.3.1 PURPOSE OF METHODS

The purpose of these methods is to:

� Characterize the distribution of product operation thresholds by sensitivity tests (either sequential or simultaneous).

� Check the appropriate probability distribution for these operating thresholds,

� Use this distribution to assess a probability of success or failure during operation of the product tested for a given confidence level.

B.3.2 CONDITIONS FOR IMPLEMENTING METHODS

B.3.2.1 Definition of test samples

The definition of test samples has to take the three following points into account:

1. Nominal definition of test specimen:

� The nominal definition of test specimen complies with a Definition File and the sample is representative of a given population (see Appendix 1).

� The test specimen can be:

o A functional object (e.g., an initiator, the couple formed by a shear and the rod to be cut, etc.),

o A defined quantity of a product.

2. Definition of the population:

� The test specimen belongs to a clearly identified population.

� It is recommended that a homogeneous batch be used : manufactured at the same time and place, using the same raw materials, methods and personnel, accordance with the methods and equipment defined (see Appendix 1).

3. Definition of the test sample:

It is chosen from the population following a sampling plan defined by:

� The type of test,

� The particular sampling scheme required to ensure the validity of the test results,

� The size of sample to be tested. It depends on the method used, as explained in the table section B.4. However it is recommended that a reserve for additional specimens is established for contingencies.

� The relationship between the tests results and the test acceptance criteria.


B.3.2.2 TEST REPRODUCTIBILITY

Test reproducibility has to take the following four points into account:

1. Identification of test support equipment:

� Consumable test support equipment compliant with a Definition File,

� Reusable test support equipment for which compliance with a Definition File and stability of functional characteristics will be checked.

2. Identification of test facilities:

� Environmental conditions [9],

� Power sources,

� Calibrated measuring equipment.

3. Control of stresses applied to the specimens:

� The uncertainty of stresses applied has to be less than the assumed standard deviation for the population.

4. Control of test conditions:

� Stable environmental and test conditions during a test sequence,

� Representative conditions and / or test specimens from the actual configuration (confinement, critical diameter, heat exchange ...),

� Test facilities,

� Procedures,

� Personnel.


B.3.2.3 Prerequisites

1. Choice of the functional parameter:

It must meet the following criteria:

� To be adjustable,

� To behave in a known and continuous manner in the field to be investigated.

2. Choice of the success/failure criterion:

� It must be clearly defined, after analysis of all possible responses of the product studied.

� It is necessary to understand how the probability of success or failure varies according to the variation rate of the chosen functional parameter.

3. Assumptions:

It is assumed that:

� The resolution of the functional parameter for the test should be approximately 1/10 of the initial evaluation of the estimated standard deviation.

� The functional threshold of the selected functional parameter is a random variable.

� The density probability of this random variable follows a normal (1) or a log normal (2) law. It should take into account experience feedback.

(1) NOTE: Concerning the assumption of normality, it shall be ensured that the selected functional parameter is leaded by only one physical phenomenon in the field trials. Indeed, some cases may be governed by several physical phenomena that lead to multimodal statistical laws, like the initiation gap between an explosive relay and a detonator:

(2) NOTE: If a log-normal probability density function is used, a change of variable will be done in order to reduce the studied case to a normal statistical law.

X = Initiation gap

1rst initiation mode 2nd initiation mode

Detonation pressure Screening plate (Detonator cap)

Small gap Big gap

V

X = Initiation gap

1rst initiation mode 2nd initiation mode

Detonation pressure Screening plate (Detonator cap)

Small gapSmall gap Big gap

V


B.4 COMPARISON OF METHODS FOR EVALUATING THE RELIABILITY OF PYROTECHNIC PRODUCTS

Table 1 below lists the advantages and drawbacks of each of the statistical methods.

Method No. of tests

Advantages Drawbacks

Probit

GTPS 11A (See Part Two

of this document)

≥ 72

Non sequential test Possible to adjust levels during tests Best estimator of standard deviation

Define at least 5 levels Significant risk that the method will fail (estimated at 16%), even under ideal test conditions.

One-shot

GTPS 11B

(see para. 2 of section A.2.1)

≥ 30

All test results can be used Choice of initial test value does not alter results accuracy Convergence toward the mean is assured and very fast for a small sample tested:

• possibly poorly known, • whose probability distribution

law is unimodal

Sequential test involving management of test constraints with levels unknown in advance

Bruceton

GTPS 11C (see para. 3 of section A.2.1)

≥ 30

Provides statistical estimators for mean and standard deviation, with good precision for the mean

Sequential test involving management of test constraints, but with a fixed pitch

Results depend on the pitch value

Severe tests

GTPS 11F

(see para. 4 of section A.2.1)

≥ 1

≤ 10

Method suitable to confirm functional margins with respect to its nominal operating point, with less than 10 trials. Analytical approach taking into account the contents of the FMECA as a complementary tool. Enables one failure in implementing the severe tests plan, through:

• either a degraded reliability assessment (compared to the initial value intended)

• or an increase in the number of specimens tested

Requires knowledge of the coefficients of variation of predominant parameters Results depend closely on the coefficients of variation associated with widely scattered parameters. Does not provide the distribution of the functional parameter being tested.


C. PART TWO: USING THE PROBIT METHOD

C.1 NOTATION CONVENTIONS aj intercept of the PROBIT line at iteration j bj slope of the PROBIT line at iteration j dof degree of freedom k number of test levels (1 ≤ j ≤ k)

ΦΦΦΦ(t) standard normal distribution function: duett u

⋅=Φ ∫ ∞−

− 22

2

1)(

π

S standard deviation of the distribution of operating thresholds for functional parameter X

Sp pre-estimator of the standard deviation of the distribution of operating thresholds for functional parameter X

uαααα/2 quantile of the normal distribution (expressed bilaterally) for confidence level 1-αααα 1-α confidence level

X Mean of the operating thresholds distribution for functional parameter X XF operating threshold for functional parameter X XNF non-operating threshold for functional parameter X Xnom nominal level of functional parameter X

pX pre-estimator of the mean of the operating thresholds distribution for functional parameter X

XRef level corresponding to the nominal requirement associated with the reliability objective And for each test level i:

xi level of the functional parameter X (possibly xi = Log10(hi) if the functional parameter H is log-normally distributed)

ni total number of tests at level xi r i: number of successful results at level xi pi success rate calculated at level xi (pi = r i / ni) T i, , Qi intermediate values for weight W i and provisional PROBIT YPi Y i empirical PROBIT at level xi Ypi provisional PROBIT at level xi W i weight assigned at level xi zi working PROBIT at level xi Z i final PROBIT at level xi.


C.2 GENERAL PRINCIPLE OF THE PROBIT METHOD For a given batch of products, and in relation with the functional parameter studied, this method can be used to evaluate the estimators of the mean and standard deviation of the probability distribution of functional thresholds, for a given confidence level. These estimators are based on the characteristics (slope and intercept) of a line known as the "PROBIT line". This line is obtained after performing non-sequential tests, through iterative calculations. C.2.1 ADVANTAGES AND LIMITATIONS OF THE PROBIT METHOD

The main advantages of this method are:

• An easy application of acceptance control tests (see example in §C.4.2);

• The ability to concatenate the results of Bruceton and/or PROBIT tests obtained from several tests (see example in §C.4.3);

• The recycling of otherwise unusable Bruceton test sequences (S/d condition outside range of 0.5 to 2) (see example in § C.4.4).

Its limitations are:

• Its sensibility to the chosen levels and to the way the batch is allocated across these levels: as far as possible, one should aim for an N/4 distribution at the limit levels and an N/6 distribution at intermediate levels (see § C2.2.1);

• The difficulty of adjusting the levels during the trials: this can only be done if there is an adequate reserve of specimens (see § C2.2.1);

• There is a risk of ending up with an unusable PROBIT test, despite adopting an appropriate test strategy (see § C2.2.1).

C.2.2 SPECIAL CONDITIONS: SELECTING TEST PARAMETERS It is necessary:

• To know the type of distribution of the functional thresholds XF or XNF (function of the functional parameter X), the method is only applicable in the case of a normal distribution (*);

• To have an initial evaluation of estimators px and Sp of the probability of the functional parameter studied, using numerical simulations, databases or a dichotomous approach (e.g. the One-Shot method, §A.2.1 reference 2).

• To decide on the test levels and the number of tests per level.

(*) Note If the functional parameter H is log-normally distributed, we change the variable X = Log10 (H), and perform the test on the transformed variable X.


C.2.2.1 Determining the levels and the number of tests per level

Px and S -p being pre-estimates of the mean and the standard deviation of the operating thresholds distribution, we recommend spreading the N test specimens over 5 test levels (where N is the number of components tested). For maximum precision, we recommend selecting the levels and the allocation of tests per level as shown in the table below:

Recommended test level values Estimated probability of success Recommended number of tests

per level

Px - Sp 16% N/4

Px - 0.5 Sp 30% N/6

Px 50% N/6

Px + 0.5 Sp 70% N/6

Px - + Sp 84% N/4

Distribution of N specimens across 5 test levels:

16%

100%

PX

2

σ+PX2

σ−PX

σ−PX σ+PX

30%

50%

70%

84%

N/6

N/4

N/6

N/6

N/4

Functional parameter X

Density of probability :

dueXXX

u⋅=Φ ∫

−

∞−

−σ

π2

2

2

1)(

dX

XdX

)()(' Φ=Φ

standard normal distribution function :

16%

100%

PX

2

σ+PX2

σ−PX

σ−PX σ+PX

30%

50%

70%

84%

N/6

N/4

N/6

N/6

N/4

Functional parameter X

Density of probability :

dueXXX

u⋅=Φ ∫

−

∞−

−σ

π2

2

2

1)(

dX

XdX

)()(' Φ=Φ

standard normal distribution function :


Remark: Occasionally, a test performed at a given level can lead to an observed success rate that is very different from what was expected, in such cases the level is called “outlier”. The most frequent instance of an outlier level is when a success rate of 0% or 100% is observed at the upper or lower limit level. This could be for a number of reasons:

• The test level is the right one: this result can be explained using a binomial distribution:

o The probability of observing a 100% success rate, at the level corresponding to an expected success rate of 84%, is P=0.2 when the number of tests is 10;

o This probability falls to P=0.04 when the number of tests is 18;

o It is therefore advisable to conduct a minimum of 18 firings at any limit level in order to resolve the outlier with certainty.

• The test level is not the right one: in this case, there is a shift in the mean, or the real standard deviation is lower than expected.

• The test level is an intermediate one: in this case, the hypotheses (see § B.3.2) need to be checked (normality of the distribution, representativeness of the sample, or operating procedure).

C.2.2.2 Performing the tests As the risk of observing an outlier level exists, it is advisable not to fire all of the samples at the same time at every test level. It is best to adopt a test conduct strategy that allows the test levels to be adjusted in response to the firing results obtained, in order to avoid outliers:

• Testing should therefore begins at the limit levels;

• If, after the 18th firing at a limit level, there is still a 100% failure rate or 100% success rate, then the level is probably (96%) an outlier;

• In this case, the levels and the allocation of tests per level need to be revised. Remark 1: This strategy of readjusting the test levels depends on having additional specimens. Remark 2: During a batch acceptance phase, specimens can be fired without worrying about results; Then the data will be analysed once the tests are complete, because:

• The product is known (verification of the compliance with nominal definition); • The procedure is set (the settings are test criteria, level and number of specimens per level); • The acceptance of the batch depends on the success of the PROBIT test.


C.3 USING TESTS C.3.1 PURPOSE The tests are used:

• To calculate the distribution parameters; • To verify the distribution hypothesis; • To evaluate the probability of success or failure based on the chosen distribution function.

C.3.2 VERIFICATION OF INITIAL HYPOTHESIS Check that the effective resolution of the functional parameter is at least 10 times smaller than the calculated standard deviation estimator. The validity of the PROBIT test results depends on the adjustment resolution of the test levels. C.3.3 DETERMINATION OF THE PROBIT LINE This paragraph describes the 8-step approach for determining the PROBIT line that corresponds to the maximum-likelihood (for details, see reference 22 in § A2.2). C.3.3.1 Step 1: Calculate the success rates For each given testing level xi the success rate is determined by the equation:

pi = r i / ni C.3.3.2 Step 2: Calculate the empirical PROBIT The empirical PROBIT can be either read directly on Table 1 in Appendix E1 or calculated using the following formula:

Y i = f(pi) = Φ-1 (r i / ni) + 5 Note: Originally, when the calculations were hand-made, the value +5 was added to keep all PROBIT values positive. C.3.3.3 Step 3: Calculate the provisional PROBIT The provisional PROBIT line is determined by linear regression (e.g. least squares approximation) between xi and Y i on each of the test levels xi:

Ypi = a0 + b0 xi

This yields a first estimate of the mean and standard deviation of the operating thresholds distribution:

0X = and s0 =


C.3.3.4 Step 4: Calculate the weights The weights are calculated using the equation:

W i = ( )ii

i

QQ

T

−1.

2

where: .2

1 T 2

)5 -( -

i

2 ipY

eπ

= and ( ) ∫∞−

=−Φ=5-

2

-

Pii . 2

15YQ

2iYp u

dueπ

W i can also be read on Table 2 in Appendix E2. C.3.3.5 Step 5: Calculate the working PROBIT The working PROBIT are refined values of the provisional PROBIT Ypi. They are obtained from the provisional PROBIT and the success rate pi:

zi = Ypi + T

)Q -(p

i

i i

zi can also be read on Table 3 in Appendix E3. C.3.3.6 Step 6: Calculate the final PROBIT The final PROBIT line is obtained by linear regression between W i xi and W i zi:

Zi = a1 + b1 xi

To calculate a1 and b1, we perform the following intermediate calculations:

∑

∑

=

==k

iii

k

iiii

Wn

xWn

X

1

1

.

..)

∑

∑

=

==k

iii

k

iiii

Wn

zWn

Z

1

1

.

..)

∑∑

∑

=

=

=

−=k

ik

iii

k

iiii

iiiXX

Wn

xWn

xWnS1

1

2

12

.

..

..

∑∑

∑∑

=

=

==

−=k

ik

iii

k

iiii

k

iiii

iiiiXZ

Wn

zWnxWn

zxWnS1

1

11

.

.....

...

The intermediate values are used to find a1 and b1:

XbZa))

.11 −= and XX

XZ

S

Sb =1

A new estimation of the mean and standard deviation of the distribution is evaluated:

1X = and s1 =

The final PROBIT represent the results of a first iteration. Others iterations could be necessary to improve the accuracy of solution.


C.3.3.7 Step 7: Convergence test A convergence test is performed on the calculated results of 1X and s1 to determine if a new iteration is necessary.

Error calculation on the values of the estimators X and s:

� The standard deviations for the estimators X and b = 1/s are:

( ) ( ) 21

1/2i i

Sxx

1bS&

)Wn ( b

1XS

==∑

where - xSxx i i

2i i i

Wn

)xWn (

2i

∑

∑= ∑ iiWn

� From this, we can deduce the intervals to +/- 1 standard deviation of the estimated values X and b. If the previous estimates are within these intervals, then the results are deemed to be sufficiently convergent:

( ) ( ))1(1

1)1(1

1& 011011 bsb

sbsb

XsXXXsX−

<<+

+<<−

C.3.3.8 Step 8: Iteration: Repeat Steps 4, 5, 6 and 7 If the convergence test failed then repeat Steps 4, 5 and 6, replacing:

• The provisional PROBIT YPi with the final PROBIT Zi • The provisional PROBIT line equation with the final PROBIT line

(a0 = a1 & b0 = b1 for the 1st iteration) After n iterations, and after passing the convergence test (Step 7), we determine the PROBIT line that corresponds to the maximum probability:

Z = an + bn X

Note: In practice, one or two iterations are generally enough. After iteration n, we calculate the estimators for the mean and standard deviation of the operating thresholds distribution:

b

a - 5X

n

n= and b1

S n

=

C.3.4 CONFIDENCE INTERVALS ON ESTIMATORS For a given confidence level 1−α1−α1−α1−α, calculate the quantile (*) of the normal distribution uαααα/2:

Confidence level 1-αααα

uαααα/2 (bilateral)

60% 0.84 90% 1.65 95% 1.96

(*) See Appendix E4


C.3.4.1 Confidence interval of the mean at confidence level 1 - αααα (bilateral risk) The mean follows a normal distribution.

