(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 285450, 13779]*) (*NotebookOutlinePosition[ 287337, 13838]*) (* CellTagsIndexPosition[ 287077, 13827]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ \(\(Get["\"]; \nNeeds["\"]; \n Needs["\"]; \n$DefaultFont\ = \ {"\", \ 8}; \n\)\)], "Input", InitializationCell->True], Cell[TextData[{ "MATHEMATICA IN ACTION for Issue 8-1 of ", StyleBox["Mathematica", FontSlant->"Italic"], " in Education and Research" }], "Text", TextAlignment->Center, FontSize->16], Cell[CellGroupData[{ Cell["Statistics in the Classroom and the Courtroom", "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The field of statistics has long been known for innovative use of \ graphics. ", StyleBox["Mathematica", FontSlant->"Italic"], "\[CloseCurlyQuote]s breadth allows us to create graphics suited to \ specific needs, from the education of students to presentations to a jury." }], "Subtitle", Evaluatable->False, TextAlignment->Left, AspectRatioFixed->True, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell["by Karl Heiner and Stan Wagon", "Subtitle", CellMargins->{{Inherited, Inherited}, {18, Inherited}}, Evaluatable->False, TextAlignment->Left, AspectRatioFixed->True, FontFamily->"Helvetica"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". Introduction" }], "Section"], Cell[TextData[{ "The abstract nature of some of the fundamental concepts in statistics \ causes difficulty for many students. Experimentation and case studies are \ pedagogical tools that facilitate the study of these concepts. Both provide \ concrete examples to which to tie the concepts and both require that students \ remain \[OpenCurlyDoubleQuote]on task\[CloseCurlyDoubleQuote] in order to \ complete the assignment. It is well known by experts in education that \ on-task behavior is a required component of effective learning. This is \ especially critical for students who have grown up in a world of sound bites \ and commercial TV. In this column we describe some approaches to teaching \ statistical concepts that show how one might use the power and versatility of \ ", StyleBox["Mathematica", FontSlant->"Italic"], " to develop functions that allow students to experiment and analyze case \ studies", StyleBox[". ", FontColor->RGBColor[1, 0, 0]], "Moreover, the examples are based on actual data sets and cases, thus \ providing a solid link from abstract theorems and distributions to the real \ world." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". Hypothesis Testing" }], "Section"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", "1", ". An Employment Discrimination Case " }], "Subsection"], Cell[TextData[{ "In the early 1990s New York State was experiencing a serious budget \ crisis. As a result, Governor Cuomo instructed each division of the state \ government to lay off approximately 10% of its employees. So the Management \ Administration Unit of the Office of Mental Health dismissed 11 of its 127 \ employees. The average age of the terminated employees \[LongDash] call it ", StyleBox["avg", FontSlant->"Italic"], " \[LongDash] was greater than than that of those not laid off. Those \ terminated filed an age discrimination suit that was argued in the Federal \ District Court in Albany. The ages of the individuals in each group are given \ below." }], "Text"], Cell[BoxData[ \(Needs["\"]; \nNeeds["\"]; \n Needs["\"]; \)], "Input"], Cell[BoxData[ \(kept = {29, 34, 36, 44, 36, 47, 33, 33, 45, 45, 36, \n\t\t26, 29, 37, 29, 30, 38, 38, 57, 40, 47, 41, 42, 45, 48, \n\t\t33, 62, 43, 44, 32, 40, 32, 25, 47, 36, 31, 24, 50, 26, \n\t\t42, 45, 32, 46, 28, 42, 53, 45, 57, 26, 62, 46, 49, 47, \n\t\t31, 40, 42, 28, 45, 30, 53, 27, 48, 29, 40, 48, 47, 42, \n\t\t44, 36, 66, 40, 40, 47, 38, 29, 31, 37, 39, 47, 42, 58, \n\t\t60, 52, 48, 43, 47, 37, 39, 21, 41, 31, 39, 39, 53, 41, \n\t\t25, 48, 51, 41, 23, 48, 48, 62, 66, 44, 48, 35, 45, 37, \n\t\t46, 34, 40, 33, 32, 40, 30} // N; \n dismissed = {42, 39, 37, 57, 33, 42, 60, 39, 56, 44, 28} // N; \n all = Join[kept, dismissed]; \)], "Input"], Cell[TextData[{ "In such cases there is an initial burden on the plaintiff to show that \ there was disparate impact on the protected class; this is usually \ accomplished by a statistical test of the hypothesis of no disparate impact. \ The following demonstration was used by K.H.", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "to introduce the concept of hypothesis testing to the jury and to \ demonstrate that obtaining an average age at least as large as ", StyleBox["avg", FontSlant->"Italic"], " when a random drawing of eleven from the combined list was not \ particularly unusual. First, we note the size of the groups and their average \ age." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Length, {kept, dismissed}]\)], "Input"], Cell[BoxData[ \({116, 11}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({avgkept, avg} = Map[Mean, {kept, dismissed}]\)], "Input"], Cell[BoxData[ \({40.6982758620689644`, 43.3636363636363686`}\)], "Output"] }, Open ]], Cell[TextData[{ "Using the ", StyleBox["RandomKSubset", "Input", FontWeight->"Plain"], " function from the ", StyleBox["VisualStats", "Input", FontWeight->"Plain"], " package we can easily get the ages of eleven randomly chosen \ individuals." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RandomKSubset[all, 11]\)], "Input"], Cell[BoxData[ \({40.9999999999999964`, 62.0000000000000017`, 30.`, 52.9999999999999982`, 40.`, 60.`, 52.0000000000000017`, 40.9999999999999964`, 62.0000000000000017`, 32.0000000000000017`, 57.0000000000000017`}\)], "Output"] }, Open ]], Cell["\<\ We can generate as many of these random layoff lists as we wish, \ say five, and find the mean age for each one.