Lösungen / Statistik 2/09

Remove["Global`*"]

1.

Idee/ Idée:

α = 0.001;  
    n = 100;   μ0 = 200;  σ = 4;

fR[α_] := 
        FindRoot[
          1/(Sqrt[2 Pi]) Integrate[
                E^(-x^2/2), {x, -c, c}] == 
            1 - α, {c, 1}];

c1[c_] := c /. fR[α];  c1[c]

g1[
          c_] := (-c1[c] == 
            Sqrt[n] (cU - μ0)/σ);
g2[
          c_] := (c1[c] == 
            Sqrt[n] (cO - μ0)/σ);

solv1 = Solve[g1[c], {cU}] //  Flatten;
cU1[n_, μ_, σ_] := 
        cU /. solv1;
cU1[n, μ, σ]

solv2 = Solve[g2[c], {cO}] //  Flatten;
cO1[n_, μ_, σ_] := 
        cO /. solv2;
cO1[n, μ, σ]

Programm

α = 0.001;  n = 10;   μ0 = 200;  σ = 4; 
    fR[α_] := 
      FindRoot[1/(Sqrt[2 Pi]) Integrate[
              E^(-x^2/2), {x, -c, c}] == 1 - α, {c, 
          1}];
c1[c_] := c /. fR[α];
g1[
          c_] := (-c1[c] == 
            Sqrt[n] (cU - μ0)/σ);
g2[
          c_] := (c1[c] == 
            Sqrt[n] (cO - μ0)/σ);
solv1 = Solve[g1[c], {cU}] //  Flatten;
cU1[n_, μ_, σ_] := 
        cU /. solv1;
solv2 = Solve[g2[c], {cO}] //  Flatten;
cO1[n_, μ_, σ_] := 
        cO /. solv2;
{cU1[n, μ, σ], cO1[n, μ, σ]}

α = 0.001;  n = 100;   μ0 = 200;  σ = 4; 
    fR[α_] := 
      FindRoot[1/(Sqrt[2 Pi]) Integrate[
              E^(-x^2/2), {x, -c, c}] == 1 - α, {c, 
          1}];
c1[c_] := c /. fR[α];
g1[
          c_] := (-c1[c] == 
            Sqrt[n] (cU - μ0)/σ);
g2[
          c_] := (c1[c] == 
            Sqrt[n] (cO - μ0)/σ);
solv1 = Solve[g1[c], {cU}] //  Flatten;
cU1[n_, μ_, σ_] := 
        cU /. solv1;
solv2 = Solve[g2[c], {cO}] //  Flatten;
cO1[n_, μ_, σ_] := 
        cO /. solv2;
{cU1[n, μ, σ], cO1[n, μ, σ]}

α = 0.001;  n = 1000;   μ0 = 200;  σ = 4; 
    fR[α_] := 
      FindRoot[1/(Sqrt[2 Pi]) Integrate[
              E^(-x^2/2), {x, -c, c}] == 1 - α, {c, 
          1}];
c1[c_] := c /. fR[α];
g1[
          c_] := (-c1[c] == 
            Sqrt[n] (cU - μ0)/σ);
g2[
          c_] := (c1[c] == 
            Sqrt[n] (cO - μ0)/σ);
solv1 = Solve[g1[c], {cU}] //  Flatten;
cU1[n_, μ_, σ_] := 
        cU /. solv1;
solv2 = Solve[g2[c], {cO}] //  Flatten;
cO1[n_, μ_, σ_] := 
        cO /. solv2;
{cU1[n, μ, σ], cO1[n, μ, σ]}

α = 0.001;  n = 100;   μ0 = 200;  σ = 0.1; 
    fR[α_] := 
      FindRoot[1/(Sqrt[2 Pi]) Integrate[
              E^(-x^2/2), {x, -c, c}] == 1 - α, {c, 
          1}];
c1[c_] := c /. fR[α];
g1[
          c_] := (-c1[c] == 
            Sqrt[n] (cU - μ0)/σ);
g2[
          c_] := (c1[c] == 
            Sqrt[n] (cO - μ0)/σ);
solv1 = Solve[g1[c], {cU}] //  Flatten;
cU1[n_, μ_, σ_] := 
        cU /. solv1;
solv2 = Solve[g2[c], {cO}] //  Flatten;
cO1[n_, μ_, σ_] := 
        cO /. solv2;
{cU1[n, μ, σ], cO1[n, μ, σ]}

2.