The standard deviation of the estimator is equivalent to: ( )1/2

iin )Wn( b

1XS

∑=

The bilateral confidence interval of m at confidence level 1 - αααα is such that:

( ) ( ) +− =+≤≤−= mXSuXmXSuXm ..22

αα

C.3.4.2 Confidence interval of the standard deviation at confidence level 1 - αααα (bilateral risk) The estimators an and bn follow normal distributions.

The standard deviation of the estimator bSn = 1 is equivalent to:

( )S b = =1

Sxxn

12

where ( )∑∑∑ −=

ii

iii

Wn

xWn2

2iiiXX xWnS

The bilateral confidence interval of bn for a chosen confidence level 1 - αααα is such that:

( ) ( )nnnn bSubbbSub ⋅+≤≤⋅−22

αα

So the bilateral confidence interval of S at confidence level 1 - αααα is such that:

( ) ( ) +=⋅

≤≤⋅

= SbS

SbS nn

2n

2n

- u - b

1

u + b

1S

αα

The logic diagram on the next page sums up the PROBIT test procedure.


Application of the PROBIT test

Final Estimation :

nn

n

bS

b

aX

15 =−=

Results on level i :- Number of successes : r i

PROBIT te st plan:- Number of tests levels : k- Number of samples : N- Number of tests per level i : n i

With : 1 ≤ i ≤ k & ∑=

=k

ii Nn

1

Step N°1 :Calculate the rates of successes per level i

Pi = ri / n i

Step N°2 :Calculate the empirical PROBIT per level i

Yi = ΦΦΦΦ -1 (r i / n i) + 5j = 0

Step N°3 :Linear regression on empirical PROBIT :

Y = a j + b j.XCalculate the provisional PROBIT per level i

Ypi = aj + b j.Xi

1st estimation

00

0

00

1

5

bS

b

aX

=

−=

Step N°5 :Calculate the working PROBIT per level i

( )i

iiPii T

QpYZ

−+=

Step N°4 :Calculate the weights per level

( )ii

ii QQ

TW

−=

1

2

Step N°6 :Linear regression on working PROBIT :

Z = a j + b j.X

Calculate the final PROBIT per level iZi = a j + b j.Xi

j = j + 1

Estimation J :

j

j

j

jj

bS

b

aX

1

5

=

−=

Step N°7 :Convergence test :

Is the accuracy of solution sufficient?j = n

Step N°8 :Iteration

jj

jj

iPi

bb

aa

ZY

=

==

−

−

1

1

No Yes

( ) ( )

( ) ( ) +−

+−

=−

≤≤+

=

=+≤≤−=

σσσαα

αα

nnnn bSubbSub

mXSuXmXSuXm

.

1

.

1:deviationStandard

..:Mean

2/2/

2/2/

Final Estimation :

nn

n

bS

b

aX

15 =−=




With : 1 ≤ i ≤ k & ∑=

=k

ii Nn

1


With : 1 ≤ i ≤ k & ∑=

=k

ii Nn

1

Step N°1 :Calculate the rates of successes per level i

Pi = ri / n i

Step N°2 :Calculate the empirical PROBIT per level i

Yi = ΦΦΦΦ -1 (r i / n i) + 5j = 0

Step N°3 :Linear regression on empirical PROBIT :

Y = a j + b j.XCalculate the provisional PROBIT per level i

Ypi = aj + b j.Xi

1st estimation

00

0

00

1

5

bS

b

aX

=

−=


( )i

iiPii T

QpYZ

−+=


( )i

iiPii T

QpYZ

−+=


( )ii

ii QQ

TW

−=

1

2


( )ii

ii QQ

TW

−=

1

2

Step N°6 :Linear regression on working PROBIT :

Z = a j + b j.X

Calculate the final PROBIT per level iZi = a j + b j.Xi

j = j + 1

Estimation J :

j

j

j

jj

bS

b

aX

1

5

=

−=

Estimation J :

j

j

j

jj

bS

b

aX

1

5

=

−=

Step N°7 :Convergence test :

Is the accuracy of solution sufficient?j = n


jj

jj

iPi

bb

aa

ZY

=

==

−

−

1

1


jj

jj

iPi

bb

aa

ZY

=

==

−

−

1

1

No Yes

( ) ( )

( ) ( ) +−

+−

=−

≤≤+

=

=+≤≤−=

σσσαα

αα

nnnn bSubbSub

mXSuXmXSuXm

.

1

.

1:deviationStandard

..:Mean

2/2/

2/2/


C.3.5 VERIFICATION OF DISTRIBUTION NORMALITY The hypothesis that the operating thresholds follow a normal distribution can be verified (see §A.2.1,

reference 8), by applying a χ2 test:

• Calculate the value of χ2:

( )Sxx

SxzSzzCalculated

22 −=χ where:

∑

∑∑

−=

iii

iiii

iiii Wn

xWn

xWnSxx

2

2

∑

∑∑

−=

iii

iiii

iiii Wn

ZWn

ZWnSxz

2

2

∑

∑∑∑

×

−=

iii

iiii

iiii

iiiii Wn

ZWnxWn

ZxWnSxz

• Compare with the theoretical value of( )2

2, −kαχ corresponding to k-2 degrees of freedom (k =

number of test levels) and a risk αααα (see Appendix E5).

If χ2calculated < χ2

theoretical then the normality hypothesis cannot be rejected at risk α.

C.3.6 EVALUATION OF SUCCESS (OR FAILURE) PROBABILITY AND THE ASSOCIATED OPERATING THRESHOLDS

Nominal level (Xnom): this is the nominal value of the predominant parameter

Reference level (Xref): this is the value of the predominant parameter corresponding to the reliability assessment (possible margins already taken on the nominal level)

According to the case, the reference level could be:

� the nominal level of the predominant functional parameter,

� the nominal value of the predominant functional parameter with a margin coefficient specified by the customer or by standards,

� an upper or lower bound of a deterministic requirement,

� the level corresponding to the limits of the tolerance interval of the nominal level,

� the level corresponding to the limits at ±3 standard deviations from the nominal level,

� ... Note: If the functional parameter follows a log-normal distribution, simply change the variable X=Log(H) , and work on the transformed variable X.


C.3.6.1 Calculate the reliability (bilateral risk)

If ΦΦΦΦ represents the standard normal distribution function, then the probability R of success at confidence level (1 - α/2)2 is calculated as follows: C.3.6.1.1 CASE 1: R INCREASES WITH THE FUNCTIONAL PARAMETER If the probability of success increases with the functional parameter, then the reference level is higher than

the upper limit of the confidence interval for the mean X (see §C.3.4.1): +> mX Réf

In this case: ( )

−Φ=

+

+

− σα

mXR Réf

2

21

C.3.6.1.2 CASE 2: R DECREASES WITH THE FUNCTIONAL PARAMETER If the probability of success decreases relative to the functional parameter, then the reference level is lower

than the lower limit of the confidence interval for the mean X (see §C.3.4.1): −< mX Réf

In this case: ( )

−Φ=

+

−

− σαRéfXm

R 2

21

C.3.6.2 Calculate the success and failure thresholds (bilateral risk)

For a given reliability R and confidence level (1-αααα/2)2, the success threshold xF and the failure threshold xNF are defined below: C.3.6.2.1 CASE 1: R INCREASES WITH THE FUNCTIONAL PARAMETER

Success threshold: xF = m+ + Φ-1(R).σ+

Failure threshold: xNF = m- - Φ-1(R).σ+ C.3.6.2.2 CASE 2: R DECREASES WITH THE FUNCTIONAL PARAMETER

Success threshold: xF = m- - Φ-1(R).σ+

Failure threshold: xNF = m+ + Φ-1(R).σ+ C.3.6.3 Recommendations All the statistical test results (see §A.2.2, reference 28) show that, for every PROBIT test, there are deviations from the “true” distribution:

� To determine reliability at level Xref, an increase (or decrease as appropriate) in the level of Xref by 10% is recommended before calculating reliability.

� To determine a success or failure threshold, an increase (or decrease as appropriate) of 10% in the value of the success or failure threshold calculated is also recommended.

For an example of how this recommendation is applied, see §C.4.1.5.2. C.3.7 SPREADSHEET A spreadsheet running French version of MICROSOFT EXCEL is available on the web site GTPS (http://www.gtps.fr).


C.4 APPLICATION EXAMPLES Preliminary remarks:

� The calculation results presented below are actual examples from the Excel spreadsheet (see §C.3.7);

� The values shown are rounded;

� The spreadsheet (§C.3.7) only uses the results of the sixth iteration in reliability calculations despite earlier convergence (the results thus obtained are more precise).

C.4.1 RELIABILITY OF AN ELECTRO-PYROTECHNIC IGNITER C.4.1.1 Case covered The aim is to evaluate the reliability of an electro-pyrotechnic igniter in terms of the intensity of its ignition current. C.4.1.2 Test conditions The conditions of application of the PROBIT test are as follows:

� The functional parameter under study is the intensity of the ignition current applied to the terminals on the igniter;

� The probability of success varies as the functional parameter under study;

� The success criterion is the ignition within 10ms of applying the current. Either an ignition after 10ms or an non-ignition is therefore counted as a failure;

� The distribution of success thresholds, in response to the intensity of the ignition current, is assumed to be normal;

� We have a total of 90 igniters, taken from a batch that meets the guidelines in § B.3.2.1;

� During the development of the igniter, the mean and the standard deviation of the success thresholds distribution were evaluated at: pX = 1.6 A and sp = 0.02 A.

C.4.1.3 Choice of test levels The following 5 test levels are chosen, with the corresponding sample numbers per level:

Test level Value of functional parameter Number of tests planned per level

pX - sp 1.63 A N/4=18

pX - 0.5 sp 1.64 A N/6=12

pX 1.65 A N/6=12

pX + 0.5 sp 1.66 A N/6=12

pX + sp 1.67 A N/4=18

For a total of 72 firings (leaving 18 igniters in reserve, in order to be able to readjust the levels if necessary).


C.4.1.4 Test procedure As the functional parameter and the probability of success vary in the same direction, the criterion chosen to conduct the tests is the success of the firing. The firings are carried out in accordance with the strategy in § C.2.2.2: the firings executed at the limit levels suggest that these levels, as currently defined, are not outliers.

Success rate per level (%)

Test level

Value of the functional parameter

Number of successes observed per level Observed Theoretically

expected

pX - sp 1.63 A 1 5.56% 16%

pX - 0.5 sp 1.64 A 4 33.33% 30%

pX 1.65 A 6 50.00% 50%

pX + 0.5 sp 1.66 A 11 91.67% 70%

pX + sp 1.67 A 16 88.89% 84%

Although the observed results are different from the expected results, the calculation of the PROBIT line is performed. C.4.1.5 Exploitation of the results C.4.1.5.1 DETERMINATION OF THE MEAN AND THE STANDARD DEVIATION The mean and standard deviation of the distribution of success thresholds are determined by exploiting the test results in accordance with the method in §C.3. The calculations are performed using the spreadsheet referenced in §C.3.7. The first linear regression (provisional PROBIT) yields the following results:

• a0 = -117.67 and b0 = 74.41

• 0X = 1.6484A and s0 = 0.0134A

Convergence test (extract from the GTPS MICROSOFT EXCEL spreadsheet – Cf. §C.3.7):

Itération Moyenne Ecart-type

i a b X S S( X ) S(b) X - S( X ) < X < X + S( X ) b - S(b) < b < b + S(b) Résultat

0 -117,66782 74,41440 1,6484E+00 1,3438E-02 - - Itération (i) Itération (i-1) Itération (i) Itération (i) Itération (i-1) Itération (i) -1 -113,38406 71,80811 1,6486E+00 1,3926E-02 2,5478E-03 1,3944E+01 1,6461E+00 1,6484E+00 1,6512E+00 5,7864E+01 7,4414E+01 8,5752E+01 Oui2 -113,63619 71,80811 1,6486E+00 1,3896E-02 2,5090E-03 1,3591E+01 1,6461E+00 1,6486E+00 1,6511E+00 5,8217E+01 7,1808E+01 8,5399E+01 Oui3 -113,62860 71,95723 1,6486E+00 1,3897E-02 2,5111E-03 1,3611E+01 1,6461E+00 1,6486E+00 1,6511E+00 5,8346E+01 7,1808E+01 8,5569E+01 Oui4 -113,62887 71,95740 1,6486E+00 1,3897E-02 2,5110E-03 1,3611E+01 1,6461E+00 1,6486E+00 1,6511E+00 5,8347E+01 7,1957E+01 8,5568E+01 Oui5 -113,62886 71,95739 1,6486E+00 1,3897E-02 2,5110E-03 1,3611E+01 1,6461E+00 1,6486E+00 1,6511E+00 5,8347E+01 7,1957E+01 8,5568E+01 Oui6 -113,62886 71,95739 1,6486E+00 1,3897E-02 2,5110E-03 1,3611E+01 1,6461E+00 1,6486E+00 1,6511E+00 5,8347E+01 7,1957E+01 8,5568E+01 Oui

La convergence du test est suffisante: Oui dès l'itération N° 1

Ce document a été édité à partir de la feuille de calcul du GTPS (Version du 07/03/2013).Son utilisation relève exclusivement de la responsabilité de son utilisateur.

Droite des PROBITS :Y = a+b*X

Ecart-types estimateurs Test de précision


The requirements for stopping the iterations are met at the first iteration:

• a1 = -113.38 and b1 = 71.81 A-1

• 1X = 1.649 A and s1 = 0.0139 A

o where the standard deviation estimators are S(1X ) = 2.55.10-3 A and S(b1) = 13.94 A-1

• Convergence test: o 1.6461= X1 − s X1( ) < X0 =1.6484< X1 + s X1( ) =1.6512

o 001728.0)(

10134.0

)(1

001166.011

011

=−

<=<+

=bsb

sbsb

• The first iteration is therefore sufficient for the calculation of X and S. Normality test:

• The results of the χ2 test were:

o χ2 calculated = 2.46 < χ2

theoretical (3dof, 5%) = 7.81

o Thus, the normal distribution hypothesis cannot be rejected, with a risk of 5%. We therefore retain the final results:

Characteristic Estimator Standard deviation of estimator

Mean X = 1.649 A )(XS = 0.00251 A

Slope b = 71.96 A-1 S(b) = 13.6 A-1

Standard deviation S = 0.0139 A -

The PROBIT line is plotted on the graph below (*):

Méthode PROBIT : Droite des PROBIT finauxInflammateur électrique

3,00

3,50

4,00

4,50

5,00

5,50

6,00

6,50

7,00

1,62E+00 1,63E+00 1,64E+00 1,65E+00 1,66E+00 1,67E+00 1,68E+00

Paramètre fonctionnel X


PR

OB

ITS

Méthode GTPS N°11A

(*) Extract from the GTPS MICROSOFT EXCEL spreadsheet (Cf. §C.3.7):


C.4.1.5.2 EVALUATION OF THE IGNITER RELIABILITY The reference value xRef of parameter x (in our case, the minimum firing current) can be used to calculate the reliability of the device at a given confidence level (1-α/2)² using the estimates of m and s at confidence level (1-α):

−Φ=

+

+

− S

mXR

Réf2)2/1( α

Where Φ(t) is the standard normal distribution function, for which the values are listed in Table 4, appendix E4. The nominal intensity of the ignition current is 2A: given the dispersion of the firing system employed, it is higher than 1.9A. Following the recommendation in §C.3.6.3, the reference level is determined by diminishing the nominal level by 10%:

• XRef = 1.9/1.1 = 1.727 A For a confidence level of 1-α = 90% (i.e. where uα/2 = 1.64), the bilateral confidence interval limits for the mean and the standard deviation are:

• m+ = +uα/2.s( ) = 1.649 + 1.64*0.00251 = 1.653 A

• s+ = 1

b − uα.s(b)= 1

71.96−1.28*13.6 = 0.0202 A

Then the reliability at confidence level (1-α/2)2 = 90.25% is:

• R90.25% = Φ1.727−1.653

0.0202

= Φ 3.6816( ) = 0.99988

This electro-pyrotechnic igniter has an ignition reliability of R = 0.99988, at a confidence level of 90.25%, with a reference ignition current equal to Xref = 1.727 A. C.4.2 ACCEPTANCE PROCEDURE FOR A BATCH OF CUTTING CORDS This case concerns the method used for the acceptance testing of cutting cords belonging to the same batch.