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Mean\ /@\ Table[RandomKSubset[all, \ 11], {5}]\)], "Input"], Cell[BoxData[ \({40.4545454545454497`, 35.6363636363636349`, 42.5454545454545485`, 42.90909090909091`, 40.1818181818181763`}\)], "Output"] }, Open ]], Cell[TextData[ "By generating 100 random layoff lists we get an idea of the likelihood of \ randomly picking a list with an average age equal to or greater than that of \ the plaintiffs (43.3636). This time 21 of the 100 means exceeded 43.3636, \ which is not rare enough for the jury to conclude disparate impact. Inasmuch \ as the plaintiffs\[CloseCurlyQuote] lawyers failed to meet this burden of \ proof, much of their remaining evidence was irrelevant."], "Text"], Cell[BoxData[ RowBox[{ StyleBox[\(SeedRandom[1]\), FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[";", FontColor->RGBColor[1, 0, 0]], "\n", \(means = Map[Mean, Table[RandomKSubset[all, \ 11], {100}]]\), ";"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(bc = BinCounts[means, {32, 50}]\)], "Input"], Cell[BoxData[ \({0, 1, 2, 3, 2, 7, 7, 15, 15, 17, 13, 7, 8, 2, 1, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Count[means, x_ /; x > avg]\)], "Input"], Cell[BoxData[ \(15\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(BarChart[bc, PlotRange \[Rule] {{0, 18}, {0, \ Max[bc] 1.1}}, \n\t BarLabels \[Rule] Range[33, 50], BarStyle \[Rule] {Gray}, Frame \[Rule] True, \n\tFrameTicks \[Rule] None, Axes \[Rule] True, \n\t Epilog \[Rule] {Text["\", {avg - 31.8, 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those not on the list was itself irrelevant, \ because by law the protected class is those over the age of 40. The usual \ statistical test employed by the plaintiff\[CloseCurlyQuote]s attorney to \ argue that the protected class has been disparately impacted is the Fisher \ exact test, which is based on the hypergeometric distribution; see section \ 3.1.2."], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ". The Fisher Exact Test and the Hypergeometric Distribution" }], "Subsection"], Cell[TextData[{ "In cases such as the preceding, ordinarily the plaintiff will construct a \ ", Cell[BoxData[ \(TraditionalForm\`2\[Times]2\)]], " table displaying the numbers of individuals included on and excluded from \ the layoff list by class status (protected vs. not protected) and attempt to \ argue that, because of their disproportionate representation on the layoff \ list., the protected class was treated unfairly. Then, using the \ hypergeometric distribution, one calculates the probability of randomly \ observing the table in question or one more extreme (i.e., more \ discriminatory), given that the marginal counts are held constant (e.g., 55 \ less than 40 years of age and 72 older, 11 listed for layoff, while 116 were \ not). If the sum of these probabilities is small, we reject the hypothesis of \ no disparate impact. This is called Fisher\[CloseCurlyQuote]s exact test. By \ observing the table for our age discrimination case (see below), it is clear \ that a lesser proportion of the protected class was listed for layoff than \ that proportion of the unprotected class. Here there is no need to calculate \ probabilities, since it is clear that there is a greater proportion of people \ over 40 in the group not laid off than there is in the laid off group. More \ about the hypergeometric distribution later." }], "Text"], Cell["\<\ Code for Table (do not print, adjust font size as appropriate)\ \>", "Text", FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[ \(fontsize = 9; \n DisplayForm[ StyleBox[GridBox[ Prepend[{\n\t\t Prepend[Append[ b1 = BinCounts[dismissed, {39.5 - 100, 39.5 + 100, 100}], Plus@@b1], "\"], \n\t\t Prepend[Append[ b2 = BinCounts[kept, {39.5 - 100, 39.5 + 100, 100}], Plus@@b2], "\"], \n\t\t Prepend[{ b1\[LeftDoubleBracket]1\[RightDoubleBracket] + b2\[LeftDoubleBracket]1\[RightDoubleBracket], b1\[LeftDoubleBracket]2\[RightDoubleBracket] + b2\[LeftDoubleBracket]2\[RightDoubleBracket], "\<\>"}, "\"]}, \n{"\<\>", "\<<40\>", "\<\[GreaterEqual]40\>", "\"}], \n\ GridFrame \[Rule] 1, RowLines \[Rule] {1, 0, 1}, ColumnLines \[Rule] 1], \nFontFamily \[Rule] Times, FontSize \[Rule] fontsize, Background \[Rule] GrayLevel[0.9]]]\)], "Input"], Cell[BoxData[ TagBox[ StyleBox[GridBox[{ {"", \( < 40\), \( \[GreaterEqual] 40\), "Total"}, {"Dismissed", "5", "6", "11"}, {"Kept", "50", "66", "116"}, {"Total", "55", "72", ""} }, GridFrame->True, RowLines->{True, False, True}, ColumnLines->True], FontFamily->"Times", FontSize->9, Background->GrayLevel[0.900008]], DisplayForm]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ". A Randomization Test Lab Activity" }], "Subsection", Evaluatable->False], Cell[TextData[{ "Scheaffer et al [Scheaffer 1996] describe a number of discrimination cases \ that lend themselves to analysis using randomization tests. For example, in a \ 1977 race discrimination case, Dendy vs. Washington Hospital Center, four of \ nine African-American nurses passed an examination while all 26 white nurses \ passed. In such cases it may be argued that employment tests are \ discriminatory if they adversely impact a protected class. To be successful, \ the plaintiff endeavors to shift the burden of proof to the defendant by \ using statistical evidence to reject the null hypothesis that the impact on \ the protected class is the same as that on the unprotected class. If the \ plaintiff is successful in establishing this disparate impact, then the \ defendant must prove that the use of the criterion is a business necessity.\n\ \nAn ordinary deck of playing cards can help us develop a reference \ distribution for assessing the likelihood of the observed event: four of nine \ African-American nurses passed an examination while all 26 white nurses \ passed. If we discard 17 black cards from the deck and shuffle it thoroughly, \ we can deal five cards representing one realization of this examination \ situation where success on the exam is unrelated to race. The dealt cards \ correspond to individuals who fail.", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "If we repeat this experiment many times we can assess the likelihood that \ all five failures are from the nine nurses belonging to the protected class.