(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing 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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Feel free to use and distribute this notebook, but \ keep this author information in any copy you use or distribute.\ \>", "SmallText"], Cell[TextData[{ "In any input cell containing ", StyleBox["xxx", FontColor->RGBColor[1, 0, 1]], " , you must replace it with your input before evaluating the cell. In \ general, anything in ", StyleBox["magenta", FontColor->RGBColor[1, 0, 1]], " is something you can, and possibly should, change." }], "Text"], Cell[CellGroupData[{ Cell["Introduction to Confidence Intervals", "Section"], Cell[TextData[{ "When we randomly sample data from a population, we can compute a sample \ mean. This value may or may not be the actual population mean. In order to \ decide how confident we are in asserting that the sample mean we computed is \ near the actual population mean, we make use of what is called a ", StyleBox["confidence interval", FontWeight->"Bold"], "." }], "Text"], Cell["\<\ We begin by setting a level of confidence, say 95%. From this we calculate \ an interval centered at the sample mean. Each time we sample data, we \ calculate a new interval. We say that the interval we calculated from a \ particular sample is a 95% confidence interval for the actual mean if the \ actual mean is contained in the interval we calculate 95% of the time. (If \ we do 100 samplings, we would expect that the actual population mean is in \ the confidence interval we compute for the sample about 95 times.)\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Normally distributed populations with known standard deviation\ \>", "Subsection"], Cell[TextData[{ "If we take a random sample of size ", StyleBox["n", FontSlant->"Italic"], " from a normally distributed population whose standard deviation \[Sigma] \ is known and whose mean is \[Mu] (probably unknown), then from Statistics we \ know that the sample mean random variable ", Cell[BoxData[ \(TraditionalForm\`X\&_\)]], " is normally distributed with expected value \[Mu] and standard deviation \ ", Cell[BoxData[ FormBox[ FractionBox["\[Sigma]", SqrtBox[ StyleBox["n", FontSlant->"Italic"]]], TraditionalForm]]], ". Because the area under the standard normal curve between -1.96 and 1.96 \ is .95, we know that" }], "Text"], Cell[TextData[{ "P(-1.96 < ", Cell[BoxData[ \(TraditionalForm\`\(X\&_ - \[Mu]\)\/\(\[Sigma]/\@n\)\)]], " < 1.96) = .95" }], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ (where P(Y) means the probability of Y). We can algebraically manipulate the \ expression\ \>", "Text"], Cell[TextData[{ " ", "-1.96 < ", Cell[BoxData[ \(TraditionalForm\`\(X\&_ - \[Mu]\)\/\(\[Sigma]/\@n\)\)]], " < 1.96" }], "Text", TextAlignment->Center, TextJustification->0], Cell[" to get", "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`X\&_\)]], " - 1.96 ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\/\@n\)]], " < \[Mu] < ", Cell[BoxData[ \(TraditionalForm\`X\&_\)]], " + 1.96 ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\/\@n\)]], "," }], "Text", TextAlignment->Center, TextJustification->0], Cell["so", "Text"], Cell[TextData[{ "P(", Cell[BoxData[ \(TraditionalForm\`X\&_\)]], " - 1.96 ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\/\@n\)]], " < \[Mu] < ", Cell[BoxData[ \(TraditionalForm\`X\&_\)]], " + 1.96 ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\/\@n\)]], ") = .95." }], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ This says that the probability that \[Mu] is in the \"random\" interval\ \>", "Text"], Cell[TextData[{ " (", Cell[BoxData[ \(TraditionalForm\`X\&_\)]], " - 1.96 ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\/\@n\)]], ", ", Cell[BoxData[ \(TraditionalForm\`X\&_\)]], " + 1.96 ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\/\@n\)]], ")" }], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ " is 95%. If we take a random sample of size ", StyleBox["n", FontSlant->"Italic"], " that has sample mean", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\&_\)]], " then our 