Cutting cord diagram

Dihedral for cutting (linear hollow charge)

Metal schell of cutting cord

Core cord (explosive)

Dihedral for cutting (linear hollow charge)

Metal schell of cutting cord

Core cord (explosive)


We are investigating the reliability of a batch of cutting cords during acceptance testing: • The success criterion corresponds to the cutting of a defined target (material, thickness, etc.)

• The batch contains 19 manufactured cords:

o They are all taken from the same batches of raw materials (explosives and metal sheath);

o They were all made on the same machine, during a single production run.

• From each cord, we take 3 segments for the acceptance firings (1 from each end and 1 from the middle), for a total of 57 firing specimens;

• The condition for acceptance of the batch of cords relates to the operating threshold:

o The operating threshold is calculated for a reliability of R=1-1E-6 with a confidence level of 90%;

o The operating threshold (depth of cut) for the batch must be equal to or greater than 3.3mm.

C.4.2.1 Test conditions The application conditions of the PROBIT test are as follows:

• The functional parameter under study is the thickness of a sample plate, to be cut by the cord;

• The segment of cord is placed on a “stepped” metal target, with several levels of thickness;

• The success criterion for each level is whether the cord completely cuts through that level of thickness on the stepped target;

• The distribution of operating thresholds, in terms of the depth of cut, is assumed to be normal;

• We have a total of 57 stepped targets and 57 detonators to ignite the cord segments;

• The thicknesses of the stepped targets are defined in advance, and are listed in the acceptance conditions for cord batches: o 13 levels, ranging from 4.3mm to 5.5mm; o constant increments of 0.1mm between successive levels.

Cutting-cord firing tests: setup diagram

Staggered target (test levels)

Cutting cord (sample)

Détonator

Staggered target (test levels)

Cutting cord (sample)

Détonator


C.4.2.2 Test procedure The tests went entirely to plan, yielding the following results:

Test levels (thickness of stepped target in mm) Cord n° Seg. Firing

n° 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50

A 1 1 1 1 1 1 0 0 0 0 0 0 0 0 B 2 1 1 1 1 1 0 0 0 0 0 0 0 0 Cord N°1 C 3 1 1 0 0 0 0 0 0 0 0 0 0 0 A 4 1 1 1 1 0 0 0 0 0 0 0 0 0 B 5 1 1 1 0 0 0 0 0 0 0 0 0 0 Cord N°2 C 6 1 1 1 1 0 0 0 0 0 0 0 0 0 A 7 1 1 1 1 1 0 0 0 0 0 0 0 0 B 8 1 1 1 0 0 0 0 0 0 0 0 0 0 Cord N°3 C 9 1 1 1 1 0 0 0 0 0 0 0 0 0 A 10 1 1 1 1 1 0 0 0 0 0 0 0 0 B 11 1 1 1 1 1 0 0 0 0 0 0 0 0 Cord N°4 C 12 1 1 1 0 0 0 0 0 0 0 0 0 0 A 13 1 1 1 0 0 0 0 0 0 0 0 0 0 B 14 1 1 1 1 1 0 0 0 0 0 0 0 0 Cord N°5 C 15 1 1 1 0 0 0 0 0 0 0 0 0 0 A 16 1 1 1 0 1 0 0 0 0 0 0 0 0 B 17 1 1 1 1 0 0 0 0 0 0 0 0 0 Cord N°6 C 18 1 0 1 0 1 0 0 0 0 0 0 0 0 A 19 1 1 1 1 1 1 1 0 1 0 0 0 0 B 20 1 1 1 1 1 1 1 1 0 0 0 0 0 Cord N°7 C 21 1 1 1 1 1 1 1 1 1 0 0 0 0 A 22 1 1 1 1 1 1 1 0 0 0 0 0 0 B 23 1 1 1 1 1 1 1 0 0 0 0 0 0 Cord N°8 C 24 1 1 1 1 1 1 1 0 0 0 0 0 0 A 25 1 1 1 1 1 1 0 1 1 0 0 0 0 B 26 1 1 1 1 1 0 1 0 0 0 0 0 0 Cord N°9 C 27 1 1 1 1 1 1 0 0 0 0 0 0 0 A 28 1 1 1 1 1 1 0 0 0 0 0 0 0 B 29 1 1 1 1 1 1 0 0 0 0 0 0 0 Cord N°10 C 30 1 1 1 1 1 1 1 0 0 0 0 0 0 A 31 1 1 1 1 1 1 1 1 1 0 0 0 0 B 32 1 1 1 1 1 0 1 0 0 0 0 0 0 Cord N°11 C 33 1 1 1 1 1 1 0 0 0 0 0 0 0 A 34 1 1 1 1 1 1 0 0 0 0 0 0 0 B 35 1 1 1 1 0 0 0 0 0 0 0 0 0 Cord N°12 C 36 1 1 1 1 1 0 0 0 0 0 0 0 0 A 37 1 1 1 1 1 1 1 0 0 0 0 0 0 B 38 1 1 0 0 1 0 0 0 0 0 0 0 0 Cord N°13 C 39 1 1 1 1 0 0 0 0 0 0 0 0 0 A 40 1 1 1 1 1 1 0 0 0 0 0 0 0 B 41 1 0 1 1 1 0 0 0 0 0 0 0 0 Cord N°14 C 42 1 1 0 0 1 0 0 0 0 0 0 0 0 A 43 1 1 1 0 0 0 0 0 0 0 0 0 0 B 44 1 1 1 0 0 0 0 0 0 0 0 0 0 Cord N°15 C 45 1 1 1 0 0 0 0 0 0 0 0 0 0 A 46 1 1 1 1 1 0 0 0 0 0 0 0 0 B 47 1 1 1 0 0 0 0 0 0 0 0 0 0 Cord N°16 C 48 1 1 0 1 0 0 0 0 0 0 0 0 0 A 49 1 1 1 1 1 0 0 0 0 0 0 0 0 B 50 1 1 1 1 1 1 0 0 0 0 0 0 0 Cord N°17 C 51 1 1 1 1 0 0 0 0 0 0 0 0 0 A 52 1 1 0 0 0 0 0 0 0 0 0 0 0 B 53 1 1 1 1 0 0 0 0 0 0 0 0 0 Cord N°18 C 54 1 1 1 0 0 0 0 0 0 0 0 0 0 A 55 1 1 1 1 0 0 0 0 0 0 0 0 0 B 56 1 1 1 1 0 0 0 0 0 0 0 0 0 Cord N°19 C 57 1 1 1 1 0 1 0 0 0 0 0 0 0

Successes are represented by 1 and failures by 0.


C.4.2.3 Exploiting the results We exploit the non-outlying levels with the PROBIT method, giving the following results on the eight exploitable levels (i.e. with success rates other than 0% or 100%):

Functional parameter

Sample size per level

Number of successes observed

Observed success rate

4.4 mm 57 55 96.49%

4.5 mm 57 52 91.23%

4.6 mm 57 41 71.93%

4.7 mm 57 33 57.89%

4.8 mm 57 18 31.58%

4.9 mm 57 11 19.30%

5.0 mm 57 4 7.02%

5.1 mm 57 4 7.02%

The calculations were performed using the spreadsheet referenced in §C.3.7. The first linear regression (provisional PROBITs) yields the following results:

• a0 = 28.80 and b0 = -5.02

• 0X = 4.74mm and s0 = 0.199mm




0 28,80003 -5,01974 4,7413E+00 1,9921E-01 - - Itération (i) Itération (i-1) Itération (i) Itération (i) Itération (i-1) Itération (i) -1 28,95593 -5,06067 4,7337E+00 1,9760E-01 1,4273E-02 3,8698E-01 4,7297E+00 4,7413E+00 4,7582E+00 4,6737E+00 5,0197E+00 5,4476E+00 oui2 28,95535 -5,06067 4,7337E+00 1,9761E-01 1,4318E-02 3,8980E-01 4,7194E+00 4,7337E+00 4,7480E+00 4,6708E+00 5,0607E+00 5,4504E+00 oui3 28,95524 -5,06057 4,7337E+00 1,9761E-01 1,4318E-02 3,8980E-01 4,7243E+00 4,7337E+00 4,7529E+00 4,6708E+00 5,0607E+00 5,4504E+00 oui4 28,95524 -5,06057 4,7337E+00 1,9761E-01 1,4318E-02 3,8979E-01 4,7243E+00 4,7337E+00 4,7529E+00 4,6708E+00 5,0606E+00 5,4504E+00 oui5 28,95524 -5,06057 4,7337E+00 1,9761E-01 1,4318E-02 3,8979E-01 4,7243E+00 4,7337E+00 4,7529E+00 4,6708E+00 5,0606E+00 5,4504E+00 oui6 28,95524 -5,06057 4,7337E+00 1,9761E-01 1,4318E-02 3,8979E-01 4,7243E+00 4,7337E+00 4,7529E+00 4,6708E+00 5,0606E+00 5,4504E+00 oui






The conditions for stopping the iterations are met as of the first iteration:

• a1 = 28.96 and b1 = -5.06

• 1X = 4.734mm and s1 = 0.1976mm

o where the standard deviations of the estimators are s( 1X ) = 0.0143mm and s(b1) = 0.39mm-1

• Convergence test:

o 4.7297= X1 − s X1( ) < X0 = 4.74< X1 + s X1( ) = 4.7582

o 0.1836= 1

b1 + s(b1)< s0 = 0.199 < 1

b1 − s(b1)= 0.2139

• The first iteration is therefore sufficient for the calculation of X and s. Normality test:




o Thus, the normal distribution hypothesis cannot be rejected, with a risk of 5%.

We therefore retain the following results (6th iteration):


Mean X = 4.734 mm ( )XS = 0.01432 mm

Slope b = -5.06 mm-1 s(b) = 0.39 mm-1

Standard deviation s = 0.1976 mm -


Méthode PROBIT : Droite des PROBIT finauxTirs de recette cordeaux découpeurs

3,00

3,50

4,00

4,50

5,00

5,50

6,00

6,50

7,00

4,30E+00 4,40E+00 4,50E+00 4,60E+00 4,70E+00 4,80E+00 4,90E+00 5,00E+00 5,10E+00 5,20E+00



PR

OB

ITS




C.4.2.3.1 EVALUATING THE OPERATING THRESHOLD OF THE CUTTING CORD The probability of correct operation and the functional parameter vary in different directions. The operating threshold is given by the following formula:

( ) +−

− Φ−= σ.1 RmXF For a confidence level of 1-α = 90%, we obtain:

• uα2

= 1.64

• ( ) 71.401432.064.1734.42

=×−=×−=− XsuXm α mm

• ( ) 2263.039.064.106.5

11

2/

=×−

=×−

=+ Xsub α

σ mm

The operating threshold is therefore (for a cutting reliability of 1-1E-6):

• ( ) ( ) 635.32263.075.471.42263.061171.4 11 =×−=×−−Φ−=×Φ−= −+

−− ERmXF σ mm

For this batch of cutting cords, the calculated operating threshold is greater than the acceptance criterion (3.3 mm), and the acceptance test is therefore passed. C.4.3 “TRUE DISTRIBUTION” PROBIT OF AN ELECTRO-PYROTECHN IC IGNITER C.4.3.1 Case study The aim is to concatenate a number of Bruceton and PROBIT tests conducted on a single batch of electro-pyrotechnic igniters. This generalised PROBIT analysis was put together during statistical method comparison tests conducted as part of the CNES PAQTE 2004 project (reference 28 in §A.2.2). It concatenates and exploits the results of:

• 5 Bruceton tests;

• 4 PROBIT tests;

• A total of 591 firings, performed on the batch of igniters. The igniter is the same as the one presented in the example in §C.4.1.


C.4.3.2 Results of the concatenated tests The results of the 591 firings at the 38 exploitable levels are concatenated in the table below:

Functional parameter Level Functional

parameter Sample size

per level Successes observed

Observed success rate

1 1.675 A 14 13 92.86% 2 1.660 A 29 16 55.17% 3 1.645 A 20 4 20.00%

Bruceton N°1 (74 firings)

4 1.630 A 7 3 42.86% 5 1.675 A 15 12 80.00% 6 1.660 A 24 11 45.83%


7 1.645 A 16 5 31.25% 8 1.680 A 15 13 86.67% 9 1.662 A 26 13 50.00% 10 1.643 A 19 6 31.58%


11 1.625 A 7 1 14.29% 12 1.671 A 8 7 87.50% 13 1.658 A 13 6 46.15% 14 1.6456 A 16 10 62.50%


15 1.633 A 13 3 23.08% 16 1.6323 A 13 3 23.08% 17 1.639 A 20 10 50.00%


18 1.6456 A 16 9 56.25% 19 1.603 A 18 1 5.56% 20 1.618 A 12 3 25.00% 21 1.6323 A 12 2 16.67% 22 1.647 A 12 4 33.33%

PROBIT N°1 (78 firings)

23 1.6612 A 18 11 61.11% 24 1.631 A 18 4 22.22% 25 1.6454 A 12 3 25.00% 26 1.657 A 12 3 25.00% 27 1.664 A 12 6 50.00%


28 1.670 A 18 17 94.44% 29 1.631 A 18 2 11.11% 30 1.6398 A 24 7 29.17% 31 1.646 A 12 5 41.67% 32 1.653 A 12 10 83.33%


33 1.6601 A 18 11 61.11% 34 1.636 A 18 6 33.33% 35 1.645 A 12 4 33.33% 36 1.653 A 12 6 50.00% 37 1.662 A 12 11 91.67%


38 1.670 A 18 16 88.89%


C.4.3.3 Exploiting the results C.4.3.3.1 DETERMINING THE MEAN AND THE STANDARD DEVIATION The mean and standard deviation of the distribution of operating thresholds are determined by exploiting the test results in accordance with the method in § C.3. The calculations were performed using the worksheet referenced in §C.3.7. The first linear regression (provisional PROBITs) yields the following results:

• a0 = -59.25 and b0 = 38.93

• 0X = 1.6505 and s0 = 0.025687A




0 -59,25350 38,92970 1,6505E+00 2,5687E-02 - - Itération (i) Itération (i-1) Itération (i) Itération (i) Itération (i-1) Itération (i) -1 -55,79406 36,79644 1,6522E+00 2,7177E-02 1,4983E-03 3,8299E+00 1,6507E+00 1,6505E+00 1,6537E+00 3,2966E+01 3,8930E+01 4,0626E+01 Non2 -55,85241 36,79644 1,6522E+00 2,7150E-02 1,4884E-03 3,7722E+00 1,6507E+00 1,6522E+00 1,6536E+00 3,3024E+01 3,6796E+01 4,0569E+01 Oui3 -55,85286 36,83229 1,6522E+00 2,7150E-02 1,4886E-03 3,7733E+00 1,6507E+00 1,6522E+00 1,6536E+00 3,3059E+01 3,6796E+01 4,0606E+01 Oui4 -55,85288 36,83230 1,6522E+00 2,7150E-02 1,4886E-03 3,7733E+00 1,6507E+00 1,6522E+00 1,6536E+00 3,3059E+01 3,6832E+01 4,0606E+01 Oui5 -55,85288 36,83230 1,6522E+00 2,7150E-02 1,4886E-03 3,7733E+00 1,6507E+00 1,6522E+00 1,6536E+00 3,3059E+01 3,6832E+01 4,0606E+01 Oui6 -55,85288 36,83230 1,6522E+00 2,7150E-02 1,4886E-03 3,7733E+00 1,6507E+00 1,6522E+00 1,6536E+00 3,3059E+01 3,6832E+01 4,0606E+01 Oui





Convergence is achieved only at the second iteration.