\n\ \nIn teaching the concepts of hypothesis testing and reference distributions \ (or working in court) it is useful to have the students (or jurors) actually \ perform the experiment with a deck of cards; but such physical experiments \ waste time and are distracting. We can use the ", StyleBox["VisualStats", "Input", FontWeight->"Plain"], " package function ", StyleBox["Deal", "Input", FontWeight->"Plain"], ". Evaluate the following ten times (or even twenty) and you will almost \ surely see no cases where five black cards turn up. The ", StyleBox["RemovedCards", "Input", FontWeight->"Plain"], " option sets the removed cards in the suits clubs, diamonds, hearts, \ spades (in that order, using ", Cell[BoxData[ \(TraditionalForm\`{1, 2, \[Ellipsis], 13}\)]], " for the cards.) 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FontWeight->"Bold"], StyleBox[CounterBox["NumberedFigure"], FontWeight->"Bold"], StyleBox[":", FontWeight->"Bold"], " Deleting 17 black cards from a deck and generating several deals of five \ cards gives students and jurors a feel for the (very small) chance that all \ five are black." }], "NumberedFigure", CellTags->"CardArray"], Cell["\<\ Of course, we could have avoided experimentation by applying our \ knowledge of the hypergeometric distribution from our study of discrete \ probability distributions. The probability that four of nine African-American \ nurses passed an examination while all 26 white nurses passed is given by the \ hypergeometric distribution (see section 3.1.2). Such a calculation shows \ that the chance of the observed event is about 1 in 2600, which provides good \ evidence that the test is discriminatory. More about the hypergeometric \ distribution in a moment.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". Some Probability Models" }], "Section"], Cell[TextData[{ "The understanding of the axioms of probability and some special \ probability distributions is essential to the understanding of hypothesis \ testing, and of course these probability models have important applications \ in their own right. In this section, we demonstrate the use of some ", StyleBox["VisualStats", "Input", FontWeight->"Plain"], " experimentation functions that we ask students to use to understand and \ reinforce how these models might naturally arise." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ". Models Arising from Sampling" }], "Subsection"], Cell["\<\ Here we describe a few experiments to help the student understand \ the similarities and differences between the binomial and hypergeometric \ random variables. These random variables arise naturally from sampling. \ Although these two distributions are a result of attribute sampling, the \ notions about sampling carry over to variable sampling.\ \>", "Text"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ".", CounterBox["Subsubsection"], ". Attribute Sampling With and Without Replacement" }], "Subsubsection", Evaluatable->False], Cell[TextData[{ "Some people distinguish between ", StyleBox["attribute sampling", FontSlant->"Italic"], ", where one samples individuals or sampling units from some population and \ then observes the presence or absence of some attribute (e.g., over 40 or \ not, black or not, success or failure, defective or not), and ", StyleBox["variable sampling,", FontSlant->"Italic"], " where one takes measurements of a more continuous nature (e.g., height, \ weight, dollars, calories). When the population from which the sample is \ drawn is finite, we distiguish between sampling with replacement and sampling \ without replacement. To illustrate, consider an urn containing ", StyleBox["r", FontSlant->"Italic"], " red balls and ", StyleBox["b", FontSlant->"Italic"], " black balls. We will choose one ball at random, record its color, and \ then sample again after having replaced the original ball (or not). A more \ general replacement strategy replaces the selected ball together with ", StyleBox["c", FontSlant->"Italic"], " balls of the color selected and ", StyleBox["d", FontSlant->"Italic"], " balls of the other color. For the hypergeometric distribution, which \ assumes sampling without replacement, ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["c", FontSlant->"Italic"], "=", \(-1\)}], TraditionalForm]]], ". The binomial distribution requires sampling with replacement, or ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["c", FontSlant->"Italic"], "=", "0"}], TraditionalForm]]], ". In both cases, ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["d", FontSlant->"Italic"], "=", "0"}], TraditionalForm]]], ".\n\nThe ", StyleBox["VisualStats", "Input", FontWeight->"Plain"], " function ", StyleBox["Urn", "Input", FontWeight->"Plain"], " provides a convenient means for performing this experiment. For more on \ urn models see [Feller, 1968]. ", StyleBox[ "Urn[{r,\[MediumSpace]b},\[MediumSpace]{c,\[MediumSpace]d},\[MediumSpace]n]\ ", "Input", FontWeight->"Plain"], " returns images of ", Cell[BoxData[ \(TraditionalForm\`n\)]], " generations of an urn that starts with ", Cell[BoxData[ FormBox[ StyleBox["r", FontSlant->"Italic"], TraditionalForm]]], " red balls and ", Cell[BoxData[ FormBox[ StyleBox["b", FontSlant->"Italic"], TraditionalForm]]], " black balls. At each generation, a ball is chosen and ", StyleBox["c", FontSlant->"Italic"], " balls of the same color of the chosen one and ", StyleBox["d", FontSlant->"Italic"], " of the other color are added to the urn (", Cell[BoxData[ FormBox[ StyleBox["c", FontSlant->"Italic"], TraditionalForm]]], " can equal ", Cell[BoxData[ \(TraditionalForm\`\(-1\)\)]], ").\n\nFor example, let\[CloseCurlyQuote]s return to the case of 26 white \ nurses and 9 African-American nurses who attempted an exam that produced five \ failures. If we assume that success on the exam is inpedendent of race and \ that 30 jobs are available (i.e., five must fail), then the process can be \ described using an urn model and sampling without replacement. Of particular \ importance is the fact that the probability of observing a red ball changes \ from trial to trial.