95% confidence interval is defined to be" }], "Text"], Cell[TextData[{ " (", Cell[BoxData[ \(TraditionalForm\`x\&_\)]], " - 1.96 ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\/\@n\)]], ", ", Cell[BoxData[ \(TraditionalForm\`x\&_\)]], " + 1.96 ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\/\@n\)]], ")." }], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ If we want a different confidence level, we simply need to replace the 1.96 \ with the appropriate value. For instance, a 99% confidence interval in this \ case would be \ \>", "Text"], Cell[TextData[{ " (", Cell[BoxData[ \(TraditionalForm\`x\&_\)]], " - 2.58 ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\/\@n\)]], ", ", Cell[BoxData[ \(TraditionalForm\`x\&_\)]], " + 2.58 ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\/\@n\)]], ")." }], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ (The area under the standard normal curve between -2.58 and 2.58 is .99.)\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Large-sample confidence intervals (population not normal or standard \ deviation unknown)\ \>", "Subsection"], Cell[TextData[{ "If the population with which we are working is not normal or if the \ standard deviation is not known, we can still compute an approximate \ confidence interval for the mean of the population from a \"large\" random \ sample of size ", StyleBox["n", FontSlant->"Italic"], ". Provided ", StyleBox["n", FontSlant->"Italic"], " is large (rule of thumb: ", StyleBox["n", FontSlant->"Italic"], " > 30) the Central Limit Theorem implies that ", Cell[BoxData[ \(TraditionalForm\`X\&_\)]], " has approximately a normal distribution. Also provided ", StyleBox["n", FontSlant->"Italic"], " is sufficiently large, replacing the actual population standard deviation \ \[Sigma] with the sample standard deviation ", Cell[BoxData[ \(TraditionalForm\`s\)]], " will make little difference. As a result, an ", StyleBox["approximate", FontSlant->"Italic"], " 95% confidence interval for the population mean \[Mu] is" }], "Text"], Cell[TextData[{ " (", Cell[BoxData[ \(TraditionalForm\`x\&_\)]], " - 1.96 ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["s", FontSlant->"Italic"], \(\@n\)], TraditionalForm]]], ", ", Cell[BoxData[ \(TraditionalForm\`x\&_\)]], " + 1.96 ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["s", FontSlant->"Italic"], \(\@n\)], TraditionalForm]]], ")." }], "Text", TextAlignment->Center, TextJustification->0] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Small-sample confidence intervals (population is normal, but the standard \ deviation is unknown)\ \>", "Subsection"], Cell[TextData[{ "If the population with which we are working is normal but the standard \ deviation is not known, we can still compute an approximate confidence \ interval for the mean of the population from a \"small\" random sample of \ size ", StyleBox["n", FontSlant->"Italic"], ". In this case, the sample mean has what is called a ", StyleBox["t distribution with ", FontWeight->"Bold"], StyleBox["n", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["-1 degrees of freedom", FontWeight->"Bold"], " rather than a normal distribution. In order to compute a 95% confidence \ interval in this case we need to replace the 1.96 with the value ", Cell[BoxData[ \(TraditionalForm\`t\_\( .025, n - 1\)\)]], " which has the property that the area under the t distribution with n-1 \ degrees of freedom between -", Cell[BoxData[ \(TraditionalForm\`t\_\( .025, n - 1\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`t\_\( .025, n - 1\)\)]], " is .95. This value is usually found in tables for t distributions. For \ instance the 95% confidence interval for the population mean obtained by \ taking a random sample of size 25 from a normal population is" }], "Text"], Cell[TextData[{ " (", Cell[BoxData[ \(TraditionalForm\`x\&_\)]], " - 2.064 ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["s", FontSlant->"Italic"], \(\@25\)], TraditionalForm]]], ", ", Cell[BoxData[ \(TraditionalForm\`x\&_\)]], " + 2.064 ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["s", FontSlant->"Italic"], \(\@25\)], TraditionalForm]]], ")." }], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "where ", StyleBox["s", FontSlant->"Italic"], " is the sample standard deviation. (The area under the t distribution \ with 24 degrees of freedom between -2.064 and 2.064 is .95.)" }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Checking the initialization.", "Section", Evaluatable->False], Cell[TextData[{ "In order to run this notebook, we must load a Mathematica package which \ contains a function that can randomly \"sample\" data from normal \ distribution. The next cell loads the package. It is marked as an \ initialization cell, so if you answered \"Yes\" to the initialization \ question when you loaded this notebook the cell should have already been \ evaluated. If it has, the output of the ", StyleBox["$Packages", FontColor->RGBColor[1, 0, 0]], " cell should include \"", StyleBox["Statistics`NormalDistributions`", FontColor->RGBColor[1, 0, 0]], ".\" If it does not, try to load the package again, or ask for help." }], "Text", Evaluatable->False], Cell[BoxData[ \(Needs["\"]\)], "Input", PageWidth->Infinity, InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \($Packages\)], "Input", PageWidth->Infinity, InitializationCell->True], Cell[BoxData[ \({"Statistics`Common`DistributionsCommon`", "Statistics`DescriptiveStatistics`", "Statistics`NormalDistribution`", "Global`", "System`"}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Random data from a normal distribution."], "Section", Evaluatable->False], Cell[TextData[{ "We begin by choosing ", StyleBox["25", FontColor->RGBColor[1, 0, 1]], " random data values from a normal distribution with mean ", StyleBox["67", FontColor->RGBColor[1, 0, 1]], " and standard deviation ", StyleBox["5", FontColor->RGBColor[1, 0, 1]], "." }], "Text", Evaluatable->False], Cell[BoxData[{\(Clear[n, \[Mu], \[Sigma], data]\), "\[IndentingNewLine]", RowBox[{ RowBox[{"n", "=", StyleBox["25", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"\[Mu]", "=", StyleBox["67", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"\[Sigma]", "=", StyleBox["5", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", \(data = Table[Random[NormalDistribution[\[Mu], \[Sigma]]], {n}]\)}], "Input", PageWidth->Infinity], Cell[TextData[{ "We can calculate the sample mean ", Cell[BoxData[ \(TraditionalForm\`x\&_\)]], " and the sample standard deviation ", Cell[BoxData[ \(TraditionalForm\`s\)]], ". We are using a shortcut formula to find ", StyleBox["s", FontSlant->"Italic"], "." }], "Text", Evaluatable->False], Cell[BoxData[{ \(Clear[s]\), "\n", \(x\&_ = Plus @@ data\/n\), "\n", \(s = N[\@\(\(Plus @@ \(data\^2\) - x\&_\^2\ n\)\/\(n - 1\)\)]\)}], "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ The first confidence interval we will find is a 95% confidence interval for \ the sample mean. We will use the known population standard deviation and \ our knowledge that the population is normally distributed. Recall that the \ value 1.96 is the number of standard deviations away from a mean that \ encompasses 95% of a normal population. If we want other % confidence \ intervals, this is the number we change.\ \>", "Text", Evaluatable->False], Cell[BoxData[ \(N[{x\&_ - \(1.96\ \[Sigma]\)\/\@n, x\&_ + \(1.96\ \[Sigma]\)\/\@n}]\)], "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ Does the actual mean (67) lie in the interval? If we repeat this experiment \ 100 times, we would expect the answer to be \"yes\" about 95 times. We will \ do this in the next section.\ \>", "Text"], Cell["\<\ The second confidence interval we will find is a large-sample \"approximately\ \" 95% confidence interval. We will use the sample standard deviation, as \ we would have to if the population standard deviation was not known. \ (Actually, according to the rule of thumb, n should be greater than 30 for a \ large-sample confidence interval if the population distribution is not normal \ or if the population standard deviation is not known, so we are cheating a \ bit here.)