� At the first iteration, we obtain the following results:

• a1 = -55.79 and b1 = 36.80

• 1X = 1.6522A and s1 = 0.02718A

o where standard deviations of the estimators are s(1X ) = 1.498.10-3 and s(b1) = 3.83

• The convergence test on the mean is not satisfied:

o 1.6507= X1 − s X1( ) > X0 =1.6505< X1 + s X1( ) =1.6537

� After the second iteration we obtain the following results:

• a2 = -55.85 and b2 = 36.80

• 2X = 1.6522A and s2 = 0.02715A

o where standard deviations of the estimators are s(2X ) = 1.498.10-3 and s(b2) = 3.77

• This time, the conditions for stopping the iterations are met at the second iteration:

o 1.6507= X2 − s X2( ) < X1 =1.6522< X2 + s X2( ) =1.6536

o 0.00246= 1

b1 + s(b1)< s0 = 0.002715 < 1

b1 − s(b1)= 0.00303

Normality test:



theoretical (36dof, 5%) = 51

o Thus, the normal distribution hypothesis cannot be rejected, with a risk of 5%.




Mean X = 1.6522 A ( )XS = 0.00149 A

Slope b = 36.83 A-1 s(b) = 3.77 A-1

Standard deviation s = 0.02715 A -


Méthode PROBIT : Droite des PROBIT finauxPROBIT "loi vraie" inflammateur électrique (CNES-PA QTE 2004)

3,00

3,50

4,00

4,50

5,00

5,50

6,00

6,50

1,59E+00 1,60E+00 1,61E+00 1,62E+00 1,63E+00 1,64E+00 1,65E+00 1,66E+00 1,67E+00 1,68E+00 1,69E+00



PR

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ITS




C.4.3.3.2 EVALUATING THE RELIABILITY OF THE IGNITER We want to evaluate the reference level that will guarantee a reliability of R=1-1E-6. The probability of correct operation and the functional parameter vary in the same direction. The operating threshold is given by the following formula:

( ) +−

+ Φ+= σ.1 RmXF For a confidence level of 1-α = 90%, we obtain:

• uα2

= 1.64

• ( ) 655.100149.064.16522.12/ =×+=×+=+ XsuXm α A

• ( ) 03265.077.364.183.36

11

2/

=×−

=×−

=+ bsub α

σ A

The operating threshold is therefore:

• ( ) ( ) 81.103265.075.4655.103265.06.11655.1 11 =×+=×−−Φ+=×Φ+= −+

−+ ERmXF σ A

This usage represents the “true distribution” of the electro-pyrotechnic igniter under study:

• Given the large number of firing results exploited, there is no need to apply a 10% margin to the calculation of the operating threshold XF (the recommendation in §C.3.6.3 can be ignored);

• We can conclude that the device has an ignition reliability of R = 1-1E-6, at a confidence level of 90.25%, with a reference priming current equal to Xref = 1.81 A.


C.4.4 RE-USING RESULTS FROM AN UNUSABLE BRUCETON TEST C.4.4.1 Case study We conducted the following closed series of Bruceton tests (as per GTPS document N°11C cited in reference 3 of §A.2.1:

= 7'

= 8'

= 9'

= 10

'

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

8,007,00 S S6,00 S E S S S S E S5,00 S E S E S E S E S E S4,00 S E E S E S E S E S3,00 S E S E E E S2,00 E E1,00

Preliminary tests Closed sequence

Functional parameter

1' 2' 3' 4' 5' 6' 7' 8' 9' 10'

The exploitation of this closed series yields the following results:

XiRank of level

test: iNumber of tests:

nn*i n*i*i

7,0000E+00 5 2 10 50

6,0000E+00 4 7 28 112

5,0000E+00 3 10 30 90

4,0000E+00 2 9 18 36

3,0000E+00 1 6 6 6

2,0000E+00 0 2 0 0

ΣΣΣΣ 36 92 294=∗=∑=

k

ii inB

1

2=∗=∑=

k

ii inA

1

==∑=

k

iiS nN

1

U = NN − 2

∗ N ∗ B− A2

N2− 1

4

=1.4673 X = 4.556 s = 2.4944

As the test increment is equal to d = 1, the standard deviation ratio comes to = 2.4944 > 2. In accordance with GTPS 11C (reference 3 in §A.2.1), we conclude that this test is unusable. C.4.4.2 Retrieving firing data for a PROBIT calculation We can retrieve the following firing results for use with the PROBIT method:

Test levels Number of tests Successes observed Calculated success rate

6.00 7 5 71.43% 5.00 10 5 50.00% 4.00 9 4 44.44% 3.00 6 2 33.33% Total 32 16 -

Giving us a total of 32 tests to use.


C.4.4.3 Exploiting the results The mean and standard deviation of the distribution of operating thresholds are determined by exploiting the test results in accordance with the method in § C.3. The calculations were performed using the spreadsheet referenced in §C.3.7. The first linear regression (provisional PROBITs) yields the following results:

• a0 = 7.32 and b0 = -0.50

• 0X = 4.64 and s0 = 2.00




0 7,31888 -0,49932 4,6441E+00 2,0027E+00 - - Itération (i) Itération (i-1) Itération (i) Itération (i) Itération (i-1) Itération (i) -1 7,18939 -0,47880 4,5727E+00 2,0886E+00 4,8488E-01 2,3576E-01 4,0878E+00 4,6441E+00 5,0575E+00 2,4304E-01 4,9932E-01 7,1456E-01 Oui2 7,18906 -0,47880 4,5729E+00 2,0890E+00 4,8311E-01 2,3386E-01 4,0898E+00 4,5727E+00 5,0560E+00 2,4494E-01 4,7880E-01 7,1266E-01 Oui3 7,18907 -0,47871 4,5729E+00 2,0890E+00 4,8310E-01 2,3385E-01 4,0898E+00 4,5729E+00 5,0560E+00 2,4485E-01 4,7880E-01 7,1256E-01 Oui4 7,18907 -0,47871 4,5729E+00 2,0890E+00 4,8310E-01 2,3385E-01 4,0898E+00 4,5729E+00 5,0560E+00 2,4485E-01 4,7871E-01 7,1256E-01 Oui5 7,18907 -0,47871 4,5729E+00 2,0890E+00 4,8310E-01 2,3385E-01 4,0898E+00 4,5729E+00 5,0560E+00 2,4485E-01 4,7871E-01 7,1256E-01 Oui6 7,18907 -0,47871 4,5729E+00 2,0890E+00 4,8310E-01 2,3385E-01 4,0898E+00 4,5729E+00 5,0560E+00 2,4485E-01 4,7871E-01 7,1256E-01 Oui





The conditions for stopping the iterations are met as of the first iteration:

• a1 = 7.19 and b1 = -0.48

• 1X = 4.57 and s1 = 2.09

o where the standard deviations of the estimators are s( 1X ) = 0.485 and s(b1) = 0.236

• Convergence test:

o 4.09= X1 − s X1( ) < X0 = 4.64< X1 + s X1( ) = 5.06

o 1.3995= 1

b1 + s(b1)< s0 = 2.00 < 1

b1 − s(b1)= 4.1146

o The first iteration is therefore sufficient for the calculation of X and s.

Normality test: The results of the χ2 test were:

• χ2 calculated = 0.29 < χ2


• Thus, the normal distribution hypothesis cannot be rejected, with a risk of 5%.




Mean X = 4.57 ( )XS = 0.483

Slope b = -0.50 s(b) = 0.234

Standard deviation s = 2.09 -

These values should be compared to the initial estimates given by the Bruceton test:

( X = 4.556 & s = 2.4944). The PROBIT line is plotted on the graph below (*):

Méthode PROBIT : Droite des PROBIT finauxRéexploitation BRUCETON non exploitable (S/d > 2)

3,00

3,50

4,00

4,50

5,00

5,50

6,00

6,50

2,00E+00 2,50E+00 3,00E+00 3,50E+00 4,00E+00 4,50E+00 5,00E+00 5,50E+00 6,00E+00 6,50E+00



PR

OB

ITS



Note that this method for re-using data from an unusable Bruceton test only applies where S/d>2, this being the only case that provides us with enough levels to exploit with the PROBIT method.


D. CONCLUSION This document describes how to apply the PROBIT method for evaluating the reliability of a one-shot device. It explains the steps to follow (the conditions of actual implementation) with a particular focus on:

• the determination of the test levels; • the conduct of the tests.

The method provides a reasonably precise estimate of the mean and dispersion of operating thresholds for a sample size of at least 72, on condition that the operating thresholds of the functional parameter being sampled follow a normal or log-normal distribution. Extended to an infinite population and associated with a given confidence level, these estimates can be used to evaluate the reliability of a one-shot device for any given value of the functional parameter. The method remains applicable to smaller samples, although with a deterioration in the quality of the results and an increased risk of obtaining an unusable test (a risk which is already significant even under ideal test conditions). The method can be used for:

• exploiting a series of tests, including for batch acceptance where the reliability level and the predetermination of test levels are part of the acceptance criteria (the most familiar application is for the acceptance of cutting cords on stepped targets);

• concatenating the results of Bruceton and/or PROBIT tests from different test campaigns in order to derive, from a large test sample, the characteristics of the “true distribution”;

• recycling data from an otherwise unusable Bruceton test series (where S/d > 2).


E. APPENDICES (TABLES): Appendix E1 Empirical PROBIT Yi as a function of observed success rate pi

Appendix E2 Weighting coefficient Wi as a function of provisional PROBIT value Ypi

Appendix E3 Working PROBIT zi as a function of observed success rate pi and provisional PROBIT value Ypi

Appendix E4 Values of the standard normal distribution function

Appendix E5 Values of ),2/(2

ναχ as a function of the number of degrees of freedom νννν and the confidence

level (1-αααα)


E.1 Empirical PROBIT Yi as a function of observed success rate pi

% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1.9098 2.1218 2.2522 2.3479 2.4242 2.4879 2.5427 2.5911 2.6344 1 2.6737 2.7096 2.7429 2.7738 2.8027 2.8299 2.8556 2.8799 2.9031 2.9252 2 2.9463 2.9665 2.9859 3.0046 3.0226 3.0400 3.0569 3.0732 3.0890 3.1043 3 3.1192 3.1337 3.1478 3.1616 3.1750 3.1881 3.2009 3.2134 3.2256 3.2376 4 3.2493 3.2608 3.2721 3.2831 3.2940 3.3046 3.3151 3.3253 3.3354 3.3454 5 3.3551 3.3648 3.3742 3.3836 3.3928 3.4018 3.4107 3.4195 3.4282 3.4368 6 3.4452 3.4536 3.4618 3.4699 3.4780 3.4859 3.4937 3.5015 3.5091 3.5167 7 3.5242 3.5316 3.5389 3.5462 3.5534 3.5605 3.5675 3.5745 3.5813 3.5882 8 3.5949 3.6016 3.6083 3.6148 3.6213 3.6278 3.6342 3.6405 3.6468 3.6531 9 3.6592 3.6654 3.6715 3.6775 3.6835 3.6894 3.6953 3.7012 3.7070 3.7127 10 3.7184 3.7241 3.7298 3.7354 3.7409 3.7464 3.7519 3.7574 3.7628 3.7681 11 3.7735 3.7788 3.7840 3.7893 3.7945 3.7996 3.8048 3.8099 3.8150 3.8200 12 3.8250 3.8300 3.8350 3.8399 3.8448 3.8497 3.8545 3.8593 3.8641 3.8689 13 3.8736 3.8783 3.8830 3.8877 3.8923 3.8969 3.9015 3.9061 3.9107 3.9152 14 3.9197 3.9242 3.9286 3.9331 3.9375 3.9419 3.9463 3.9506 3.9549 3.9593 15 3.9636 3.9678 3.9721 3.9763 3.9806 3.9848 3.9890 3.9931 3.9973 4.0014 16 4.0055 4.0096 4.0137 4.0178 4.0218 4.0259 4.0299 4.0339 4.0379 4.0419 17 4.0458 4.0498 4.0537 4.0576 4.0615 4.0654 4.0693 4.0731 4.0770 4.0808 18 4.0846 4.0884 4.0922 4.0960 4.0998 4.1035 4.1073 4.1110 4.1147 4.1184 19 4.1221 4.1258 4.1294 4.1331 4.1368 4.1404 4.1440 4.1476 4.1512 4.1548 20 4.1584 4.1619 4.1655 4.1690 4.1726 4.1761 4.1796 4.1831 4.1866 4.1901 21 4.1936 4.1970 4.2005 4.2039 4.2074 4.2108 4.2142 4.2176 4.2210 4.2244 22 4.2278 4.2312 4.2345 4.2379 4.2412 4.2446 4.2479 4.2512 4.2546 4.2579 23 4.2612 4.2644 4.2677 4.2710 4.2743 4.2775 4.2808 4.2840 4.2872 4.2905 24 4.2937 4.2969 4.3001 4.3033 4.3065 4.3097 4.3129 4.3160 4.3192 4.3224 25 4.3255 4.3287 4.3318 4.3349 4.3380 4.3412 4.3443 4.3474 4.3505 4.3536 26 4.3567 4.3597 4.3628 4.3659 4.3689 4.3720 4.3750 4.3781 4.3811 4.3842 27 4.3872 4.3902 4.3932 4.3962 4.3992 4.4022 4.4052 4.4082 4.4112 4.4142 28 4.4172 4.4201 4.4231 4.4260 4.4290 4.4319 4.4349 4.4378 4.4408 4.4437 29 4.4466 4.4495 4.4524 4.4554 4.4583 4.4612 4.4641 4.4670 4.4698 4.4727 30 4.4756 4.4785 4.4813 4.4842 4.4871 4.4899 4.4928 4.4956 4.4985 4.5013 31 4.5042 4.5070 4.5098 4.5126 4.5155 4.5183 4.5211 4.5239 4.5267 4.5295 32 4.5323 4.5351 4.5379 4.5407 4.5435 4.5462 4.5490 4.5518 4.5546 4.5573 33 4.5601 4.5628 4.5656 4.5684 4.5711 4.5739 4.5766 4.5793 4.5821 4.5848 34 4.5875 4.5903 4.5930 4.5957 4.5984 4.6011 4.6039 4.6066 4.6093 4.6120 35 4.6147 4.6174 4.6201 4.6228 4.6255 4.6281 4.6308 4.6335 4.6362 4.6389 36 4.6415 4.6442 4.6469 4.6495 4.6522 4.6549 4.6575 4.6602 4.6628 4.6655 37 4.6681 4.6708 4.6734 4.6761 4.6787 4.6814 4.6840 4.6866 4.6893 4.6919 38 4.6945 4.6971 4.6998 4.7024 4.7050 4.7076 4.7102 4.7129 4.7155 4.7181 39 4.7207 4.7233 4.7259 4.7285 4.7311 4.7337 4.7363 4.7389 4.7415 4.7441 40 4.7467 4.7492 4.7518 4.7544 4.7570 4.7596 4.7622 4.7647 4.7673 4.7699 41 4.7725 4.7750 4.7776 4.7802 4.7827 4.7853 4.7879 4.7904 4.7930 4.7955 42 4.7981 4.8007 4.8032 4.8058 4.8083 4.8109 4.8134 4.8160 4.8185 4.8211 43 4.8236 4.8262 4.8287 4.8313 4.8338 4.8363 4.8389 4.8414 4.8440 4.8465 44 4.8490 4.8516 4.8541 4.8566 4.8592 4.8617 4.8642 4.8668 4.8693 4.8718 45 4.8743 4.8769 4.8794 4.8819 4.8844 4.8870 4.8895 4.8920 4.8945 4.8970 46 4.8996 4.9021 4.9046 4.9071 4.9096 4.9122 4.9147 4.9172 4.9197 4.9222 47 4.9247 4.9272 4.9298 4.9323 4.9348 4.9373 4.9398 4.9423 4.9448 4.9473 48 4.9498 4.9524 4.9549 4.9574 4.9599 4.9624 4.9649 4.9674 4.9699 4.9724 49 4.9749 4.9774 4.9799 4.9825 4.9850 4.9875 4.9900 4.9925 4.9950 4.9975 50 5.0000 5.0025 5.0050 5.0075 5.0100 5.0125 5.0150 5.0175 5.0201 5.0226