\n\nThe code that follows generates a 5-generation movie \ of an urn evolution (see Figure ", CounterBox["NumberedFigure", "UrnArray"], ")." }], "Text"], Cell[BoxData[ \(\(Urn[{26, 9}, {\(-1\)\ , 0}, 5]; \)\)], "Input"], Cell[TextData[{ "Code for Figure ", CounterBox["NumberedFigure", "UrnArray"] }], "Text", FontWeight->"Bold"], Cell[BoxData[ \(\(SeedRandom[1]; \n images\ = \ Block[{$DisplayFunction\ = \ Identity}, \n\t\t \((\ uu = Urn[{26, 9}, {\(-1\), 0}, 6, \ BallColors -> {White, Black}, \n LabelBackground -> White, \ DefaultFont\ -> \ {"\", \ 8}, StepSize -> 1])\); \n\t\t tt = Table[ uu[\([i]\)]\ /. \ \ \((PlotLabel -> z_)\)\ :> \ \n\t\t\t\t \((PlotLabel\ -> DisplayForm@\n\t\t\t\t\t\t\ \ \((StyleBox[ GridBox[ Map[MakeBoxes[#]\ &, \n\t\t\t\t\t\t\t\t\t\t Partition[ Cases[z, \ StringForm[x_, y__]\ :> \ StringForm[ StringReplace[x, \ "\" -> "\"], y], \ \[Infinity]], 2], 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", "Hypergeometric Distribution" }], "Subsubsection", Evaluatable->False], Cell[TextData[{ "Suppose we had a population of size ", Cell[BoxData[ FormBox[ StyleBox["N", FontSlant->"Italic"], TraditionalForm]]], " consisting of ", Cell[BoxData[ \(TraditionalForm\`k\)], FontSlant->"Italic"], " successes, and consequently ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["N", FontSlant->"Italic"], "-", StyleBox["k", FontSlant->"Italic"]}], TraditionalForm]]], " failures. If we draw a sample of size ", Cell[BoxData[ FormBox[ StyleBox["n", FontSlant->"Italic"], TraditionalForm]]], " without replacement, the probability of exactly ", Cell[BoxData[ FormBox[ StyleBox["x", FontSlant->"Italic"], TraditionalForm]]], " successes and ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["n", FontSlant->"Italic"], "-", StyleBox["x", FontSlant->"Italic"]}], TraditionalForm]]], " failures is" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ { StyleBox["k", FontSlant->"Italic"]}, { StyleBox["x", FontSlant->"Italic"]} }], "\[NegativeThinSpace]", ")"}], RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ { RowBox[{ StyleBox["N", FontSlant->"Italic"], "-", StyleBox["k", FontSlant->"Italic"]}]}, { RowBox[{ StyleBox["n", FontSlant->"Italic"], "-", StyleBox["x", FontSlant->"Italic"]}]} }], "\[NegativeThinSpace]", ")"}]}], RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ { StyleBox["N", FontSlant->"Italic"]}, { StyleBox["n", FontSlant->"Italic"]} }], "\[NegativeThinSpace]", ")"}]], "."}], TextForm]], "Text",\ TextAlignment->Center, TextJustification->0], Cell["\<\ If a sample of size five is drawn without replacement from a \ population of size 35 consisting of 9 black balls and 26 red ones, the \ probability of five blacks, and consequently 0 reds, is \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[\(Binomial[9, 5]\ Binomial[26, 0]\)\/Binomial[35, 5]]\)], "Input"], Cell[BoxData[ \(0.000388131792306365408`\)], "Output"] }, Open ]], Cell[TextData[{ "Or, we may use functions from ", StyleBox["Mathematica\[CloseCurlyQuote]", FontSlant->"Italic"], "s standard package ", StyleBox["Statistics`DiscreteDistributions`", "Input", FontWeight->"Plain"], ". The hypergeometric distribution is the probability function returning \ the probability of exactly ", StyleBox["x", FontSlant->"Italic"], " successes in a sample of size ", StyleBox["n", FontSlant->"Italic"], " drawn without replacement from a population of size N comprised of k \ successes and ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["N", FontSlant->"Italic"], "-", StyleBox["k", FontSlant->"Italic"]}], TraditionalForm]]], " failures. The function ", StyleBox[ "PDF[HypergeometricDistribution[n,\[ThinSpace]k,\[ThinSpace]N],\[ThinSpace]\ x]", "Input", FontWeight->"Plain"], " from the ", StyleBox["DiscreteDistributions", "Input", FontWeight->"Plain"], " package provides this probability mass." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[PDF[HypergeometricDistribution[5, 9, 35], 5]]\)], "Input"], Cell[BoxData[ \(0.000388131792306365408`\)], "Output"] }, Open ]], Cell["\<\ As expected from our investigation into the nurse-exam simulation, \ this is an extremely rare event.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ".", CounterBox["Subsubsection"], ". The Binomial Experiment" }], "Subsubsection", Evaluatable->False], Cell[TextData[{ "We can return to the urn model and now require that the selected ball be \ replaced. Unlike sampling without replacement, in this situation the \ probability of obtaining a red ball remains the same from trial to trial. By \ specifying a fixed number of draws or trials, ", Cell[BoxData[ \(TraditionalForm\`n\)]], ", and counting the number of successes (red balls) observed, ", Cell[BoxData[ \(TraditionalForm\`x\)], FontSlant->"Italic"], ", we have the binomial experiment. The probability of a success on each \ trial, ", Cell[BoxData[ \(TraditionalForm\`p\)], FontSlant->"Italic"], ", is merely the relative frequency of red balls in the urn; ", StyleBox["p", FontSlant->"Italic"], " can of course be any number from zero to one, ", Cell[BoxData[ FormBox[ RowBox[{"0", "\[LessEqual]", StyleBox["p", FontSlant->"Italic"], "\[LessEqual]", "1"}], TraditionalForm]]], ". Sampling with replacement in the urn model can be seen to be equivalent \ to observing the number of heads in a sequence of coin flips.\n", StyleBox["\n", "Input"], "The function ", StyleBox["CoinFlips[n,\[ThinSpace]1]", "Input", FontWeight->"Plain"], " used next performs a useful experiment. Students can experiment with it \ by changing the number of tosses or the relative frequency (use the ", StyleBox["HeadProbability", "Input", FontWeight->"Plain"], " option) of red balls.