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In Search of The Biggest Bang for the Buck Differential Statistical www.Entrac-Petroleum.com R E (Gene) Ballay SPWLA UAE October 2009

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In Search of The Biggest Bang for the Buck Geoscientists are accustomed to dealing with uncertainty While minimum and maximum variations (for example) are common, they may be incomplete, and even non-representative There are two basic alternatives Partial derivatives Statistical simulation The Devil’s Promenade, SW Missouri

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In Search of The Biggest Bang for the Buck Why quantify the uncertainty? Time and Budget There is typically a limited timeframe and budget. Where do we focus? Lack of data and/or previous experiences Even with ‘focus’, we seldom have all the data we would like A previous bad experience, by either our self or someone else on the Team, will compound the unease

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In Search of The Biggest Bang for the Buck Concepts (and spreadsheet) applicable to many common oilfield issues The various, discrete measurements that make up routine and special core analyses The various attributes that are required to convert lab capillary pressure to a reservoir conditions Reservoir volumetrics Once we have mastered one calculation, it is straight-forward to address another

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In Search of The Biggest Bang for the Buck As carbonate petrophysicists, we are typically faced with Sw(Archie) estimates, in the face of uncertainty Mineralogy Porosity Formation brine resistivity Formation resistivity “m” and “n” exponents Exhibit following

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In Search of The Biggest Bang for the Buck As carbonate petrophysicists, we are typically faced with …… Mineralogy In some (KSA Shuaiba, for example) reservoirs, mineralogy is nearly uniform, and hence well known. In other (probably most) locales, uncertainty exists in this basic information In West Texas one may be faced with three minerals (limestone – dolostone – anhydrite), and only two logging tools (density-neutron) Anhydrite may be present as obvious nodules, or as (more subtle) cement Porosity Porosity is often thought of as +/- “x” pu of uncertainty, but in actual fact the uncertainty may be a function of the amount of porosity present Tool measurement techniques and statistical noise Formation brine resistivity We’ve all worked Fields for which we had high confidence in Rw, but that is not always the case Exhibit following

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In Search of The Biggest Bang for the Buck Formation resistivity The 6FF40, as an example, has a Skin Effect limitation at the low resistivity end, and a signal-to-noise issue at high resistivity The mud resistivity (and hence borehole & invasion effects) may change from one well to the next, indeed one logging interval to the next “m” and “n” exponents How many of us have ever been “really sure” of our exponents? In carbonates, wettability (and hence “n”) may vary with Sw, and present a significant challenge in the transition zone Sw(Archie) involves the combination of all these attributes, and their respective uncertainties Is it any wonder that “one size may not fit all feet”?

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The Differential Approach Over-view Chen & Fang have taken an In Depth Differential Look at Sw(Archie) Uncertainty, in terms of input attribute values and respective uncertainties Sensitivity Analysis of the Parameters in Archie‘s Water Saturation Equation. The Log Analyst. Sept – Oct 1986

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The Differential Approach Over-view Chen & Fang produced generic charts to facilitate locale-specific evaluation At ~ 20 pu, for stated conditions, the saturation exponent is a relatively minor issue Sw(Archie) attributes with associated Best Estimate and Uncertainties More on this later For attributes specified above, and in the case of ~ 20 pu, “n” is a relatively minor issue

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The Differential Approach Over-view Chen & Fang’s results have been coded to an Excel spreadsheet, to facilitate locale specific, digital evaluation

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The Differential Approach Over-view Chen & Fang’s results have been coded to an Excel spreadsheet, to facilitate locale specific, digital evaluation Sw(Archie) attributes with associated Best Estimate and Uncertainties For attributes specified above, and in the case of ~ 30 pu, the priorities change. Now ‘m’ & ‘n’ are of about equal importance

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The Differential Approach Over-view At ~ 10 pu, tortuosity in the pore system is far more important than “n” variations These relations may be coded into nearly any petrophysical s/w package, to facilitate foot-by-foot evaluation

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The Differential Approach Over-view The differential approach is applicable to many common oilfield issues ArchieUncertainty.pdf for derivation & additional considerations How to calculate the derivatives How to calculate the propagation of error Additional Details

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The Statistical Approach Over-view Many phenomena in nature are not Gaussian distributed. The probabilistic method allows one to model non- Gaussian distributions More difficult to set up Not included in the petrophysical s/w that is in use

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The Statistical Approach Over-view An advantage of Monte Carlo is that many types of distributions can be used to characterize the uncertainty of a parameter Normal, log normal, rectangular, triangular, etc

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The Statistical Approach Over-view The Monte Carlo method relies on repeated random sampling to model results This approach is also attractive when it is infeasible or impossible to compute an exact result with a deterministic algorithm.