Appendix E1 (ctd.): Empirical PROBIT Yi as a function of observed success rate pi

% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 51 5.0251 5.0276 5.0301 5.0326 5.0351 5.0376 5.0401 5.0426 5.0451 5.0476 52 5.0502 5.0527 5.0552 5.0577 5.0602 5.0627 5.0652 5.0677 5.0702 5.0728 53 5.0753 5.0778 5.0803 5.0828 5.0853 5.0878 5.0904 5.0929 5.0954 5.0979 54 5.1004 5.1030 5.1055 5.1080 5.1105 5.1130 5.1156 5.1181 5.1206 5.1231 55 5.1257 5.1282 5.1307 5.1332 5.1358 5.1383 5.1408 5.1434 5.1459 5.1484 56 5.1510 5.1535 5.1560 5.1586 5.1611 5.1637 5.1662 5.1687 5.1713 5.1738 57 5.1764 5.1789 5.1815 5.1840 5.1866 5.1891 5.1917 5.1942 5.1968 5.1993 58 5.2019 5.2045 5.2070 5.2096 5.2121 5.2147 5.2173 5.2198 5.2224 5.2250 59 5.2275 5.2301 5.2327 5.2353 5.2378 5.2404 5.2430 5.2456 5.2482 5.2508 60 5.2533 5.2559 5.2585 5.2611 5.2637 5.2663 5.2689 5.2715 5.2741 5.2767 61 5.2793 5.2819 5.2845 5.2871 5.2898 5.2924 5.2950 5.2976 5.3002 5.3029 62 5.3055 5.3081 5.3107 5.3134 5.3160 5.3186 5.3213 5.3239 5.3266 5.3292 63 5.3319 5.3345 5.3372 5.3398 5.3425 5.3451 5.3478 5.3505 5.3531 5.3558 64 5.3585 5.3611 5.3638 5.3665 5.3692 5.3719 5.3745 5.3772 5.3799 5.3826 65 5.3853 5.3880 5.3907 5.3934 5.3961 5.3989 5.4016 5.4043 5.4070 5.4097 66 5.4125 5.4152 5.4179 5.4207 5.4234 5.4261 5.4289 5.4316 5.4344 5.4372 67 5.4399 5.4427 5.4454 5.4482 5.4510 5.4538 5.4565 5.4593 5.4621 5.4649 68 5.4677 5.4705 5.4733 5.4761 5.4789 5.4817 5.4845 5.4874 5.4902 5.4930 69 5.4958 5.4987 5.5015 5.5044 5.5072 5.5101 5.5129 5.5158 5.5187 5.5215 70 5.5244 5.5273 5.5302 5.5330 5.5359 5.5388 5.5417 5.5446 5.5476 5.5505 71 5.5534 5.5563 5.5592 5.5622 5.5651 5.5681 5.5710 5.5740 5.5769 5.5799 72 5.5828 5.5858 5.5888 5.5918 5.5948 5.5978 5.6008 5.6038 5.6068 5.6098 73 5.6128 5.6158 5.6189 5.6219 5.6250 5.6280 5.6311 5.6341 5.6372 5.6403 74 5.6433 5.6464 5.6495 5.6526 5.6557 5.6588 5.6620 5.6651 5.6682 5.6713 75 5.6745 5.6776 5.6808 5.6840 5.6871 5.6903 5.6935 5.6967 5.6999 5.7031 76 5.7063 5.7095 5.7128 5.7160 5.7192 5.7225 5.7257 5.7290 5.7323 5.7356 77 5.7388 5.7421 5.7454 5.7488 5.7521 5.7554 5.7588 5.7621 5.7655 5.7688 78 5.7722 5.7756 5.7790 5.7824 5.7858 5.7892 5.7926 5.7961 5.7995 5.8030 79 5.8064 5.8099 5.8134 5.8169 5.8204 5.8239 5.8274 5.8310 5.8345 5.8381 80 5.8416 5.8452 5.8488 5.8524 5.8560 5.8596 5.8632 5.8669 5.8706 5.8742 81 5.8779 5.8816 5.8853 5.8890 5.8927 5.8965 5.9002 5.9040 5.9078 5.9116 82 5.9154 5.9192 5.9230 5.9269 5.9307 5.9346 5.9385 5.9424 5.9463 5.9502 83 5.9542 5.9581 5.9621 5.9661 5.9701 5.9741 5.9782 5.9822 5.9863 5.9904 84 5.9945 5.9986 6.0027 6.0069 6.0110 6.0152 6.0194 6.0237 6.0279 6.0322 85 6.0364 6.0407 6.0451 6.0494 6.0537 6.0581 6.0625 6.0669 6.0714 6.0758 86 6.0803 6.0848 6.0893 6.0939 6.0985 6.1031 6.1077 6.1123 6.1170 6.1217 87 6.1264 6.1311 6.1359 6.1407 6.1455 6.1503 6.1552 6.1601 6.1650 6.1700 88 6.1750 6.1800 6.1850 6.1901 6.1952 6.2004 6.2055 6.2107 6.2160 6.2212 89 6.2265 6.2319 6.2372 6.2426 6.2481 6.2536 6.2591 6.2646 6.2702 6.2759 90 6.2816 6.2873 6.2930 6.2988 6.3047 6.3106 6.3165 6.3225 6.3285 6.3346 91 6.3408 6.3469 6.3532 6.3595 6.3658 6.3722 6.3787 6.3852 6.3917 6.3984 92 6.4051 6.4118 6.4187 6.4255 6.4325 6.4395 6.4466 6.4538 6.4611 6.4684 93 6.4758 6.4833 6.4909 6.4985 6.5063 6.5141 6.5220 6.5301 6.5382 6.5464 94 6.5548 6.5632 6.5718 6.5805 6.5893 6.5982 6.6072 6.6164 6.6258 6.6352 95 6.6449 6.6546 6.6646 6.6747 6.6849 6.6954 6.7060 6.7169 6.7279 6.7392 96 6.7507 6.7624 6.7744 6.7866 6.7991 6.8119 6.8250 6.8384 6.8522 6.8663 97 6.8808 6.8957 6.9110 6.9268 6.9431 6.9600 6.9774 6.9954 7.0141 7.0335 98 7.0537 7.0748 7.0969 7.1201 7.1444 7.1701 7.1973 7.2262 7.2571 7.2904 99 7.3263 7.3656 7.4089 7.4573 7.5121 7.5758 7.6521 7.7478 7.8782 8.0902


E.2 Weighting coefficient Wi as a function of provisional PROBIT value Ypi

Provisional PROBIT Yp 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1 0.00057 0.00082 0.00118 0.00167 0.00235 0.00327 0.00451 0.00614 0.00828 0.01104

2 0.01457 0.01903 0.02458 0.03143 0.03977 0.04979 0.06168 0.07564 0.09179 0.11026

3 0.13112 0.15436 0.17994 0.20774 0.23754 0.26907 0.30199 0.33589 0.37031 0.40474

4 0.43863 0.47144 0.50260 0.53159 0.55788 0.58099 0.60052 0.61609 0.62742 0.63431

5 0.63662 0.63431 0.62742 0.61609 0.60052 0.58099 0.55788 0.53159 0.50260 0.47144

6 0.43863 0.40474 0.37031 0.33589 0.30199 0.26907 0.23754 0.20774 0.17994 0.15436

7 0.13112 0.11026 0.09179 0.07564 0.06168 0.04979 0.03977 0.03143 0.02458 0.01903

8 0.01457 0.01104 0.00828 0.00614 0.00451 0.00327 0.00235 0.00167 0.00118 0.00082

9 0.00057


E.3 Working PROBIT zi = Fn (observed success rate pi, provisional PROBIT Ypi)

%p i Provisional PROBIT YPI obs. 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

0 1.6954 1.7865 1.8772 1.9673 2.0568 2.1457 2.2339 2.3214 2.4081 2.4938 1 3.9518 3.4665 3.1405 2.9269 2.7930 2.7162 2.6805 2.6744 2.6900 2.7212 2 6.2082 5.1465 4.4039 3.8865 3.5293 3.2867 3.1270 3.0275 2.9719 2.9486 3 8.4646 6.8264 5.6672 4.8461 4.2655 3.8573 3.5736 3.3805 3.2537 3.1759 4 8.5064 6.9306 5.8057 5.0017 4.4278 4.0201 3.7335 3.5356 3.4033 5 8.1939 6.7653 5.7379 4.9983 4.4666 4.0865 3.8175 3.6306 6 9.4573 7.7249 6.4741 5.5688 4.9132 4.4395 4.0994 3.8580 7 8.6845 7.2103 6.1393 5.3597 4.7926 4.3813 4.0853 8 9.6442 7.9466 6.7098 5.8062 5.1456 4.6632 4.3127 9 8.6828 7.2803 6.2528 5.4986 4.9451 4.5401 10 9.4190 7.8508 6.6993 5.8516 5.2270 4.7674 11 8.4213 7.1459 6.2046 5.5089 4.9948 12 8.9918 7.5924 6.5577 5.7908 5.2221 13 9.5623 8.0389 6.9107 6.0727 5.4495 14 8.4855 7.2637 6.3546 5.6768 15 8.9320 7.6167 6.6365 5.9042 16 9.3785 7.9697 6.9183 6.1316 17 9.8251 8.3228 7.2002 6.3589 18 8.6758 7.4821 6.5863 19 9.0288 7.7640 6.8136 20 9.3818 8.0459 7.0410 21 9.7348 8.3278 7.2683 22 8.6097 7.4957 23 8.8916 7.7231 24 9.1735 7.9504 25 9.4554 8.1778 26 9.7373 8.4051 27 8.6325 28 8.8598 29 9.0872 30 9.3146 31 9.5419 32 9.7693 33 9.9966 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50


Appendix E3 (ctd.): Working PROBIT zi = Fn (observed success rate pi, provisional PROBIT Ypi)


0 2.5786 2.6624 2.7449 2.8261 2.9060 2.9842 3.0606 3.1351 3.2074 3.2773 1 2.7638 2.8148 2.8716 2.9325 2.9961 3.0614 3.1274 3.1935 3.2589 3.3232 2 2.9491 2.9672 2.9982 3.0388 3.0863 3.1386 3.1942 3.2518 3.3104 3.3691 3 3.1343 3.1196 3.1249 3.1451 3.1764 3.2158 3.2610 3.3102 3.3619 3.4150 4 3.3195 3.2720 3.2515 3.2515 3.2666 3.2930 3.3278 3.3685 3.4134 3.4609 5 3.5047 3.4244 3.3782 3.3578 3.3567 3.3702 3.3946 3.4269 3.4649 3.5068 6 3.6899 3.5768 3.5049 3.4641 3.4469 3.4474 3.4614 3.4853 3.5164 3.5527 7 3.8751 3.7292 3.6315 3.5704 3.5370 3.5247 3.5282 3.5436 3.5679 3.5986 8 4.0604 3.8816 3.7582 3.6768 3.6272 3.6019 3.5949 3.6020 3.6194 3.6445 9 4.2456 4.0340 3.8849 3.7831 3.7173 3.6791 3.6617 3.6603 3.6709 3.6904 10 4.4308 4.1864 4.0115 3.8894 3.8075 3.7563 3.7285 3.7187 3.7224 3.7363 11 4.6160 4.3388 4.1382 3.9957 3.8977 3.8335 3.7953 3.7770 3.7739 3.7822 12 4.8012 4.4912 4.2648 4.1021 3.9878 3.9107 3.8621 3.8354 3.8254 3.8281 13 4.9864 4.6436 4.3915 4.2084 4.0780 3.9879 3.9289 3.8937 3.8769 3.8740 14 5.1717 4.7960 4.5182 4.3147 4.1681 4.0651 3.9957 3.9521 3.9284 3.9199 15 5.3569 4.9484 4.6448 4.4211 4.2583 4.1423 4.0625 4.0104 3.9799 3.9658 16 5.5421 5.1008 4.7715 4.5274 4.3484 4.2195 4.1292 4.0688 4.0314 4.0117 17 5.7273 5.2532 4.8982 4.6337 4.4386 4.2967 4.1960 4.1271 4.0829 4.0576 18 5.9125 5.4056 5.0248 4.7400 4.5287 4.3740 4.2628 4.1855 4.1344 4.1035 19 6.0977 5.5580 5.1515 4.8464 4.6189 4.4512 4.3296 4.2439 4.1859 4.1494 20 6.2830 5.7104 5.2781 4.9527 4.7090 4.5284 4.3964 4.3022 4.2374 4.1953 21 6.4682 5.8628 5.4048 5.0590 4.7992 4.6056 4.4632 4.3606 4.2889 4.2412 22 6.6534 6.0152 5.5315 5.1654 4.8894 4.6828 4.5300 4.4189 4.3404 4.2871 23 6.8386 6.1676 5.6581 5.2717 4.9795 4.7600 4.5968 4.4773 4.3919 4.3330 24 7.0238 6.3200 5.7848 5.3780 5.0697 4.8372 4.6636 4.5356 4.4434 4.3789 25 7.2090 6.4724 5.9115 5.4843 5.1598 4.9144 4.7303 4.5940 4.4949 4.4248 26 7.3943 6.6248 6.0381 5.5907 5.2500 4.9916 4.7971 4.6523 4.5463 4.4707 27 7.5795 6.7772 6.1648 5.6970 5.3401 5.0688 4.8639 4.7107 4.5978 4.5166 28 7.7647 6.9296 6.2914 5.8033 5.4303 5.1461 4.9307 4.7690 4.6493 4.5625 29 7.9499 7.0820 6.4181 5.9096 5.5204 5.2233 4.9975 4.8274 4.7008 4.6084 30 8.1351 7.2344 6.5448 6.0160 5.6106 5.3005 5.0643 4.8857 4.7523 4.6543 31 8.3203 7.3868 6.6714 6.1223 5.7007 5.3777 5.1311 4.9441 4.8038 4.7002 32 8.5055 7.5392 6.7981 6.2286 5.7909 5.4549 5.1979 5.0025 4.8553 4.7461 33 8.6908 7.6916 6.9248 6.3350 5.8811 5.5321 5.2646 5.0608 4.9068 4.7920 34 8.8760 7.8440 7.0514 6.4413 5.9712 5.6093 5.3314 5.1192 4.9583 4.8379 35 9.0612 7.9964 7.1781 6.5476 6.0614 5.6865 5.3982 5.1775 5.0098 4.8838 36 9.2464 8.1488 7.3047 6.6539 6.1515 5.7637 5.4650 5.2359 5.0613 4.9298 37 9.4316 8.3012 7.4314 6.7603 6.2417 5.8409 5.5318 5.2942 5.1128 4.9757 38 9.6168 8.4536 7.5581 6.8666 6.3318 5.9181 5.5986 5.3526 5.1643 5.0216 39 9.8021 8.6060 7.6847 6.9729 6.4220 5.9954 5.6654 5.4109 5.2158 5.0675 40 9.9873 8.7584 7.8114 7.0792 6.5121 6.0726 5.7322 5.4693 5.2673 5.1134 41 8.9108 7.9380 7.1856 6.6023 6.1498 5.7989 5.5276 5.3188 5.1593 42 9.0632 8.0647 7.2919 6.6924 6.2270 5.8657 5.5860 5.3703 5.2052 43 9.2156 8.1914 7.3982 6.7826 6.3042 5.9325 5.6443 5.4218 5.2511 44 9.3681 8.3180 7.5046 6.8728 6.3814 5.9993 5.7027 5.4733 5.2970 45 9.5205 8.4447 7.6109 6.9629 6.4586 6.0661 5.7611 5.5248 5.3429 46 9.6729 8.5714 7.7172 7.0531 6.5358 6.1329 5.8194 5.5763 5.3888 47 9.8253 8.6980 7.8235 7.1432 6.6130 6.1997 5.8778 5.6278 5.4347 48 9.9777 8.8247 7.9299 7.2334 6.6902 6.2665 5.9361 5.6793 5.4806 49 8.9513 8.0362 7.3235 6.7675 6.3333 5.9945 5.7308 5.5265 50 9.0780 8.1425 7.4137 6.8447 6.4000 6.0528 5.7823 5.5724