\ \>", "Text"], Cell[BoxData[ \(N[{x\&_ - \(1.96\ s\)\/\@n, x\&_ + \(1.96\ s\)\/\@n}]\)], "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ Does the actual mean (67) lie in the interval? If we repeat this experiment \ 100 times, we would expect the answer to be \"yes\" about 95 times, although \ we might expect a bit more variation since we are not using the actual \ population standard deviation. We will do this in the next section.\ \>", "Text"], Cell["\<\ It might be interesting to change the sample size n, the population mean \ \[Mu], and/or the population standard deviation \[Sigma], and see how the \ changes affect the confidence intervals.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Repeated experiments using random data."], "Section", Evaluatable->False], Cell[CellGroupData[{ Cell["Normal population, known standard deviation.", "Subsection", Evaluatable->False], Cell["\<\ We will automate the process we used above. This will let us test the \"95%\ \" part of the confidence interval. We will run the experiment 100 times. \ It should be the case that we get a confidence interval which contains the \ actual population mean (which we have initially set to 67) in about 95 of the \ experiments. Each line of output will record one experiment. From left to \ right, the reported information is the sample mean, the confidence interval, \ and the report of the test as to whether or not the actual mean is in the \ confidence interval. Check and see if there are about 95 \"Trues\" (and \ about 5 \"Falses\").\ \>", "Text", Evaluatable->False], Cell[BoxData[{\(Clear[n, \[Mu], \[Sigma], data]\), "\[IndentingNewLine]", RowBox[{ RowBox[{"n", "=", StyleBox["25", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"\[Mu]", "=", StyleBox["67", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"\[Sigma]", "=", StyleBox["5", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", \(Do[ data = Table[Random[NormalDistribution[\[Mu], \[Sigma]]], {n}]; x\&_ = Plus @@ data\/n; conf = N[{x\&_ - \(1.96\ \[Sigma]\)\/\@n, x\&_ + \(1.96\ \[Sigma]\)\/\@n}]; Print[x\&_, "\< \>", conf, "\< \>", conf\[LeftDoubleBracket]1\[RightDoubleBracket] \[LessEqual] \[Mu] \ \[LessEqual] conf\[LeftDoubleBracket]2\[RightDoubleBracket]], {100}]\)}], "Input",\ PageWidth->Infinity], Cell["\<\ It might be interesting to change the sample size n, the population mean \ \[Mu], and/or the population standard deviation \[Sigma], and run the \ experiment again. The most interesting thing to change might be the sample \ size. Does the sample size have a great effect on the number of \"Trues?\"\ \>", "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Normal population, standard deviation \"unknown,\" large sample CI\ \>", "Subsection"], Cell[TextData[{ "This time we will pretend not to know the population standard deviation. \ (Given collected data, this would be more realistic.) We will test the \ \"95%\" part of the large-sample approximate confidence interval. We will \ run the experiment 100 times. We want to see if it is the case that we get a \ confidence interval which contains the actual population mean in about 95 of \ the experiments. Each line of output will record one experiment. From left \ to right, the reported information is the sample mean, the sample standard \ deviation, the large-sample confidence interval, and the report of the test \ as to whether or not the actual mean is in the confidence interval. Check \ and see if there are about 95 \"Trues\" (and about 5 \"Falses\"). Keep in \ mind that our initial sample size (", StyleBox["25", FontColor->RGBColor[1, 0, 1]], ") is below the 30 that it should be for large-sample confidence \ intervals." }], "Text", Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[{\(Clear[n, \[Mu], \[Sigma], s, data]\), "\[IndentingNewLine]", RowBox[{ RowBox[{"n", "=", StyleBox["25", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"\[Mu]", "=", StyleBox["67", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"\[Sigma]", "=", StyleBox["5", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", \(Do[ data = Table[Random[NormalDistribution[\[Mu], \[Sigma]]], {n}]; x\&_ = Plus @@ data\/n; s = N[\@\(\(Plus @@ \(data\^2\) - x\&_\^2\ n\)\/\(n - 