\n\nRealistic examples are useful. Suppose that one \ of the products produced by a computer manufacturer is very new and requires \ a component, a particular computer chip, that is very difficult to produce in \ large quantities. However, a supplier of the chip has been found, but the \ supplier will only assure that 85% of the chips function reliably. In chip \ manufacturing this is known as the ", StyleBox["process yield", FontSlant->"Italic"], ". Defective chips will be detected at one stage of the manufacturing \ process and the cost of replacing the defective chips in terms of time and \ materials has been factored into published prices. It is critical that \ shipments of the chips contain no more than 15% defective, so procurement has \ recommended a lot acceptance sampling procedure that will test some number \ \[LongDash] say ", StyleBox["n", FontSlant->"Italic"], " \[LongDash] of chips from each lot. In order to eliminate the cost of \ returned shipments, the testing will take place at the supplier\ \[CloseCurlyQuote]s facility.\n\nIf the manufacturer directs the supplier to \ randomly choose seven chips for testing and refuses to accept a lot if the \ seven tested chips include more than one defective chip, what proportion of \ the time will procurement refuse a shipment with 85% good chips? Here we may \ ask whether the sampling should be with or without replacement, which \ distribution is appropriate, and when does the choice of distribution \ matter." }], "Text"], Cell[TextData[ "Execute the next cell for an example of what might happen when the results \ of the tests are recorded. On the inspector\[CloseCurlyQuote]s recordkeeping \ form there is box representing each chip to be inspected. If a chip is \ defective, he darkens the corresponding box on his form. Try this a few \ times. We may change the number of chips inspected and/or the proportion \ defective."], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox[\(SeedRandom[1]\), FontColor->RGBColor[1, 0, 0]], StyleBox[";", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\n", StyleBox[" ", FontColor->RGBColor[1, 0, 0]], \(CoinFlips[7, 1, MovieOnly \[Rule] True, HeadProbability \[Rule] 0.15] \), ";"}]], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .14286 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 8 scalefont setfont % Scaling calculations -0.112245 0.136054 0.139456 0.136054 [ [ 0 0 0 0 ] [ 1 .14286 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath .5 g .43197 .0034 m .43197 .13946 L .56803 .13946 L .56803 .0034 L F 0 g .5 Mabswid [ ] 0 setdash .15986 .13946 m .15986 .0034 L s .29592 .13946 m .29592 .0034 L s .43197 .13946 m .43197 .0034 L s .56803 .13946 m .56803 .0034 L s .70408 .13946 m .70408 .0034 L s .84014 .13946 m .84014 .0034 L s .02381 .0034 m .97619 .0034 L .97619 .13946 L .02381 .13946 L .02381 .0034 L s 0 0 m 1 0 L 1 .14286 L 0 .14286 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 41.125}, ImageMargins->{{62, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False, ImageRangeCache->{{{0, 287}, {40.125, 0}} -> {0.74475, -1.02501, 0.0261691, 0.0261691}}] }, Open ]], Cell[TextData[{ StyleBox["Figure ", FontWeight->"Bold"], StyleBox[CounterBox["NumberedFigure"], FontWeight->"Bold"], StyleBox[":", FontWeight->"Bold"], " A typical case of a sample of 7 chips, with the defective ones marked by \ dark squares." }], "NumberedFigure", CellTags->"SevenCoinFlips"], Cell[TextData[{ "This problem is analogous to flipping seven coins. Experiments similar to \ a coin-tossing experiment occur frequently. And the ", StyleBox["CoinFlips", "Input", FontWeight->"Plain"], " function can be used in several ways to study such binomial experiments. \ By setting the ", StyleBox["MovieOnly", "Input", FontWeight->"Plain"], " option to ", StyleBox["False", "Input", FontWeight->"Plain"], ", the default, and by using the second argument to indicate the number of \ times to repeat the binomial experiment, we introduce the notion of a \ sampling distribution and the central limit theorem. Notice that to the right \ of each realization of the binomial experiment is printed the numbers of \ heads, which are in turn tabulated in the histogram on the right. The ", StyleBox["HeadProbability", "Input", FontWeight->"Plain"], " option can be used when the probability of a head is different than one \ half.\n\nThrough experimentation, the student should discover that as the \ number of coins gets large, the histogram displaying the number of heads \ resembles the Gaussian distribution; this is the DeMoivre\[Dash]LaPlace \ theorem, a special case of the central limit theorem. 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Distribution Associated with Waiting Times" }], "Subsection"], Cell["\<\ Experimenting with randomly occurring events in time allows us to \ explore the Poisson and exponential distributions. The Poisson random \ variable is a discrete random variable; the exponential random variable is \ continuous. We discuss them together because in some applications they are \ intimately related. In such cases, each of these random variables is defined \ on the same sample space, the sample space resulting from a Poisson \ experiment. These two random variables are not always naturally connected and \ may be applied in a variety of situations, but here they will be introduced \ together as random variables evolving from the Poisson process.\ \>", "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ".", CounterBox["Subsubsection"], ".", " ", "The Poisson Process" }], "Subsubsection", Evaluatable->False], Cell["\<\ The Poisson process describes randomly occurring events in a \ continuum, such as time or space (of 1, 2, or 3 dimensions). For the moment \ we will use time as an example of a continuum in which random events may \ occur. In the demonstration that follows, events are randomly generated in \ time. You can think of these events as customers arriving at a bank, or cars \ arriving at a toll booth. Note that we may view this experiment as random \ events in time and are interested in how many events occur during some time \ period, or the amount of ellapsed time between arrivals.