0 3.3443 3.4083 3.4687 3.5251 3.5770 3.6236 3.6643 3.6982 3.7241 3.7407 1 3.3856 3.4458 3.5032 3.5571 3.6070 3.6520 3.6915 3.7244 3.7496 3.7659 2 3.4270 3.4834 3.5377 3.5892 3.6370 3.6804 3.7186 3.7506 3.7752 3.7911 3 3.4683 3.5210 3.5722 3.6212 3.6670 3.7088 3.7458 3.7768 3.8008 3.8163 4 3.5096 3.5586 3.6068 3.6532 3.6970 3.7373 3.7729 3.8030 3.8263 3.8415 5 3.5510 3.5962 3.6413 3.6852 3.7270 3.7657 3.8001 3.8293 3.8519 3.8667 6 3.5923 3.6338 3.6758 3.7173 3.7570 3.7941 3.8273 3.8555 3.8775 3.8919 7 3.6336 3.6713 3.7103 3.7493 3.7870 3.8225 3.8544 3.8817 3.9031 3.9171 8 3.6749 3.7089 3.7448 3.7813 3.8171 3.8509 3.8816 3.9079 3.9286 3.9423 9 3.7163 3.7465 3.7794 3.8133 3.8471 3.8793 3.9087 3.9341 3.9542 3.9675 10 3.7576 3.7841 3.8139 3.8454 3.8771 3.9077 3.9359 3.9604 3.9798 3.9927 11 3.7989 3.8217 3.8484 3.8774 3.9071 3.9361 3.9630 3.9866 4.0054 4.0178 12 3.8402 3.8592 3.8829 3.9094 3.9371 3.9645 3.9902 4.0128 4.0309 4.0430 13 3.8816 3.8968 3.9174 3.9414 3.9671 3.9929 4.0173 4.0390 4.0565 4.0682 14 3.9229 3.9344 3.9520 3.9735 3.9971 4.0213 4.0445 4.0652 4.0821 4.0934 15 3.9642 3.9720 3.9865 4.0055 4.0271 4.0497 4.0716 4.0915 4.1076 4.1186 16 4.0056 4.0096 4.0210 4.0375 4.0571 4.0781 4.0988 4.1177 4.1332 4.1438 17 4.0469 4.0472 4.0555 4.0695 4.0871 4.1065 4.1260 4.1439 4.1588 4.1690 18 4.0882 4.0847 4.0900 4.1016 4.1171 4.1349 4.1531 4.1701 4.1844 4.1942 19 4.1295 4.1223 4.1246 4.1336 4.1472 4.1633 4.1803 4.1963 4.2099 4.2194 20 4.1709 4.1599 4.1591 4.1656 4.1772 4.1917 4.2074 4.2226 4.2355 4.2446 21 4.2122 4.1975 4.1936 4.1976 4.2072 4.2201 4.2346 4.2488 4.2611 4.2698 22 4.2535 4.2351 4.2281 4.2297 4.2372 4.2485 4.2617 4.2750 4.2867 4.2950 23 4.2948 4.2727 4.2626 4.2617 4.2672 4.2769 4.2889 4.3012 4.3122 4.3202 24 4.3362 4.3102 4.2972 4.2937 4.2972 4.3053 4.3160 4.3274 4.3378 4.3453 25 4.3775 4.3478 4.3317 4.3257 4.3272 4.3337 4.3432 4.3537 4.3634 4.3705 26 4.4188 4.3854 4.3662 4.3578 4.3572 4.3621 4.3703 4.3799 4.3889 4.3957 27 4.4602 4.4230 4.4007 4.3898 4.3872 4.3905 4.3975 4.4061 4.4145 4.4209 28 4.5015 4.4606 4.4352 4.4218 4.4172 4.4189 4.4246 4.4323 4.4401 4.4461 29 4.5428 4.4981 4.4698 4.4538 4.4473 4.4473 4.4518 4.4585 4.4657 4.4713 30 4.5841 4.5357 4.5043 4.4859 4.4773 4.4758 4.4790 4.4848 4.4912 4.4965 31 4.6255 4.5733 4.5388 4.5179 4.5073 4.5042 4.5061 4.5110 4.5168 4.5217 32 4.6668 4.6109 4.5733 4.5499 4.5373 4.5326 4.5333 4.5372 4.5424 4.5469 33 4.7081 4.6485 4.6078 4.5819 4.5673 4.5610 4.5604 4.5634 4.5680 4.5721 34 4.7494 4.6861 4.6423 4.6140 4.5973 4.5894 4.5876 4.5896 4.5935 4.5973 35 4.7908 4.7236 4.6769 4.6460 4.6273 4.6178 4.6147 4.6159 4.6191 4.6225 36 4.8321 4.7612 4.7114 4.6780 4.6573 4.6462 4.6419 4.6421 4.6447 4.6476 37 4.8734 4.7988 4.7459 4.7100 4.6873 4.6746 4.6690 4.6683 4.6702 4.6728 38 4.9148 4.8364 4.7804 4.7421 4.7173 4.7030 4.6962 4.6945 4.6958 4.6980 39 4.9561 4.8740 4.8149 4.7741 4.7474 4.7314 4.7233 4.7207 4.7214 4.7232 40 4.9974 4.9115 4.8495 4.8061 4.7774 4.7598 4.7505 4.7470 4.7470 4.7484 41 5.0387 4.9491 4.8840 4.8381 4.8074 4.7882 4.7776 4.7732 4.7725 4.7736 42 5.0801 4.9867 4.9185 4.8702 4.8374 4.8166 4.8048 4.7994 4.7981 4.7988 43 5.1214 5.0243 4.9530 4.9022 4.8674 4.8450 4.8320 4.8256 4.8237 4.8240 44 5.1627 5.0619 4.9875 4.9342 4.8974 4.8734 4.8591 4.8518 4.8493 4.8492 45 5.2040 5.0995 5.0221 4.9662 4.9274 4.9018 4.8863 4.8781 4.8748 4.8744 46 5.2454 5.1370 5.0566 4.9983 4.9574 4.9302 4.9134 4.9043 4.9004 4.8996 47 5.2867 5.1746 5.0911 5.0303 4.9874 4.9586 4.9406 4.9305 4.9260 4.9248 48 5.3280 5.2122 5.1256 5.0623 5.0174 4.9870 4.9677 4.9567 4.9515 4.9500 49 5.3694 5.2498 5.1601 5.0943 5.0475 5.0154 4.9949 4.9829 4.9771 4.9751 50 5.4107 5.2874 5.1947 5.1264 5.0775 5.0438 5.0220 5.0092 5.0027 5.0003




0 3.7467 3.7401 3.7187 3.6798 3.6203 3.5360 3.4220 3.2724 3.0794 2.8335 1 3.7718 3.7653 3.7443 3.7061 3.6474 3.5644 3.4521 3.3044 3.1139 2.8711 2 3.7968 3.7905 3.7698 3.7323 3.6746 3.5928 3.4821 3.3364 3.1484 2.9087 3 3.8219 3.8156 3.7954 3.7585 3.7017 3.6212 3.5121 3.3684 3.1829 2.9463 4 3.8470 3.8408 3.8210 3.7847 3.7289 3.6496 3.5421 3.4005 3.2174 2.9839 5 3.8720 3.8660 3.8465 3.8109 3.7560 3.6780 3.5721 3.4325 3.2520 3.0215 6 3.8971 3.8912 3.8721 3.8372 3.7832 3.7064 3.6021 3.4645 3.2865 3.0590 7 3.9222 3.9164 3.8977 3.8634 3.8103 3.7348 3.6321 3.4965 3.3210 3.0966 8 3.9472 3.9416 3.9233 3.8896 3.8375 3.7632 3.6621 3.5286 3.3555 3.1342 9 3.9723 3.9668 3.9488 3.9158 3.8647 3.7916 3.6921 3.5606 3.3900 3.1718 10 3.9973 3.9920 3.9744 3.9420 3.8918 3.8200 3.7221 3.5926 3.4246 3.2094 11 4.0224 4.0172 4.0000 3.9683 3.9190 3.8484 3.7522 3.6247 3.4591 3.2469 12 4.0475 4.0424 4.0256 3.9945 3.9461 3.8768 3.7822 3.6567 3.4936 3.2845 13 4.0725 4.0676 4.0511 4.0207 3.9733 3.9052 3.8122 3.6887 3.5281 3.3221 14 4.0976 4.0928 4.0767 4.0469 4.0004 3.9336 3.8422 3.7207 3.5626 3.3597 15 4.1227 4.1179 4.1023 4.0731 4.0276 3.9620 3.8722 3.7528 3.5972 3.3973 16 4.1477 4.1431 4.1278 4.0994 4.0547 3.9904 3.9022 3.7848 3.6317 3.4349 17 4.1728 4.1683 4.1534 4.1256 4.0819 4.0188 3.9322 3.8168 3.6662 3.4724 18 4.1979 4.1935 4.1790 4.1518 4.1090 4.0473 3.9622 3.8488 3.7007 3.5100 19 4.2229 4.2187 4.2046 4.1780 4.1362 4.0757 3.9922 3.8809 3.7352 3.5476 20 4.2480 4.2439 4.2301 4.2042 4.1633 4.1041 4.0222 3.9129 3.7698 3.5852 21 4.2731 4.2691 4.2557 4.2305 4.1905 4.1325 4.0523 3.9449 3.8043 3.6228 22 4.2981 4.2943 4.2813 4.2567 4.2177 4.1609 4.0823 3.9769 3.8388 3.6603 23 4.3232 4.3195 4.3069 4.2829 4.2448 4.1893 4.1123 4.0090 3.8733 3.6979 24 4.3483 4.3447 4.3324 4.3091 4.2720 4.2177 4.1423 4.0410 3.9078 3.7355 25 4.3733 4.3699 4.3580 4.3353 4.2991 4.2461 4.1723 4.0730 3.9424 3.7731 26 4.3984 4.3951 4.3836 4.3616 4.3263 4.2745 4.2023 4.1050 3.9769 3.8107 27 4.4235 4.4203 4.4091 4.3878 4.3534 4.3029 4.2323 4.1371 4.0114 3.8483 28 4.4485 4.4454 4.4347 4.4140 4.3806 4.3313 4.2623 4.1691 4.0459 3.8858 29 4.4736 4.4706 4.4603 4.4402 4.4077 4.3597 4.2923 4.2011 4.0804 3.9234 30 4.4987 4.4958 4.4859 4.4664 4.4349 4.3881 4.3223 4.2331 4.1150 3.9610 31 4.5237 4.5210 4.5114 4.4927 4.4620 4.4165 4.3524 4.2652 4.1495 3.9986 32 4.5488 4.5462 4.5370 4.5189 4.4892 4.4449 4.3824 4.2972 4.1840 4.0362 33 4.5739 4.5714 4.5626 4.5451 4.5164 4.4733 4.4124 4.3292 4.2185 4.0737 34 4.5989 4.5966 4.5881 4.5713 4.5435 4.5017 4.4424 4.3612 4.2530 4.1113 35 4.6240 4.6218 4.6137 4.5975 4.5707 4.5301 4.4724 4.3933 4.2875 4.1489 36 4.6491 4.6470 4.6393 4.6238 4.5978 4.5585 4.5024 4.4253 4.3221 4.1865 37 4.6741 4.6722 4.6649 4.6500 4.6250 4.5869 4.5324 4.4573 4.3566 4.2241 38 4.6992 4.6974 4.6904 4.6762 4.6521 4.6153 4.5624 4.4893 4.3911 4.2617 39 4.7243 4.7226 4.7160 4.7024 4.6793 4.6437 4.5924 4.5214 4.4256 4.2992 40 4.7493 4.7477 4.7416 4.7286 4.7064 4.6721 4.6224 4.5534 4.4601 4.3368 41 4.7744 4.7729 4.7672 4.7549 4.7336 4.7005 4.6525 4.5854 4.4947 4.3744 42 4.7995 4.7981 4.7927 4.7811 4.7607 4.7289 4.6825 4.6174 4.5292 4.4120 43 4.8245 4.8233 4.8183 4.8073 4.7879 4.7573 4.7125 4.6495 4.5637 4.4496 44 4.8496 4.8485 4.8439 4.8335 4.8150 4.7858 4.7425 4.6815 4.5982 4.4871 45 4.8747 4.8737 4.8694 4.8597 4.8422 4.8142 4.7725 4.7135 4.6327 4.5247 46 4.8997 4.8989 4.8950 4.8860 4.8694 4.8426 4.8025 4.7455 4.6673 4.5623 47 4.9248 4.9241 4.9206 4.9122 4.8965 4.8710 4.8325 4.7776 4.7018 4.5999 48 4.9499 4.9493 4.9462 4.9384 4.9237 4.8994 4.8625 4.8096 4.7363 4.6375 49 4.9749 4.9745 4.9717 4.9646 4.9508 4.9278 4.8925 4.8416 4.7708 4.6751 50 5.0000 4.9997 4.9973 4.9908 4.9780 4.9562 4.9225 4.8736 4.8053 4.7126




0 2.5229 2.1325 1.6429 1.0295 0.2606 1 2.5643 2.1784 1.6944 1.0878 0.3273 2 2.6056 2.2243 1.7459 1.1462 0.3941 3 2.6469 2.2702 1.7974 1.2046 0.4609 4 2.6883 2.3161 1.8489 1.2629 0.5277 5 2.7296 2.3620 1.9004 1.3213 0.5945 6 2.7709 2.4079 1.9519 1.3796 0.6613 7 2.8122 2.4538 2.0034 1.4380 0.7281 8 2.8536 2.4997 2.0549 1.4963 0.7949 9 2.8949 2.5456 2.1063 1.5547 0.8617 10 2.9362 2.5915 2.1578 1.6130 0.9284 0.0670 11 2.9775 2.6374 2.2093 1.6714 0.9952 0.1442 12 3.0189 2.6833 2.2608 1.7297 1.0620 0.2214 13 3.0602 2.7292 2.3123 1.7881 1.1288 0.2986 14 3.1015 2.7751 2.3638 1.8464 1.1956 0.3758 15 3.1429 2.8210 2.4153 1.9048 1.2624 0.4530 16 3.1842 2.8669 2.4668 1.9632 1.3292 0.5302 17 3.2255 2.9128 2.5183 2.0215 1.3960 0.6074 18 3.2668 2.9587 2.5698 2.0799 1.4627 0.6846 19 3.3082 3.0046 2.6213 2.1382 1.5295 0.7618 20 3.3495 3.0505 2.6728 2.1966 1.5963 0.8391 21 3.3908 3.0964 2.7243 2.2549 1.6631 0.9163 22 3.4322 3.1423 2.7758 2.3133 1.7299 0.9935 0.0620 23 3.4735 3.1882 2.8273 2.3716 1.7967 1.0707 0.1522 24 3.5148 3.2341 2.8788 2.4300 1.8635 1.1479 0.2423 25 3.5561 3.2800 2.9303 2.4883 1.9303 1.2251 0.3325 26 3.5975 3.3259 2.9818 2.5467 1.9970 1.3023 0.4226 27 3.6388 3.3718 3.0333 2.6050 2.0638 1.3795 0.5128 28 3.6801 3.4178 3.0848 2.6634 2.1306 1.4567 0.6029 29 3.7214 3.4637 3.1363 2.7218 2.1974 1.5339 0.6931 30 3.7628 3.5096 3.1878 2.7801 2.2642 1.6111 0.7832 31 3.8041 3.5555 3.2393 2.8385 2.3310 1.6884 0.8734 32 3.8454 3.6014 3.2908 2.8968 2.3978 1.7656 0.9635 33 3.8868 3.6473 3.3423 2.9552 2.4646 1.8428 1.0537 0.0499 34 3.9281 3.6932 3.3938 3.0135 2.5314 1.9200 1.1439 0.1562 35 3.9694 3.7391 3.4453 3.0719 2.5981 1.9972 1.2340 0.2626 36 4.0107 3.7850 3.4968 3.1302 2.6649 2.0744 1.3242 0.3689 37 4.0521 3.8309 3.5483 3.1886 2.7317 2.1516 1.4143 0.4752 38 4.0934 3.8768 3.5998 3.2469 2.7985 2.2288 1.5045 0.5816 39 4.1347 3.9227 3.6513 3.3053 2.8653 2.3060 1.5946 0.6879 40 4.1760 3.9686 3.7028 3.3636 2.9321 2.3832 1.6848 0.7942 41 4.2174 4.0145 3.7543 3.4220 2.9989 2.4605 1.7749 0.9005 42 4.2587 4.0604 3.8057 3.4804 3.0657 2.5377 1.8651 1.0069 43 4.3000 4.1063 3.8572 3.5387 3.1324 2.6149 1.9552 1.1132 0.0354 44 4.3414 4.1522 3.9087 3.5971 3.1992 2.6921 2.0454 1.2195 0.1620 45 4.3827 4.1981 3.9602 3.6554 3.2660 2.7693 2.1356 1.3258 0.2887 46 4.4240 4.2440 4.0117 3.7138 3.3328 2.8465 2.2257 1.4322 0.4153 47 4.4653 4.2899 4.0632 3.7721 3.3996 2.9237 2.3159 1.5385 0.5420 48 4.5067 4.3358 4.1147 3.8305 3.4664 3.0009 2.4060 1.6448 0.6687 49 4.5480 4.3817 4.1662 3.8888 3.5332 3.0781 2.4962 1.7512 0.7953 50 4.5893 4.4276 4.2177 3.9472 3.6000 3.1553 2.5863 1.8575 0.9220