β = 0.01;  α = β;   n = 100;   μ0 = 
      200;  σ = 4; 
    fR[α_] := 
      FindRoot[1/(Sqrt[2 Pi]) Integrate[
              E^(-x^2/2), {x, -c, c}] == 1 - α, {c, 
          1}];
c1[c_] := c /. fR[α];
g1[
          c_] := (-c1[c] == 
            Sqrt[n] (cU - μ0)/σ);
g2[
          c_] := (c1[c] == 
            Sqrt[n] (cO - μ0)/σ);
solv1 = Solve[g1[c], {cU}] //  Flatten;
cU1[n_, μ_, σ_] := 
        cU /. solv1;
solv2 = Solve[g2[c], {cO}] //  Flatten;
cO1[n_, μ_, σ_] := 
        cO /. solv2;
{cU1[n, μ, σ], cO1[n, μ, σ]}

α = 0.001;  n = 100;   μ0 = 200;  σ = 4; 
    fR[α_] := 
      FindRoot[1/(Sqrt[2 Pi]) Integrate[
              E^(-x^2/2), {x, -c, c}] == 1 - α, {c, 
          1}];
c1[c_] := c /. fR[α];
g1[
          c_] := (-c1[c] == 
            Sqrt[n] (cU - μ0)/σ);
g2[
          c_] := (c1[c] == 
            Sqrt[n] (cO - μ0)/σ);
solv1 = Solve[g1[c], {cU}] //  Flatten;
cU1[n_, μ_, σ_] := 
        cU /. solv1;
solv2 = Solve[g2[c], {cO}] //  Flatten;
cO1[n_, μ_, σ_] := 
        cO /. solv2;
{cU1[n, μ, σ], cO1[n, μ, σ]}

3.

Selbststudium

4.

Remove["Global`*"]

a

Sqrt[8000] // N

90 Einsätzt  zufällig herausgreifen erfüllt die gegebene Bedingung.

b

<< Statistics`DescriptiveStatistics`

data={{1, 7.6},{2, 7.7},{6, 7.8},{6 ,7.9},{10 ,8.0},{13, 8.1},{14, 8.2},{13, 8.3},{10, 8.4},{7, 8.5},{5 ,8.6},{1, 8.7},{2 ,8.8}}

data[[5]][[2]]

Table[data[[5]][[2]], {k, 
        1, data[[5]][[1]]}]

tab = 
      Table[Table[data[[n]][[2]], {k, 
              1, data[[n]][[1]]}], {n, 1, Length[data]}] // 
        Flatten

sum = Apply[Plus, tab]

len = Length[tab]

locRep = LocationReport[tab]

dispRep = DispersionReport[tab]

c

Δx = 1/len len 0.1

d

μ = Mean /. locRep

μMittelw = Mean /. locRep

e

σ = StandardDeviation /. dispRep

σMittelw = StandardDeviation/Sqrt[len] /. dispRep

f

dataTrans = 
      Table[{data[[k]][[2]], data[[k]][[1]]}, {k, 1, 
          Length[data]}]

p1 = 
        ListPlot[dataTrans, PlotStyle -> PointSize[0.02]];

f[x_] := 
        1/Sqrt[2 Pi σ^2] E^(-1/
                  2 ((x - μ)/σ)^2);
p2 = Plot[f[x], {x, 7.4, 8.9}];

streck = 9;
p2 = Plot[streck f[x], {x, 7.4, 8.9}];

Show[p2, p1];

Sieht optisch anständig aus.

g

μMittelw

σ95 =  2  σMittelw

untereGrenze95  =  μMittelw  - σ95

obereGrenze95  =  μMittelw  + σ95

h

Durch 3 σMittelw ist bei einer Normalverteilung ein 99.73-% Intervall gegeben.

μMittelw

σ9973 =  3  σMittelw

untereGrenze9973  =  μMittelw  - σ9973

obereGrenze9973  =  μMittelw  + σ9973