\ \>", "Text", Evaluatable->False], Cell[TextData[ "The input cell below generates a series of graphics cells that form an \ animation demonstrating the Poisson process, a process where events appear \ randomly in time. The images may be interpreted as people walking up to a \ teller\[CloseCurlyQuote]s window. The number of arrivals in any time period \ is a Poisson random variable. The distribution of the time intervals between \ arrivals will follow the exponential distribution."], "Text", Evaluatable->False], Cell[BoxData[ \(WaitingTimes[20, Movie \[Rule] True]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ".", CounterBox["Subsubsection"], ". The Poisson Random Variable and Its Probability Mass Function" }], "Subsubsection", Evaluatable->False], Cell[TextData[{ "In the Poisson process where events occur randomly in time, the number of \ events occuring in some fixed time interval follows the Poisson probability \ distribution. That is, if we fix the time interval over which we intend to \ observe the process, the random variable ", StyleBox["X", FontSlant->"Italic"], " that counts the number of events occuring in this interval is the \ discrete random variable called the Poisson random variable. The probability \ mass function for this random variable is", " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["f", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")"}], "=", RowBox[{ SuperscriptBox[ StyleBox["e", FontSlant->"Italic"], \(-\[Lambda]\)], " ", RowBox[{ SuperscriptBox["\[Lambda]", StyleBox["x", FontSlant->"Italic"]], "/", " ", RowBox[{ StyleBox["x", FontSlant->"Italic"], "!"}]}]}]}], TraditionalForm]]], ", where \[Lambda] is the average number of events occurring in a time \ interval of this length.\n\nSuppose that customers arrive randomly at a bank \ at a rate of 30 per hour. Thus if we are interested in observing the process \ for one hour, \[Lambda] is 30. On the other hand if we wish to observe for \ one half hour, ", Cell[BoxData[ \(TraditionalForm\`\[Lambda] = 15\)]], ". What is the probability of exactly 10 arrivals in a one half hour?" }], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(N[15\^10\/\(E\^15\ \(10!\)\)]\)], "Input"], Cell[BoxData[ \(0.048610750829605358`\)], "Output"] }, Open ]], Cell[TextData[{ "In its ", StyleBox["DiscreteDistributions", "Input"], " package, ", StyleBox["Mathematica", FontSlant->"Italic"], " provides a function for the Poisson distribution. We can view its \ equation for the probability mass (or density) function (", StyleBox["pdf", FontSlant->"Italic"], "), by evaluating the cell below." }], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData["PDF[PoissonDistribution[\[Lambda]],x]"], "Input"], Cell[BoxData[ \(\(E\^\(-\[Lambda]\)\ \[Lambda]\^x\)\/\(x!\)\)], "Output"] }, Open ]], Cell[TextData[{ "And we may use these functions to obtain the probability of exactly ", StyleBox["x", FontSlant->"Italic"], " events. 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The vertical \ scale is a percentage. " }], "NumberedFigure", CellTags->"PoissonBarChart"], Cell[TextData[{ "To obtain the probability of no more than 10 arrivals, we must sum the \ probabilities of ", Cell[BoxData[ \(TraditionalForm\`0, 1, \[Ellipsis], 10\)]], "." }], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(Sum[N[PDF[PoissonDistribution[15], x]], {x, 0, 10}]\)], "Input"], Cell[BoxData[ \(0.118464411529015167`\)], "Output"] }, Open ]], Cell[TextData[{ "\tOf course, what we were asking for in the cell above was the cumulative \ distribution function or cdf of 10, ", StyleBox["F", FontSlant->"Italic"], "(10)." }], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(N[CDF[PoissonDistribution[15], 10]]\)], "Input"], Cell[BoxData[ \(0.118464411529015078`\)], "Output"] }, Open ]], Cell[TextData[ "The mean and the variance of the Poisson distribution are both \[Lambda]. If \ \[Lambda] gets large, the shape of the Poisson distribution gets more \ symmetric and we get the impression that we could use the normal distribution \ to approximate it, using the appropriate correction for continuity."], "Text",\ Evaluatable->False], Cell[CellGroupData[{ Cell[TextData[ "Clear[\[Lambda]];\n{Mean[PoissonDistribution[\[Lambda]]],\n\ Variance[PoissonDistribution[\[Lambda]]]}"], "Input"], Cell[BoxData[ \({\[Lambda], \[Lambda]}\)], "Output"] }, Open ]], Cell["\<\ The Poisson random variable is a discrete random variable since \ there is only a countable infinity of possible values of the count of the \ randomly occuring events in time. If, on the other hand, we are interested in \ the continuous random variable that measures the distance between randomly \ occurring events in the Poisson process, the exponential distribution may be \ derived.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ".", CounterBox["Subsubsection"], ".", " ", "The Exponential Random Variable and Its Probability Density Function" }], "Subsubsection", Evaluatable->False], Cell[TextData[{ "When observing a Poisson process, we can, instead of counting events in \ some fixed time interval, record the time between successive events. The \ probability distribution of the continuous random variable ", StyleBox["T", FontSlant->"Italic"], ", the time between successive events, frequently follows the exponential \ probability distribution, whose probability density function, ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ StyleBox["t", FontSlant->"Italic"], ",", RowBox[{ StyleBox["f", FontSlant->"Italic"], "(", StyleBox["t", FontSlant->"Italic"], ")"}]}], "}"}], TraditionalForm]]], " is defined by the relationship ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["f", FontSlant->"Italic"], "(", StyleBox["t", FontSlant->"Italic"], ")"}], "=", RowBox[{\(1\/\[Beta]\), " ", FormBox[ SuperscriptBox["e", RowBox[{ StyleBox["t", FontSlant->"Italic"], "/", "\[Beta]"}]], "TraditionalForm"]}]}], TraditionalForm]]], " for ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["t", FontSlant->"Italic"], ">", "0"}], TraditionalForm]]], ". Here \[Beta] represents the average time between arrivals of these \ randomly occurring events in time, and therefore is the reciprocal of \ \[Lambda], the mean of the Poisson distribution. This fact may become more \ obvious if we consider our customer arrivals at the bank that averaged 30 per \ hour. The expected time between arrivals is thus 2 minutes: ", Cell[BoxData[ \(TraditionalForm\`\[Beta] = 1/\[Lambda]\)]], ".