%p i Provisional PROBIT YPI obs. 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 51 9.2047 8.2488 7.5038 6.9219 6.4668 6.1112 5.8338 5.6183 52 9.3313 8.3552 7.5940 6.9991 6.5336 6.1695 5.8853 5.6642 53 9.4580 8.4615 7.6841 7.0763 6.6004 6.2279 5.9368 5.7101 54 9.5847 8.5678 7.7743 7.1535 6.6672 6.2862 5.9883 5.7560 55 9.7113 8.6742 7.8644 7.2307 6.7340 6.3446 6.0398 5.8019 56 9.8380 8.7805 7.9546 7.3079 6.8008 6.4029 6.0913 5.8478 57 9.9646 8.8868 8.0448 7.3851 6.8676 6.4613 6.1428 5.8937 58 8.9931 8.1349 7.4623 6.9343 6.5196 6.1943 5.9396 59 9.0995 8.2251 7.5395 7.0011 6.5780 6.2457 5.9855 60 9.2058 8.3152 7.6168 7.0679 6.6364 6.2972 6.0314 61 9.3121 8.4054 7.6940 7.1347 6.6947 6.3487 6.0773 62 9.4184 8.4955 7.7712 7.2015 6.7531 6.4002 6.1232 63 9.5248 8.5857 7.8484 7.2683 6.8114 6.4517 6.1691 64 9.6311 8.6758 7.9256 7.3351 6.8698 6.5032 6.2150 65 9.7374 8.7660 8.0028 7.4019 6.9281 6.5547 6.2609 66 9.8438 8.8561 8.0800 7.4686 6.9865 6.6062 6.3068 67 9.9501 8.9463 8.1572 7.5354 7.0448 6.6577 6.3527 68 9.0365 8.2344 7.6022 7.1032 6.7092 6.3986 69 9.1266 8.3116 7.6690 7.1615 6.7607 6.4445 70 9.2168 8.3889 7.7358 7.2199 6.8122 6.4904 71 9.3069 8.4661 7.8026 7.2782 6.8637 6.5363 72 9.3971 8.5433 7.8694 7.3366 6.9152 6.5822 73 9.4872 8.6205 7.9362 7.3950 6.9667 6.6282 74 9.5774 8.6977 8.0030 7.4533 7.0182 6.6741 75 9.6675 8.7749 8.0697 7.5117 7.0697 6.7200 76 9.7577 8.8521 8.1365 7.5700 7.1212 6.7659 77 9.8478 8.9293 8.2033 7.6284 7.1727 6.8118 78 9.9380 9.0065 8.2701 7.6867 7.2242 6.8577 79 9.0837 8.3369 7.7451 7.2757 6.9036 80 9.1609 8.4037 7.8034 7.3272 6.9495 81 9.2382 8.4705 7.8618 7.3787 6.9954 82 9.3154 8.5373 7.9201 7.4302 7.0413 83 9.3926 8.6040 7.9785 7.4817 7.0872 84 9.4698 8.6708 8.0368 7.5332 7.1331 85 9.5470 8.7376 8.0952 7.5847 7.1790 86 9.6242 8.8044 8.1536 7.6362 7.2249 87 9.7014 8.8712 8.2119 7.6877 7.2708 88 9.7786 8.9380 8.2703 7.7392 7.3167 89 9.8558 9.0048 8.3286 7.7907 7.3626 90 9.9330 9.0716 8.3870 7.8422 7.4085 91 9.1383 8.4453 7.8937 7.4544 92 9.2051 8.5037 7.9451 7.5003 93 9.2719 8.5620 7.9966 7.5462 94 9.3387 8.6204 8.0481 7.5921 95 9.4055 8.6787 8.0996 7.6380 96 9.4723 8.7371 8.1511 7.6839 97 9.5391 8.7954 8.2026 7.7298 98 9.6059 8.8538 8.2541 7.7757 99 9.6727 8.9122 8.3056 7.8216 100 9.7394 8.9705 8.3571 7.8675



%p i Provisional PROBIT YPI obs. 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 51 5.4520 5.3249 5.2292 5.1584 5.1075 5.0722 5.0492 5.0354 5.0283 5.0255 52 5.4933 5.3625 5.2637 5.1904 5.1375 5.1006 5.0763 5.0616 5.0538 5.0507 53 5.5347 5.4001 5.2982 5.2224 5.1675 5.1290 5.1035 5.0878 5.0794 5.0759 54 5.5760 5.4377 5.3327 5.2545 5.1975 5.1574 5.1306 5.1140 5.1050 5.1011 55 5.6173 5.4753 5.3673 5.2865 5.2275 5.1858 5.1578 5.1403 5.1306 5.1263 56 5.6586 5.5129 5.4018 5.3185 5.2575 5.2142 5.1850 5.1665 5.1561 5.1515 57 5.7000 5.5504 5.4363 5.3505 5.2875 5.2427 5.2121 5.1927 5.1817 5.1767 58 5.7413 5.5880 5.4708 5.3826 5.3175 5.2711 5.2393 5.2189 5.2073 5.2019 59 5.7826 5.6256 5.5053 5.4146 5.3475 5.2995 5.2664 5.2451 5.2328 5.2271 60 5.8240 5.6632 5.5399 5.4466 5.3776 5.3279 5.2936 5.2714 5.2584 5.2523 61 5.8653 5.7008 5.5744 5.4786 5.4076 5.3563 5.3207 5.2976 5.2840 5.2774 62 5.9066 5.7383 5.6089 5.5107 5.4376 5.3847 5.3479 5.3238 5.3096 5.3026 63 5.9479 5.7759 5.6434 5.5427 5.4676 5.4131 5.3750 5.3500 5.3351 5.3278 64 5.9893 5.8135 5.6779 5.5747 5.4976 5.4415 5.4022 5.3762 5.3607 5.3530 65 6.0306 5.8511 5.7125 5.6067 5.5276 5.4699 5.4293 5.4025 5.3863 5.3782 66 6.0719 5.8887 5.7470 5.6388 5.5576 5.4983 5.4565 5.4287 5.4119 5.4034 67 6.1132 5.9263 5.7815 5.6708 5.5876 5.5267 5.4836 5.4549 5.4374 5.4286 68 6.1546 5.9638 5.8160 5.7028 5.6176 5.5551 5.5108 5.4811 5.4630 5.4538 69 6.1959 6.0014 5.8505 5.7348 5.6476 5.5835 5.5380 5.5073 5.4886 5.4790 70 6.2372 6.0390 5.8850 5.7669 5.6777 5.6119 5.5651 5.5336 5.5141 5.5042 71 6.2786 6.0766 5.9196 5.7989 5.7077 5.6403 5.5923 5.5598 5.5397 5.5294 72 6.3199 6.1142 5.9541 5.8309 5.7377 5.6687 5.6194 5.5860 5.5653 5.5546 73 6.3612 6.1517 5.9886 5.8629 5.7677 5.6971 5.6466 5.6122 5.5909 5.5797 74 6.4025 6.1893 6.0231 5.8950 5.7977 5.7255 5.6737 5.6384 5.6164 5.6049 75 6.4439 6.2269 6.0576 5.9270 5.8277 5.7539 5.7009 5.6647 5.6420 5.6301 76 6.4852 6.2645 6.0922 5.9590 5.8577 5.7823 5.7280 5.6909 5.6676 5.6553 77 6.5265 6.3021 6.1267 5.9910 5.8877 5.8107 5.7552 5.7171 5.6931 5.6805 78 6.5678 6.3397 6.1612 6.0231 5.9177 5.8391 5.7823 5.7433 5.7187 5.7057 79 6.6092 6.3772 6.1957 6.0551 5.9477 5.8675 5.8095 5.7695 5.7443 5.7309 80 6.6505 6.4148 6.2302 6.0871 5.9778 5.8959 5.8367 5.7958 5.7699 5.7561 81 6.6918 6.4524 6.2648 6.1191 6.0078 5.9243 5.8638 5.8220 5.7954 5.7813 82 6.7332 6.4900 6.2993 6.1512 6.0378 5.9527 5.8910 5.8482 5.8210 5.8065 83 6.7745 6.5276 6.3338 6.1832 6.0678 5.9812 5.9181 5.8744 5.8466 5.8317 84 6.8158 6.5651 6.3683 6.2152 6.0978 6.0096 5.9453 5.9006 5.8722 5.8569 85 6.8571 6.6027 6.4028 6.2472 6.1278 6.0380 5.9724 5.9269 5.8977 5.8821 86 6.8985 6.6403 6.4374 6.2793 6.1578 6.0664 5.9996 5.9531 5.9233 5.9072 87 6.9398 6.6779 6.4719 6.3113 6.1878 6.0948 6.0267 5.9793 5.9489 5.9324 88 6.9811 6.7155 6.5064 6.3433 6.2178 6.1232 6.0539 6.0055 5.9744 5.9576 89 7.0225 6.7531 6.5409 6.3753 6.2478 6.1516 6.0810 6.0317 6.0000 5.9828 90 7.0638 6.7906 6.5754 6.4074 6.2779 6.1800 6.1082 6.0580 6.0256 6.0080 91 7.1051 6.8282 6.6100 6.4394 6.3079 6.2084 6.1353 6.0842 6.0512 6.0332 92 7.1464 6.8658 6.6445 6.4714 6.3379 6.2368 6.1625 6.1104 6.0767 6.0584 93 7.1878 6.9034 6.6790 6.5035 6.3679 6.2652 6.1897 6.1366 6.1023 6.0836 94 7.2291 6.9410 6.7135 6.5355 6.3979 6.2936 6.2168 6.1628 6.1279 6.1088 95 7.2704 6.9785 6.7480 6.5675 6.4279 6.3220 6.2440 6.1891 6.1535 6.1340 96 7.3117 7.0161 6.7826 6.5995 6.4579 6.3504 6.2711 6.2153 6.1790 6.1592 97 7.3531 7.0537 6.8171 6.6316 6.4879 6.3788 6.2983 6.2415 6.2046 6.1844 98 7.3944 7.0913 6.8516 6.6636 6.5179 6.4072 6.3254 6.2677 6.2302 6.2095 99 7.4357 7.1289 6.8861 6.6956 6.5479 6.4356 6.3526 6.2939 6.2557 6.2347 100 7.4771 7.1665 6.9206 6.7276 6.5780 6.4640 6.3797 6.3202 6.2813 6.2599



%p i Provisional PROBIT YPI obs. 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 51 5.0251 5.0249 5.0229 5.0171 5.0051 4.9846 4.9525 4.9057 4.8399 4.7502 52 5.0501 5.0500 5.0485 5.0433 5.0323 5.0130 4.9826 4.9377 4.8744 4.7878 53 5.0752 5.0752 5.0740 5.0695 5.0594 5.0414 5.0126 4.9697 4.9089 4.8254 54 5.1003 5.1004 5.0996 5.0957 5.0866 5.0698 5.0426 5.0017 4.9434 4.8630 55 5.1253 5.1256 5.1252 5.1219 5.1137 5.0982 5.0726 5.0338 4.9779 4.9005 56 5.1504 5.1508 5.1507 5.1482 5.1409 5.1266 5.1026 5.0658 5.0125 4.9381 57 5.1755 5.1760 5.1763 5.1744 5.1680 5.1550 5.1326 5.0978 5.0470 4.9757 58 5.2005 5.2012 5.2019 5.2006 5.1952 5.1834 5.1626 5.1298 5.0815 5.0133 59 5.2256 5.2264 5.2275 5.2268 5.2224 5.2118 5.1926 5.1619 5.1160 5.0509 60 5.2507 5.2516 5.2530 5.2530 5.2495 5.2402 5.2226 5.1939 5.1505 5.0885 61 5.2757 5.2768 5.2786 5.2793 5.2767 5.2686 5.2526 5.2259 5.1851 5.1260 62 5.3008 5.3020 5.3042 5.3055 5.3038 5.2970 5.2827 5.2579 5.2196 5.1636 63 5.3259 5.3272 5.3298 5.3317 5.3310 5.3254 5.3127 5.2900 5.2541 5.2012 64 5.3509 5.3524 5.3553 5.3579 5.3581 5.3538 5.3427 5.3220 5.2886 5.2388 65 5.3760 5.3775 5.3809 5.3841 5.3853 5.3822 5.3727 5.3540 5.3231 5.2764 66 5.4011 5.4027 5.4065 5.4104 5.4124 5.4106 5.4027 5.3860 5.3577 5.3139 67 5.4261 5.4279 5.4320 5.4366 5.4396 5.4390 5.4327 5.4181 5.3922 5.3515 68 5.4512 5.4531 5.4576 5.4628 5.4667 5.4674 5.4627 5.4501 5.4267 5.3891 69 5.4763 5.4783 5.4832 5.4890 5.4939 5.4958 5.4927 5.4821 5.4612 5.4267 70 5.5013 5.5035 5.5088 5.5152 5.5210 5.5242 5.5227 5.5141 5.4957 5.4643 71 5.5264 5.5287 5.5343 5.5415 5.5482 5.5527 5.5527 5.5462 5.5302 5.5019 72 5.5515 5.5539 5.5599 5.5677 5.5754 5.5811 5.5828 5.5782 5.5648 5.5394 73 5.5765 5.5791 5.5855 5.5939 5.6025 5.6095 5.6128 5.6102 5.5993 5.5770 74 5.6016 5.6043 5.6111 5.6201 5.6297 5.6379 5.6428 5.6422 5.6338 5.6146 75 5.6267 5.6295 5.6366 5.6463 5.6568 5.6663 5.6728 5.6743 5.6683 5.6522 76 5.6517 5.6547 5.6622 5.6726 5.6840 5.6947 5.7028 5.7063 5.7028 5.6898 77 5.6768 5.6798 5.6878 5.6988 5.7111 5.7231 5.7328 5.7383 5.7374 5.7273 78 5.7019 5.7050 5.7133 5.7250 5.7383 5.7515 5.7628 5.7703 5.7719 5.7649 79 5.7269 5.7302 5.7389 5.7512 5.7654 5.7799 5.7928 5.8024 5.8064 5.8025 80 5.7520 5.7554 5.7645 5.7774 5.7926 5.8083 5.8228 5.8344 5.8409 5.8401 81 5.7771 5.7806 5.7901 5.8037 5.8197 5.8367 5.8528 5.8664 5.8754 5.8777 82 5.8021 5.8058 5.8156 5.8299 5.8469 5.8651 5.8829 5.8984 5.9100 5.9153 83 5.8272 5.8310 5.8412 5.8561 5.8740 5.8935 5.9129 5.9305 5.9445 5.9528 84 5.8523 5.8562 5.8668 5.8823 5.9012 5.9219 5.9429 5.9625 5.9790 5.9904 85 5.8773 5.8814 5.8924 5.9085 5.9284 5.9503 5.9729 5.9945 6.0135 6.0280 86 5.9024 5.9066 5.9179 5.9348 5.9555 5.9787 6.0029 6.0265 6.0480 6.0656 87 5.9275 5.9318 5.9435 5.9610 5.9827 6.0071 6.0329 6.0586 6.0826 6.1032 88 5.9525 5.9570 5.9691 5.9872 6.0098 6.0355 6.0629 6.0906 6.1171 6.1408 89 5.9776 5.9822 5.9946 6.0134 6.0370 6.0639 6.0929 6.1226 6.1516 6.1783 90 6.0027 6.0073 6.0202 6.0396 6.0641 6.0923 6.1229 6.1546 6.1861 6.2159 91 6.0277 6.0325 6.0458 6.0659 6.0913 6.1207 6.1529 6.1867 6.2206 6.2535 92 6.0528 6.0577 6.0714 6.0921 6.1184 6.1491 6.1829 6.2187 6.2552 6.2911 93 6.0778 6.0829 6.0969 6.1183 6.1456 6.1775 6.2130 6.2507 6.2897 6.3287 94 6.1029 6.1081 6.1225 6.1445 6.1727 6.2059 6.2430 6.2827 6.3242 6.3662 95 6.1280 6.1333 6.1481 6.1707 6.1999 6.2343 6.2730 6.3148 6.3587 6.4038 96 6.1530 6.1585 6.1737 6.1970 6.2271 6.2627 6.3030 6.3468 6.3932 6.4414 97 6.1781 6.1837 6.1992 6.2232 6.2542 6.2912 6.3330 6.3788 6.4278 6.4790 98 6.2032 6.2089 6.2248 6.2494 6.2814 6.3196 6.3630 6.4108 6.4623 6.5166 99 6.2282 6.2341 6.2504 6.2756 6.3085 6.3480 6.3930 6.4429 6.4968 6.5542 100 6.2533 6.2593 6.2759 6.3018 6.3357 6.3764 6.4230 6.4749 6.5313 6.5917