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The Statistical Approach Over-view A limitation of Monte Carlo is that special software is often utilized Not included with many commercially available petrophysics s/w packages When available, the implementation of a MC Model over a large section of log data can be very time consuming

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The Statistical Approach Over-view The Monte Carlo method makes use of probability distributions to produce a cumulative probability result The determination of a probability distribution is superior to the single deterministic value Insight is gained into the “upside” and “downside” + / - 1 will encompass ~ 68% of the distribution + / - 2 ~ 95 % of the distribution http://en.wikipedia.org/wiki/Standard_deviation

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The Statistical Approach Over-view Many Oilfield analyses can be accomplished with Excel This eliminates both the expense of, and need to learn, commercial programs @Risk, Crystal Ball, etc If you have the time, and budget, to use @Risk, etc…Great! If not, and you want to leverage your Excel skill set, consider that option Oilfield applications typically use “normal”, “log normal”, rectangular and “triangular” statistical distributions. Relevant Excel functions Rand, NormDist, NormInv, LogNormDist, LogInv, TriangleInv

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The Statistical Approach Over-view Additional Details The statistical approach is applicable to many common oilfield issues If a “reference” spreadsheet is available, Monte Carlo may be easier to implement than derivatives MonteCarloModeling.pdf and MonteCarloIllustrations.pdf for additional details and considerations Additional Details

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The Statistical Approach Over-view Monte Carlo simulation can be used to characterize the uncertainty in Sw, for the attributes (& associated uncertainties) tabulated at right “a”, Rw and Rt are assumed to be well-known, reflected here by no STD specification This simulation is approximating an Sw interpretation for which the porosity, “m” & “n” estimates are each subject to individually specified uncertainty Porosity (for example) is described by a Gaussian distribution, centered on 20 pu with a standard deviation of 1 pu Illustrative Monte Carlo Application

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The Statistical Approach Over-view Sw(Archie) Uncertainty One issue of interest is the dependence of Sw upon individual attribute values / uncertainties With the specifications at right Sw(mean) = 0.357 (Sw) = 0.038 There is a 95% likelihood that Sw is contained within + / - 2 (0.357 – 0.076) < Sw < (0.357 + 0.076) 0.28 < Sw < 0.433 Be aware of how Excel ‘bins’ data and ‘non- linear effects’ Supplemental material for details

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Statistical vs Best / Worst There is a 95% likelihood that Sw is contained within + / - 2 0.28 < Sw < 0.433 The Best / Worst case would yield considerably more uncertainty 0.239 < Sw < 0.50 In practice, it’s unlikely (but not impossible) that Best / Worst of all attributes would occur simultaneously, so that Sw(Monte Carlo) provides a better representation

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Derivatives vs Statistics Over-view Chen & Fang identify the attribute resulting in the greatest Sw uncertainty, with a derivate approach In the case at right, “m” is the dominant issue This same issue can be addressed with Monte Carlo simulation (below) The Base Case is at lower left, with each simulation towards the right reflecting an individual improvement in “Phi”, “m” and “n” precision by 10 %. Exhibit following

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Chen & Fang identify “m” as the dominant issue The Monte Carlo Base Case is at lower left, with each simulation towards the right reflecting an improvement in “Phi”, “m” and “n” individual precision by 10 %. Improved definition of cementation exponent results in the most efficient improvement in Monte Carlo Sw estimation (lowest standard deviation in Sw) per Monte Carlo simulation Monte Carlo is consistent (as expected) with Derivatives Derivatives vs Statistics Over-view

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The Spreadsheet Spreadsheet may be modified for locally specific conditions Set up to illustrate various distributions, and to model Phi(Rhob), “m” & Sw(Archie) Be aware that illustrations are from Excel 2007, in the backwards compatible mode, and Excel 2003 screens may be different Illustrative application follows