\n\nWe can plot this probability density function and notice that there \ is an uncountably infinite number of possible values of our random variable ", StyleBox["T", FontSlant->"Italic"], ". 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" }], "NumberedFigure", CellTags->"PoissonBarChart"], Cell[TextData[{ "The expected value and variance of the the exponential distribution are \ \[Beta] and ", Cell[BoxData[ \(TraditionalForm\`\[Beta]\^2\)]], ", respectively." }], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[TextData[ "Clear[\[Lambda]];\n{Mean[ExponentialDistribution[\[Lambda]]],\n\ Variance[ExponentialDistribution[\[Lambda]]]}"], "Input"], Cell[BoxData[ \({1\/\[Lambda], 1\/\[Lambda]\^2}\)], "Output"] }, Open ]], Cell[TextData[{ "We may wish to convince ourselves that ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["e", RowBox[{ RowBox[{"-", StyleBox["t", FontSlant->"Italic"]}], "/", "\[Beta]"}]], "/", "\[Beta]"}], TraditionalForm]]], " is truely a pdf (nonnegative and the total area under this curve is 1)." }], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ PDF[ExponentialDistribution[1\/\[Beta]], t], {t, 0, \[Infinity]}, Assumptions \[Rule] Re[\[Beta]] > 0]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[TextData[{ "We can find the probability that ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["T", FontSlant->"Italic"], "<", StyleBox["c", FontSlant->"Italic"]}], TraditionalForm]]], ", for any positive number ", StyleBox["c", FontSlant->"Italic"], ". We merely find the area under the pdf below the point ", StyleBox["c", FontSlant->"Italic"], " on the ", StyleBox["t", FontSlant->"Italic"], "-axis." }], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(Clear[\[Beta]]; \n \[Integral]\_0\%c PDF[ExponentialDistribution[1\/\[Beta]], t] \[DifferentialD]t\)], "Input"], Cell[BoxData[ \(1 - E\^\(-\(c\/\[Beta]\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "Recall that we call this the ", StyleBox["cumulative distribution function", FontSlant->"Italic"], " or ", StyleBox["cdf", FontSlant->"Italic"], " of ", StyleBox["c", FontSlant->"Italic"], ", or ", StyleBox["F", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ")." }], "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(CDF[ExponentialDistribution[1/\[Beta]], c]\)], "Input"], Cell[BoxData[ \(1 - E\^\(-\(c\/\[Beta]\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "So, we discover that we do not need to use calculus to find areas under \ the exponential pdf; we merely make use of the fact that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["F", FontSlant->"Italic"], "(", StyleBox["c", FontSlant->"Italic"], ")"}], "=", RowBox[{"1", "-", SuperscriptBox[ StyleBox["e", FontSlant->"Italic"], RowBox[{ RowBox[{"-", StyleBox["c", FontSlant->"Italic"]}], "/", "\[Beta]"}]]}]}], TraditionalForm]]], ". The next code generates an exponential pdf and shades the area below the \ value 1.5. The shaded area is ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ StyleBox["F", FontSlant->"Italic"], "(", "1.5", ")"}], "=", RowBox[{"1", "-", FormBox[ SuperscriptBox[ StyleBox["e", FontSlant->"Italic"], \(\(-1.5\)/\[Lambda]\)], "TraditionalForm"]}]}], TraditionalForm]]], ". 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In this simulation, 10 customers arrive at a \ server, say a drive-up teller. The time to serve these customers is 1.5 times \ the interarrival time. For each of the 10 bars produced in this graphic, \ there is a black segment at the left end of the bar which is the same for \ each customer and represents the time to get from the street to the end of \ the line. On the right end of each bar is a black (or green) segment \ depicting the service time for that customer. Between the drive-up segment \ and the service segment, there is the potential of a gray (or red) \ waiting-time segment. In this example notice that the mean service time is \ 1.5 times the mean interarrival time, so a waiting line tends to develop. The \ vertical line indicates when the window closes. 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A Queueing Case Study" }], "Subsubsection", Evaluatable->False], Cell[TextData[{ "This case study describes how a real-world situation was turned into a \ valuable project for students. The students were able to use the ", StyleBox["WaitingTimes", "Input", FontWeight->"Plain"], " function in a large business meeting to convince the principals that \ their planned communication links were in error by orders of magnitude." }], "Text"], Cell[TextData[{ "Revenue Management Services (RMS) is a company specializing in technical \ assistance, both regulatory and operational, to providers of medical \ services. One of its ventures involves establishing a wide area computer \ network (WAN) for processing Medicaid applications for current and \ prospective nursing home residents. The applications themselves are two or \ three pages of responses to questions pertaining to a resident\ \[CloseCurlyQuote]s health and financial situation. However, supporting \ documentation, including recent bank statements, involves 150 pages on \ average. The applications with supporting documentation are copied at the \ nursing home and forwarded to New York City Human Resources Administration \ (NYC HRA) for a determination of eligibility. For a number of reasons having \ to do with the sheer volume of the paperwork, the eligibility determinations \ take months to complete, delaying placement decisions and tying up cash.