%p i Provisional PROBIT YPI obs. 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 51 4.6306 4.4735 4.2692 4.0055 3.6667 3.2325 2.6765 1.9638 1.0487 52 4.6720 4.5194 4.3207 4.0639 3.7335 3.3098 2.7666 2.0701 1.1753 0.0223 53 4.7133 4.5653 4.3722 4.1222 3.8003 3.3870 2.8568 2.1765 1.3020 0.1747 54 4.7546 4.6112 4.4237 4.1806 3.8671 3.4642 2.9469 2.2828 1.4286 0.3271 55 4.7960 4.6571 4.4752 4.2389 3.9339 3.5414 3.0371 2.3891 1.5553 0.4795 56 4.8373 4.7030 4.5267 4.2973 4.0007 3.6186 3.1272 2.4954 1.6820 0.6319 57 4.8786 4.7489 4.5782 4.3557 4.0675 3.6958 3.2174 2.6018 1.8086 0.7844 58 4.9199 4.7948 4.6297 4.4140 4.1343 3.7730 3.3076 2.7081 1.9353 0.9368 59 4.9613 4.8407 4.6812 4.4724 4.2011 3.8502 3.3977 2.8144 2.0620 1.0892 60 5.0026 4.8866 4.7327 4.5307 4.2678 3.9274 3.4879 2.9208 2.1886 1.2416 61 5.0439 4.9325 4.7842 4.5891 4.3346 4.0046 3.5780 3.0271 2.3153 1.3940 62 5.0852 4.9784 4.8357 4.6474 4.4014 4.0819 3.6682 3.1334 2.4419 1.5464 63 5.1266 5.0243 4.8872 4.7058 4.4682 4.1591 3.7583 3.2397 2.5686 1.6988 64 5.1679 5.0702 4.9387 4.7641 4.5350 4.2363 3.8485 3.3461 2.6953 1.8512 65 5.2092 5.1162 4.9902 4.8225 4.6018 4.3135 3.9386 3.4524 2.8219 2.0036 66 5.2506 5.1621 5.0417 4.8808 4.6686 4.3907 4.0288 3.5587 2.9486 2.1560 67 5.2919 5.2080 5.0932 4.9392 4.7354 4.4679 4.1189 3.6650 3.0752 2.3084 68 5.3332 5.2539 5.1447 4.9975 4.8021 4.5451 4.2091 3.7714 3.2019 2.4608 69 5.3745 5.2998 5.1962 5.0559 4.8689 4.6223 4.2993 3.8777 3.3286 2.6132 70 5.4159 5.3457 5.2477 5.1143 4.9357 4.6995 4.3894 3.9840 3.4552 2.7656 71 5.4572 5.3916 5.2992 5.1726 5.0025 4.7767 4.4796 4.0904 3.5819 2.9180 72 5.4985 5.4375 5.3507 5.2310 5.0693 4.8539 4.5697 4.1967 3.7086 3.0704 73 5.5398 5.4834 5.4022 5.2893 5.1361 4.9312 4.6599 4.3030 3.8352 3.2228 74 5.5812 5.5293 5.4537 5.3477 5.2029 5.0084 4.7500 4.4093 3.9619 3.3752 75 5.6225 5.5752 5.5051 5.4060 5.2697 5.0856 4.8402 4.5157 4.0885 3.5276 76 5.6638 5.6211 5.5566 5.4644 5.3364 5.1628 4.9303 4.6220 4.2152 3.6800 77 5.7052 5.6670 5.6081 5.5227 5.4032 5.2400 5.0205 4.7283 4.3419 3.8324 78 5.7465 5.7129 5.6596 5.5811 5.4700 5.3172 5.1106 4.8346 4.4685 3.9848 79 5.7878 5.7588 5.7111 5.6394 5.5368 5.3944 5.2008 4.9410 4.5952 4.1372 80 5.8291 5.8047 5.7626 5.6978 5.6036 5.4716 5.2910 5.0473 4.7219 4.2896 81 5.8705 5.8506 5.8141 5.7561 5.6704 5.5488 5.3811 5.1536 4.8485 4.4420 82 5.9118 5.8965 5.8656 5.8145 5.7372 5.6260 5.4713 5.2600 4.9752 4.5944 83 5.9531 5.9424 5.9171 5.8729 5.8040 5.7033 5.5614 5.3663 5.1018 4.7468 84 5.9944 5.9883 5.9686 5.9312 5.8708 5.7805 5.6516 5.4726 5.2285 4.8992 85 6.0358 6.0342 6.0201 5.9896 5.9375 5.8577 5.7417 5.5789 5.3552 5.0516 86 6.0771 6.0801 6.0716 6.0479 6.0043 5.9349 5.8319 5.6853 5.4818 5.2040 87 6.1184 6.1260 6.1231 6.1063 6.0711 6.0121 5.9220 5.7916 5.6085 5.3564 88 6.1598 6.1719 6.1746 6.1646 6.1379 6.0893 6.0122 5.8979 5.7352 5.5088 89 6.2011 6.2178 6.2261 6.2230 6.2047 6.1665 6.1023 6.0043 5.8618 5.6612 90 6.2424 6.2637 6.2776 6.2813 6.2715 6.2437 6.1925 6.1106 5.9885 5.8136 91 6.2837 6.3096 6.3291 6.3397 6.3383 6.3209 6.2827 6.2169 6.1151 5.9660 92 6.3251 6.3555 6.3806 6.3980 6.4051 6.3981 6.3728 6.3232 6.2418 6.1184 93 6.3664 6.4014 6.4321 6.4564 6.4718 6.4753 6.4630 6.4296 6.3685 6.2708 94 6.4077 6.4473 6.4836 6.5147 6.5386 6.5526 6.5531 6.5359 6.4951 6.4232 95 6.4490 6.4932 6.5351 6.5731 6.6054 6.6298 6.6433 6.6422 6.6218 6.5756 96 6.4904 6.5391 6.5866 6.6315 6.6722 6.7070 6.7334 6.7485 6.7485 6.7280 97 6.5317 6.5850 6.6381 6.6898 6.7390 6.7842 6.8236 6.8549 6.8751 6.8804 98 6.5730 6.6309 6.6896 6.7482 6.8058 6.8614 6.9137 6.9612 7.0018 7.0328 99 6.6144 6.6768 6.7411 6.8065 6.8726 6.9386 7.0039 7.0675 7.1284 7.1852 100 6.6557 6.7227 6.7926 6.8649 6.9394 7.0158 7.0940 7.1739 7.2551 7.3376



%p i Provisional PROBIT YPI obs. 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 51 52 53 54 55 56 57 58 59 60 0.0127 61 0.1979 62 0.3832 63 0.5684 64 0.7536 65 0.9388 66 1.1240 67 1.3092 0.0034 68 1.4945 0.2307 69 1.6797 0.4581 70 1.8649 0.6854 71 2.0501 0.9128 72 2.2353 1.1402 73 2.4205 1.3675 74 2.6057 1.5949 0.2627 75 2.7910 1.8222 0.5446 76 2.9762 2.0496 0.8265 77 3.1614 2.2769 1.1084 78 3.3466 2.5043 1.3903 79 3.5318 2.7317 1.6722 0.2652 80 3.7170 2.9590 1.9541 0.6182 81 3.9023 3.1864 2.2360 0.9712 82 4.0875 3.4137 2.5179 1.3242 83 4.2727 3.6411 2.7998 1.6772 0.1749 84 4.4579 3.8684 3.0817 2.0303 0.6215 85 4.6431 4.0958 3.3635 2.3833 1.0680 86 4.8283 4.3232 3.6454 2.7363 1.5145 87 5.0136 4.5505 3.9273 3.0893 1.9611 0.4377 88 5.1988 4.7779 4.2092 3.4423 2.4076 1.0082 89 5.3840 5.0052 4.4911 3.7954 2.8541 1.5787 90 5.5692 5.2326 4.7730 4.1484 3.3007 2.1492 0.5810 91 5.7544 5.4599 5.0549 4.5014 3.7472 2.7197 1.3172 92 5.9396 5.6873 5.3368 4.8544 4.1938 3.2902 2.0534 0.3558 93 6.1249 5.9147 5.6187 5.2074 4.6403 3.8607 2.7897 1.3155 94 6.3101 6.1420 5.9006 5.5605 5.0868 4.4312 3.5259 2.2751 0.5427 95 6.4953 6.3694 6.1825 5.9135 5.5334 5.0017 4.2621 3.2347 1.8061 96 6.6805 6.5967 6.4644 6.2665 5.9799 5.5722 4.9983 4.1943 3.0694 1.4936 97 6.8657 6.8241 6.7463 6.6195 6.4264 6.1427 5.7345 5.1539 4.3328 3.1736 98 7.0509 7.0514 7.0281 6.9725 6.8730 6.7133 6.4707 6.1135 5.5961 4.8535 99 7.2362 7.2788 7.3100 7.3256 7.3195 7.2838 7.2070 7.0731 6.8595 6.5335 100 7.4214 7.5062 7.5919 7.6786 7.7661 7.8543 7.9432 8.0327 8.1228 8.2135


E.4 Values of the standard normal distribution function

( ) duett

u

∫∞−

−=Φ 22

2

1

π

t 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000


E.5 Values of ),2/(2

ναχ = Fn (number of degrees of freedom νννν, confidence level 1-αααα)

Confidence level: 1−α 1−α 1−α 1−α Degrees of freedom

νννν 0.99 0.95 0.9 0.8 0.7 0.6 0.5

1 0.00004 0.00098 0.00393 0.01579 0.03577 0.06418 0.10153 2 0.01002 0.05064 0.10259 0.21072 0.32504 0.44629 0.57536 3 0.07172 0.21579 0.35185 0.58438 0.79777 1.00517 1.21253 4 0.20698 0.48442 0.71072 1.06362 1.36648 1.64878 1.92256 5 0.41175 0.83121 1.14548 1.61031 1.99381 2.34253 2.67460 6 0.67573 1.23734 1.63538 2.20413 2.66127 3.07009 3.45460 7 0.98925 1.68986 2.16735 2.83311 3.35828 3.82232 4.25485 8 1.34440 2.17972 2.73263 3.48954 4.07820 4.59357 5.07064 9 1.73491 2.70039 3.32512 4.16816 4.81653 5.38006 5.89882 10 2.15585 3.24696 3.94030 4.86518 5.57006 6.17908 6.73720 11 2.60320 3.81574 4.57481 5.57779 6.33643 6.98867 7.58414 12 3.07379 4.40378 5.22603 6.30380 7.11384 7.80733 8.43842 13 3.56504 5.00874 5.89186 7.04150 7.90083 8.63386 9.29906 14 4.07466 5.62872 6.57063 7.78954 8.69630 9.46733 10.16531 15 4.60087 6.26212 7.26093 8.54675 9.49928 10.30696 11.03654 16 5.14216 6.90766 7.96164 9.31224 10.30902 11.15212 11.91222 17 5.69727 7.56418 8.67175 10.08518 11.12486 12.00226 12.79192 18 6.26477 8.23074 9.39045 10.86494 11.94625 12.85695 13.67529 19 6.84392 8.90651 10.11701 11.65091 12.77272 13.71579 14.56200 20 7.43381 9.59077 10.85080 12.44260 13.60386 14.57844 15.45177 21 8.03360 10.28291 11.59132 13.23960 14.43930 15.44461 16.34439 22 8.64268 10.98233 12.33801 14.04149 15.27875 16.31404 17.23962 23 9.26038 11.68853 13.09051 14.84795 16.12192 17.18650 18.13729 24 9.88620 12.40115 13.84842 15.65868 16.96855 18.06180 19.03725 25 10.51965 13.11971 14.61140 16.47341 17.81844 18.93975 19.93934 26 11.16022 13.84388 15.37916 17.29188 18.67138 19.82019 20.84343 27 11.80765 14.57337 16.15139 18.11389 19.52720 20.70298 21.74940 28 12.46128 15.30785 16.92788 18.93924 20.38573 21.58797 22.65716 29 13.12107 16.04705 17.70838 19.76774 21.24682 22.47505 23.56659 30 13.78668 16.79076 18.49267 20.59924 22.11034 23.36411 24.47760 31 14.45774 17.53872 19.28056 21.43357 22.97617 24.25506 25.39014 32 15.13402 18.29079 20.07191 22.27059 23.84419 25.14778 26.30411 33 15.81518 19.04666 20.86652 23.11019 24.71430 26.04221 27.21944 34 16.50130 19.80624 21.66428 23.95225 25.58640 26.93827 28.13608 35 17.19173 20.56938 22.46501 24.79665 26.46042 27.83588 29.05396 36 17.88675 21.33587 23.26862 25.64329 27.33625 28.73496 29.97305 37 18.58588 22.10562 24.07494 26.49209 28.21383 29.63547 30.89326 38 19.28882 22.87849 24.88389 27.34296 29.09306 30.53734 31.81456 39 19.99583 23.65430 25.69538 28.19579 29.97392 31.44051 32.73692 40 20.70658 24.43306 26.50930 29.05052 30.85632 32.34495 33.66029 41 21.42075 25.21452 27.32556 29.90708 31.74020 33.25060 34.58463 42 22.13838 25.99866 28.14405 30.76542 32.62553 34.15740 35.50992 43 22.85957 26.78537 28.96471 31.62546 33.51221 35.06533 36.43608 44 23.58362 27.57454 29.78750 32.48713 34.40023 35.97435 37.36313 45 24.31098 28.36618 30.61226 33.35038 35.28954 36.88441 38.29101 46 25.04130 29.16002 31.43900 34.21517 36.18009 37.79548 39.21971 47 25.77450 29.95616 32.26761 35.08142 37.07184 38.70752 40.14919 48 26.51067 30.75450 33.09807 35.94914 37.96477 39.62051 41.07943 49 27.24937 31.55493 33.93029 36.81823 38.85881 40.53441 42.01040 50 27.99082 32.35738 34.76424 37.68864 39.75392 41.44921 42.94208

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Statistical Method of PROBIT · the value of the functional parameter for which the probability of success is equal to R. The PROBIT ( Prob ability Un it ) represents the value plus