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The Monte Carlo method is attractive when it is infeasible or impossible to compute an exact result with a deterministic algorithm. Excel can handle common probability distributions, and can thus serve as a Monte Carlo simulator. Quantitative estimation of uncertainty allows one to determine where time / money is most effectively spent, and to further avoid the trap of being misled as a result of a previous bad experience with a poorly defined parameter The importance of the various input parameters will change, according to the various magnitudes. There is a linkage in that one parameter becomes more or less important as another parameter value is changed, so the simulation must be executed for locally specific conditions. Additional Details as a Text Document The Statistical Approach

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The Differential Approach An alternative, deterministic approach to error analysis is accomplished by taking the derivative of Sw(Archie) with respect to each attribute S w n = a R w / ( m R t ) The same approach will suffice for a shaly sand equation The various terms in the derivative expression quantify the individual impact of uncertainty in each term, upon the result The relative magnitude then allows one to recognize where the biggest bang for the buck, in terms of a core analyses program, suite of potential logs, etc is to be found (Figure 1). Additional Details as a Text Document

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Monte Carlo Simulation of Sw(Archie) Monte Carlo simulation can be used to characterize the uncertainty in Sw, for the attributes (and associated uncertainties) tabulated at right “a”, Rw and Rt are assumed to be well-known, reflected here by no STD specification This simulation is approximating a foot-by-foot Sw interpretation, for which the porosity, “m” & “n” estimates are subject to uncertainty Porosity (for example) is described by a Gaussian distribution, centered on 20 pu with a standard deviation of 1 pu Exhibit following Monte Carlo Simulation of Sw(Archie)

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Porosity is described by a Gaussian distribution, centered on 20 pu with a standard deviation of 1 pu Two Thousand calculations are performed and the results are accumulated Exhibit following

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Monte Carlo Simulation of Sw(Archie) Monte Carlo simulation can be used to characterize the uncertainty in Sw, for the attributes (and associated uncertainties) tabulated at right “a”, Rw and Rt are assumed to be well-known, reflected here by no STD specification “m” and “n” are also Gaussian distributed, and centered on 2.0, with standard deviations of 0.1 “m” & “n” may be varied independently by simply modifying the above Table Calculations are live-linked “a”, Rw and Rt may also be varied in the MC model by a straight-forward extension Exhibit following Monte Carlo Simulation of Sw(Archie)

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Porosity, “m” & “n” are determined with individual Random Number inputs to NormInv, and are not driven by a single, common, random number source Exhibit following Excel Details

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Monte Carlo Simulation of Sw(Archie) Calculations are in the Sw_Random worksheet, live-linked to the Sw_Results worksheet Be Careful to enter values appropriately, and not over-write live links

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Monte Carlo Simulation of Sw(Archie) To modify spreadsheet to accommodate a different three parameter simulation, simply re-title the various attributes and adjust the individual calculations as required

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Monte Carlo Simulation of Phi(Rhob) To modify spreadsheet to accommodate a different three parameter simulation, simply re-title the various attributes and adjust the individual calculations as required. This worksheet models Phi(Rhob).

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Monte Carlo Simulation of “m(Dual Porosity)” To modify spreadsheet to accommodate a different three parameter simulation, simply re-title the various attributes and adjust the individual calculations as required. This worksheet models the Wang & Lucia Dual Porosity “m” Exponent

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Monte Carlo Simulation of Sw(Archie) Porosity is specified as a Gaussian distribution, centered on 20 pu with a standard deviation of 1 pu 2,000 calculations are done, and the result “checked’ by means of histograming the resulting porosity distribution and calculating the resulting statistics Exhibit following Cross Check

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Monte Carlo Simulation of Sw(Archie) Porosity is specified as a Gaussian distribution, centered on 20 pu with a standard deviation of 1 pu 2,000 calculations are done, and the result “checked’ by means of histograming the resulting porosity distribution and calculating the resulting statistics The MC simulation is observed to reproduce the specified inputs, digitally and graphically Random number “mean” & “Std” converge to input values Exhibit following Cross Check