\n\n\ RMS has contracted with a software company specializing in optical imaging to \ develop software that would take a resident\[CloseCurlyQuote]s Medicaid \ application electronically and attach scanned images of the supporting \ documentation to a record in a database residing on the network\ \[CloseCurlyQuote]s server to be located north of NYC. Each nursing home \ subscribing to this service would have a personal computer (PC) with an \ attached scanner. The nursing home computers would be attached to the server \ via one of several possible telecommunication options. Matt, the president of \ the software company, has recommended a Frame Relay solution that would \ involve leasing a T1 trunk composed of 24 lines at $980 per month, a $1400 \ per month access charge, and $40 per month for each nursing home for \ unlimited calling. With the Frame Relay solution, the nursing home\ \[CloseCurlyQuote]s computer would never encounter a busy signal, because the \ communication carrier\[CloseCurlyQuote]s communications computers would \ merely store information sent by the nursing home until the network server \ was ready to accept it. This solution is also endorsed by David, RMS\ \[CloseCurlyQuote]s programmer assigned to this project, and RMS\ \[CloseCurlyQuote]s marketing managers, Marty and Lew, who like the idea that \ the customer would never encounter a busy signal.\n\nTom, a communications \ consultant, and Karl, a statistical consultant, oppose the Frame Relay \ solution arguing that is unnecessarily costly for the amount of traffic that \ the network is likely to see during its first year of operations. Marty has \ estimated that by the end of the year there will be 40 nursing homes \ subscribing. Marty and Lew anticipate that the larger homes would tend to be \ the first to subscribe. The solution favored by Tom and Karl involves using \ high-speed modems capable of receiving 10 scanned pages per minute at a cost \ to the nursing home of 8 cents per minute and a cost to RMS of $980 per \ month.\n\nCompare the two solutions with respect to cost and various waiting \ line characteristics, such as probability of busy, average number of \ customers in the system, and so on. State your assumptions and vary the \ parameters of your model to show under what extreme conditions your \ recommendation still holds; that is, conduct a sensitivity analysis. Simulate \ your model using ", StyleBox["WaitingTimes", "Input"], ", which will provide a graphical description of the way the proposed \ system would operate. Your report should include an executive summary \ accessible to a nontechnical reader, which explains your figures, graphs, and \ conclusions." }], "Text", Evaluatable->False] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". Summary" }], "Section"], Cell[TextData[{ "There are a number of very good software tools available to statisticians \ for performing data analysis. But ", StyleBox["Mathematica\[CloseCurlyQuote]", FontSlant->"Italic"], "s tremendous breadth will generally allow one to do much more. The \ functions in the ", StyleBox["VisualStats", "Input", FontWeight->"Plain"], " package illustrated here were quite straightforward to program; readers \ with better visualization ideas in this area should have no difficulty \ implementing them in ", StyleBox["Mathematica", FontSlant->"Italic"], ". Indeed, statistics is ripe for this sort of work because the community \ of statisticians has done a lot of excellent work on visualizing data sets. \ Another area that we have not touched on is the use of the front end to \ create flashy notebooks for student use. Again, there are a lot of materials \ on the use of computing in statistics education, and we imagine that many of \ the activities would be well suited to ", StyleBox["Mathematica", FontSlant->"Italic"], " demonstrations." }], "Text", FontWeight->"Plain"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ "[Feller 1968] Feller, W., ", StyleBox["An Introduction to Probability Theory and Its Applications,", FontSlant->"Italic"], " Wiley, New York, 1968.\n\n[Scheaffer 1996] Scheaffer, R. L., \ Gnanadesikan, M., Watkins, A., and Witmer, J. A., ", StyleBox["Activity-Based Statistics: Instructor Resources,", FontSlant->"Italic"], " Springer, New York. \n\n[Wagon 1995] Wagon, S., B\[EAcute]zier curves, ", StyleBox["Mathematica in Education and Research,", FontSlant->"Italic"], " 4:1 (48\[Dash]53)." }], "Text"], Cell[CellGroupData[{ Cell["About the Authors", "Subsection"], Cell["\<\ Karl Heiner is an associate professor in the Department of Business Administration at the State University of New York at New Paltz. His \ doctorate is from Columbia University, in teaching of college statistics. He uses Mathematica in teaching introductory statistics courses, in his consulting \ work, which includes preparing evidence as an expert witness, and in his research \ into more effective estimation in statistical auditing and in the monitoring \ health care quality. Karl Heiner Department of Business Administration SUNY at New Paltz New Paltz, NY12561 kwh@world.com \ \>", "Text"], Cell[TextData[{ "Stan Wagon uses ", StyleBox["Mathematica", FontSlant->"Italic"], " extensively in his research, teaching, and exposition, and is especially \ appreciative of the new ways ", StyleBox["Mathematica", FontSlant->"Italic"], " allows us to look at mathematical objects. He is author or coauthor of \ several books, including ", StyleBox["Mathematica in Action", FontSlant->"Italic"], ", ", StyleBox["Animating Calculus", FontSlant->"Italic"], ", ", StyleBox["VisualDSolve", FontSlant->"Italic"], ", and ", StyleBox["Which Way Did the Bicycle Go?", FontSlant->"Italic"], ". He also teaches a high-altitude course, ", StyleBox["Rocky Mountain Mathematica", FontSlant->"Italic"], ", every summer in the mountains of Colorado.\n\nStan Wagon\nDepartment of \ Mathematics and Computer Science\nMacalester College\nSt. Paul, MN 55105\n\ wagon@macalester.edu" }], "Text"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1280}, {0, 1024}}, AutoGeneratedPackage->None, ScreenStyleEnvironment->"Working", WindowToolbars->{}, WindowSize->{735, 677}, WindowMargins->{{Automatic, 157}, {Automatic, 70}}, Magnification->1 ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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