1\)\)]; conf = N[{x\&_ - \(1.96\ s\)\/\@n, x\&_ + \(1.96\ s\)\/\@n}]; Print[x\&_, "\< \>", s, "\< \>", conf, "\< \>", conf\[LeftDoubleBracket]1\[RightDoubleBracket] \[LessEqual] \[Mu] \ \[LessEqual] conf\[LeftDoubleBracket]2\[RightDoubleBracket]], {100}]\)}], "Input",\ PageWidth->Infinity], Cell[TextData[{ "Again, it might be interesting to change the sample size ", StyleBox["n", FontSlant->"Italic"], ", the population mean \[Mu], and/or the population standard deviation \ \[Sigma], and re-run the experiment. It would be particularly nice to see \ what happens as we adjust the sample size. Does this affect the number of \ \"Trues?\" (Recall that our rule of thumb says for cases where the \ population distribution is not normal or if the population standard deviation \ is unknown, ", StyleBox["n", FontSlant->"Italic"], " should be greater than 30. At least in this experiment we can be sure \ the population is normal.)" }], "Text", Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Normal population, standard deviation \"unknown,\" small sample CI\ \>", "Subsection"], Cell[TextData[{ "This time we will use a small sample approximate confidence interval \ instead of a large sample one. We will once again test the \"95%\" \ confidence level. We will run the experiment 100 times. We want to see if it \ is the case that we get a confidence interval which contains the actual \ population mean in about 95 of the experiments. Each line of output will \ record one experiment. From left to right, the reported information is the \ sample mean, the sample standard deviation, the large-sample confidence \ interval, and the report of the test as to whether or not the actual mean is \ in the confidence interval. Check and see if there are about 95 \"Trues\" \ (and about 5 \"Falses\"). If you change the value for ", StyleBox["n", FontColor->RGBColor[0, 1, 0]], ", you will also need to change the value for ", StyleBox["t\[Alpha]\[Nu]", FontColor->RGBColor[0, 1, 0]], ". The appropriate value can be found in tables for t distributions." }], "Text", Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[BoxData[{\(Clear[n, \[Mu], \[Sigma], s, data, \ t\[Alpha]\[Nu]]\), "\[IndentingNewLine]", RowBox[{ RowBox[{"n", "=", StyleBox["25", FontColor->RGBColor[0, 1, 0]]}], ";"}], "\n", RowBox[{ RowBox[{"t\[Alpha]\[Nu]", "=", StyleBox["2.064", FontColor->RGBColor[0, 1, 0]]}], ";"}], "\n", RowBox[{ RowBox[{"\[Mu]", "=", StyleBox["67", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", RowBox[{ RowBox[{"\[Sigma]", "=", StyleBox["5", FontColor->RGBColor[1, 0, 1]]}], ";"}], "\n", \(Do[ data = Table[Random[NormalDistribution[\[Mu], \[Sigma]]], {n}]; x\&_ = Plus @@ data\/n; s = N[\@\(\(Plus @@ \(data\^2\) - x\&_\^2\ n\)\/\(n - 1\)\)]; conf = N[{x\&_ - \(t\[Alpha]\[Nu]\ s\)\/\@n, x\&_ + \(t\[Alpha]\[Nu]\ s\)\/\@n}]; Print[x\&_, "\< \>", s, "\< \>", conf, "\< \>", conf\[LeftDoubleBracket]1\[RightDoubleBracket] \[LessEqual] \[Mu] \ \[LessEqual] conf\[LeftDoubleBracket]2\[RightDoubleBracket]], {100}]\)}], "Input",\ PageWidth->Infinity], Cell[TextData[{ "Again, it might be interesting to change the sample size ", StyleBox["n", FontSlant->"Italic"], ", the population mean \[Mu], and/or the population standard deviation \ \[Sigma], and re-run the experiment. It would be particularly nice to see \ what happens as we adjust the sample size. Does this affect the number of \ \"Trues?\" You also might want to compare results here with what happened in \ the preceeding section where we used the large sample CI." }], "Text", Evaluatable->False, FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontColor->GrayLevel[0], Background->GrayLevel[1], FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}] }, Closed]] }, Closed]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 720}}, AutoGeneratedPackage->None, WindowToolbars->"EditBar", WindowSize->{608, 676}, WindowMargins->{{2, Automatic}, {Automatic, 3}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False} ] (*********************************************************************** Cached data follows. 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