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Monte Carlo Simulation of Sw(Archie) As a QC device, the distribution of Excel random numbers used to drive the Monte Carlo simulation, are ‘binned’ from zero to one With 2000 simulation performed, we expect to find Frequency ~ 200 in each of the ten bins The Excel Random() function has approached the ideal distribution QC the Monte Carlo

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Monte Carlo Simulation of Sw(Archie) Sw(Archie) Uncertainty One issue of interest is the dependence of Sw upon individual attribute values / uncertainties With the specifications at right Sw(mean) = 0.357 (Sw) = 0.038 There is a 95% likelihood that Sw is contained within + / - 2 (0.357 – 0.076) < Sw < (0.357 + 0.076) 0.28 < Sw < 0.433 Be aware of how Excel ‘bins’ data Exhibit following

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Monte Carlo Simulation of Sw(Archie) Excel Bins The Freq of a specific Bin is not a centered value Sw_Bin=0.25 0.225.LT. Sw.LE..25 Sw_Bin=0.275 0.25.LT. Sw.LE..275 Sw_Bin=0.30 0.275.LT. Sw.LE..30 Sw_Bin=0.325 0.30.LT. Sw.LE..325 Sw(Mean)=0.357 but Bin Freq, and associated graphic, peaks to the high side of this value Bin Values are shifted with respect to actual distribution Be aware of how Excel ‘bins’ data

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Nonlinear relationship affects probability distributions. Oilfield Review. Autumn 2002. Ian Bryant, Alberto Malinverno, Michael Prange, Mauro Gonfalini, James Moffat, Dennis Swager, Philippe Theys, Francesca Verga. The Archie relationship for a given formation, with constant “a”, “m”, “n” and Rt results in a hyperbolic relationship of Sw to porosity (blue) This relationship distorts the frequency distributions, which are shown along the axes A normal uncertainty distribution about a given porosity (green) becomes a log- normal distribution for the resulting Sw uncertainty (red) The mean value (dashed yellow) and three sigma points (dashed purple) show the skewed Sw distribution Sw distributions determined with Monte Carlo modeling are distorted at high values, since Sw cannot exceed 100%. Nonlinear Relation Is An Additional Issue

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Monte Carlo vs Best / Worse Case There is a 95% likelihood that Sw is contained within + / - 2 0.28 < Sw < 0.433 The Best / Worst case corresponding to + / - 2 , would yield considerably more uncertainty 0.239 < Sw < 0.50 In practice, it’s unlikely (but not impossible) that Best / Worst of all attributes would occur simultaneously, so that Sw(Monte Carlo) provides a better representation

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In Search of The Biggest Bang for the Buck One issue of interest is the dependence of Sw upon individual attribute values / uncertainties With the values specified at right Sw(mean) = 0.357 (Sw) = 0.038 Monte Carlo simulation also allows one to determine for any possible (improved) input distribution, and to therefore identify where the ‘biggest bang for the buck” in reducing Sw uncertainty is at Exhibit following Where Is The Biggest Bang For The Buck?

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Uncertainty in Archie’s Equation Where to spend time, and money, in search of an improved Sw estimate Identifying the Biggest Bang for the Buck Both options have attractive features The analytical approach can be easily coded into a foot-by- foot petrophysical evaluation Monte Carlo simulation can address non-Gaussian distributions. Both options are adaptable to the spreadsheet environment In addition to quantifying the uncertainty in a single Sw(Archie) estimate, one will likely want to prioritize time and budget, in a manner that most efficiently reduces net Sw uncertainty This can be done with either derivatives or MC simulation Derivatives vs Monte Carlo

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In Search of The Biggest Bang for the Buck Chen & Fang’s results have been coded to an Excel spreadsheet, to facilitate locale specific, digital evaluation At ~ 30 pu, for stated conditions, the three issues approach one another, in importance As porosity increases, “m” or “pore system tortuosity” becomes less important, and the tortuosity of the conductive phase, represented by “n”, deserves increased attention, relative to “m” Exhibit following The Derivative Approach

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In Search of The Biggest Bang for the Buck Chen & Fang’s results have been coded to an Excel spreadsheet, to facilitate locale specific, digital evaluation Spreadsheet may be modified for locally specific conditions

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In Search of The Biggest Bang for the Buck At ~ 10 pu, tortuosity in the pore system is far more important than “n” variations As porosity decreases, “m” or “pore system tortuosity” becomes the dominant issue In these circumstances the tortuosity of the conductive phase, represented by “n”, can be nearly neglected (in a relative sense) The Derivative Approach

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In Search of The Biggest Bang for the Buck Chen & Fang’s results should be apparent in a similarly specified Monte Carlo simulation Consider the below three situations, with corresponding Chen & Fang results at right “a” = 1: Uncertain(a) = 0% Rw = 0.02 : Uncertain (Rw) = 0% Rt = 40 : Uncertain (Rt) = 0% Phi = 10, 20 & 30 pu : Uncertain (Phi) = 15% “m” = 2.0 : Uncertain (m) = 10% “n” = 2 : Uncertain (n) = 5% In each case, the dominant parameter is identified with the red arrow (at right), per derivatives Exhibit following Derivatives vs Monte Carlo

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In Search of The Biggest Bang for the Buck Derivatives identify the attribute resulting in the greatest Sw uncertainty In the case at right ( ~10 pu), “m” is the dominant issue This same issue can be addressed with Monte Carlo simulation Recalling that + / - 1 will encompass ~ 68% of the distribution, and 2 ~ 95 % of the distribution, we approximate the various relative uncertainties per the following exhibit http://en.wikipedia.org/wiki/Standard_deviation

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In Search of The Biggest Bang for the Buck Approximate the various relative uncertainties as follows + / - 1 will encompass ~ 68% of the distribution +/- 2 ~ 95 % of the distribution, Represent the “Phi” relative uncertainty of 15% @ 10 pu (1.5 pu) as 2 ~ 1.5 pu => ~ 0.75 pu for Monte Carlo Simulation purposes That is, 2 encompasses 95 % of the spread in the distribution, and we’ll set it equal to the 15 % uncertainty in the Chen & Fang analyses Exhibit following Uncertainty Specification Derivative Monte Carlo Total Uncertainty vs Standard Deviations

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In Search of The Biggest Bang for the Buck + / - 1 will encompass ~ 68% of the distribution +/- 2 ~ 95 % of the distribution, Approximate the “Phi” relative uncertainty of 15% @ 10 pu (1.5 pu) as 2 ~ 1.5 pu => ~ 0.75 pu for Monte Carlo Simulation purposes Approximate the “m” relative uncertainty of 10% @ 2.00 (0.2) as 2 ~ 0.2 => ~ 0.1 for Monte Carlo Simulation purposes Approximate the “n” relative uncertainty of 5% @ 2.00 (0.1) as 2 ~ 0.1 => ~ 0.05 for Monte Carlo Simulation purposes These mean values and standard deviations form the Base Case, against which one compares the MC results Exhibit following “m” & “n” Uncertainty

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In Search of The Biggest Bang for the Buck Derivatives identify the attribute resulting in the greatest Sw uncertainty In the case at right, “m” is the dominant issue This same issue can be addressed with Monte Carlo simulation (below) The Base Case is at lower left, with each simulation towards the right reflecting an improvement in “Phi”, “m” and “n” individual precision by 10 %. Exhibit following Derivatives vs Monte Carlo

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In Search of The Biggest Bang for the Buck Derivatives identify “m” as the dominant issue The Monte Carlo Base Case is at lower left, with each simulation towards the right reflecting an improvement in “Phi”, “m” and “n” individual precision by 10 %. Improved definition of cementation exponent results in the most efficient improvement in Monte Carlo Sw estimation (lowest standard deviation in Sw) per Monte Carlo simulation Monte Carlo is consistent (as expected) with Derivatives Derivatives vs Monte Carlo

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Monte Carlo vs Best / Worst Case One also observes that the Best / Worst numerical evaluation of Sw(Archie) is considerably more pessimistic than is the +/- 2 Monte Carlo simulation The Best and Worst of all parameters are unlikely to occur simultaneously Best Case Worst Case

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In Search of The Biggest Bang for the Buck If porosity were to be 25 pu, rather than 10, the focus changes “m” and “n” uncertainties have been set equal in this simulation The Base Case is at lower left, with each simulation towards the right reflecting an improvement in “Phi”, “m” and “n” precision by 10 %. With the improved porosity, focus shifts to “n”, the tortuosity of the conductive (brine) phase in the presence of a non-conductive (hydrocarbon) phase, as offering the Biggest Bang For The Buck. Exhibit following

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In Search of The Biggest Bang for the Buck Sw uncertainty is a dynamic issue Should be evaluated for each set of circumstances Attributes are ‘linked’ A change in one can cause another to become more, or less, important Concepts (and spreadsheet) applicable to many common oilfield issues Allow us to focus time and budget in the most efficient manner

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Comments by Carlos Torres-Verdin, University of Texas Phillipe Theys has a book which addresses this issue, as does Darwin. The book by Darwin especially considers the effects on porosity (also of the shaly-sand case). He was the first to perform a similar analysis. One should be sure to recognize that resistivity logs are affected by invasion and shoulder-bed effects This can cause significant errors in the estimation of Sw because Rt is not representative. The latter effects can be more significant that the ones you have explicitly addressed. Electrical anisotropy effects could have a significant impact on Rt in deviated and horizontal wells. The most significant example is that of thinly-bedded sequences, where nuclear and resistivity logs are significantly biased (in addition to the clay effect) because of shoulder-bed and invasion effects. Additional Considerations

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Carlos Torres-Verdin: Be sure to recognize that logs are affected by various issues Ozark Mtns Road Cut, SW Missouri

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Quantifying Petrophysical Uncertainties S J Adams 2005 Asia Pacific Oil & Gas Conference, Jakarta, Indonesia, 5 - 7 April 2005 Quantitative uncertainty definition is more than just using Monte Carlo simulation to vary the inputs to the interpretation model The largest source of uncertainty may be the interpretation model itself In graphic at right, (Rhob) is calculated with, and without, light hydrocarbon corrections (D-N) and (Dt) are not LHC corrected Accounting for LHC effects shifts (D) to over-lay the core distribution Additional Considerations

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We appreciate the unidentified LSU faculty who posted their material (located via Google) to http://www.enrg.lsu.edu/pttc/ Additional Considerations

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http://www.statsoft.com/textbook/stathome.html Additional Considerations

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Useful Links http://en.wikipedia.org/wiki/Monte_carlo_simulation http://www.ipp.mpg.de/de/for/bereiche/stellarator/Comp_sci/CompScience/csep/csep1. phy.ornl.gov/mc/mc.html http://www.sitmo.com/eqcat/15 http://www.riskglossary.com/link/monte_carlo_method.htm http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html http://office.microsoft.com/en-us/excel/HA011118931033.aspx What happens when I enter =Rand() in a cell? How can I simulate values of a discrete random variable? How can I simulate values of a normal random variable? Problems You can download the sample files that relate to excerpts from Microsoft Excel Data Analysis and Business Modeling from Microsoft Office Online. This article uses the files RandDemo.xls, Discretesim.xls, NormalSim.xls, and Valentine.xls.

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Early morning carbonate bluffs along Steel Creek, Arkansas Thank you for your Time and Interest

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R. E. (Gene) Ballay’s 32 years in petrophysics include research and operations assignments in Houston (Shell Research), Texas; Anchorage (ARCO), Alaska; Dallas (Arco Research), Texas; Jakarta (Huffco), Indonesia; Bakersfield (ARCO), California; and Dhahran, Saudi Arabia. His carbonate experience ranges from individual Niagaran reefs in Michigan to the Lisburne in Alaska to Ghawar, Saudi Arabia (the largest oilfield in the world). He holds a PhD in Theoretical Physics with double minors in Electrical Engineering & Mathematics, has taught physics in two universities, mentored Nationals in Indonesia and Saudi Arabia, published numerous technical articles and been designated co-inventor on both American and European patents. Chattanooga shale Mississippian limestone At retirement from the Saudi Arabian Oil Company he was the senior technical petrophysicist in the Reservoir Description Division and had represented petrophysics in three multi-discipline teams bringing on-line three (one clastic, two carbonate) multi-billion barrel increments. Subsequent to retirement from Saudi Aramco he established Robert E Ballay LLC, which provides physics - petrophysics training & consulting services. He served in the U.S. Army as a Microwave Repairman and in the U.S. Navy as an Electronics Technician, and he is a USPA Parachutist and a PADI Dive Master.

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