Teil a (2)
(%i1) "Parameter der Binomialverteilung"$
(%i2) bin_n : 5$
(%i3) bin_p : 0.166666667$
(%i4) bin_q : 1-bin_p;
(%o4) 0.833333333
(%i5) "Erwartungswert ew"$
(%i6) bin_ew : bin_n * bin_p;
(%o6) 0.833333335
(%i7) "Standardabweichung sigma"$
(%i8) bin_sigma : ev(sqrt(bin_n * bin_p * bin_q),numer);
(%o8) 0.833333334
(%i9) sumBin(n,links,rechts,p) := block( sum_ : 0, q_ : 1 - p, if (links < 0) then (links : 0), if (rechts > n) then (rechts : n), for j_: links thru rechts do ( sum_ : sum_ + p^j_ * q_^(n-j_) * binomial(n,j_) ), print("P(",links, "<= X <=", rechts,") = ",sum_) );
(%o9) sumBin(n, links, rechts, p) := block(sum_ : 0, q_ : 1 - p,
if links < 0 then links : 0, if rechts > n then rechts : n,
j_ n - j_
for j_ from links thru rechts do sum_ : sum_ + p q_ binomial(n, j_),
print("P(", links, "<= X <=", rechts, ") = ", sum_))
(%i10) "P(ew-sigma <= X <= ew+sigma) (1-Sigma-Umgebung)"$
(%i11) sumBin(bin_n,ceiling(bin_ew-bin_sigma), floor(bin_ew+bin_sigma), bin_p)$
P( 1 <= X <= 1 ) = 0.40187757217721
(%i12) "P(ew-2*sigma <= X <= ew+2*sigma) (2-Sigma-Umgebung)"$
(%i13) sumBin(bin_n,ceiling(bin_ew-2*bin_sigma), floor(bin_ew+2*bin_sigma), bin_p)$
P( 0 <= X <= 2 ) = 0.96450617264661
(%i14) "P(ew-3*sigma <= X <= ew+3*sigma) (3-Sigma-Umgebung)"$
(%i15) sumBin(bin_n,ceiling(bin_ew-3*bin_sigma), floor(bin_ew+3*bin_sigma), bin_p)$
P( 0 <= X <= 3 ) = 0.9966563785751
(%i16) xy:makelist([x,binomial(bin_n, x)*bin_p^x*bin_q^(bin_n-x)],x,0,bin_n);
(%o16) [[0, 0.40187757121271], [1, 0.40187757217721], [2, 0.16075102925669],
[3, 0.032150205928498], [4, 0.0032150206005658], [5, 1.2860082433127569E-4]]
(%i17) plot2d([discrete,xy],[gnuplot_curve_titles,"title 'Binomialverteilung (n=5, p=0.166666667)'"],[gnuplot_curve_styles,["with boxes fill solid 0.3"]])$
Teil a (3)
(%i18) "P(k1 <= X <= k2) = P(1 <= X <= 5 )"$
(%i19) sumBin(5,1,5,0.166666667)$
P( 1 <= X <= 5 ) = 0.59812242878729
(%i20) "P(k1 <= X <= k2) = P(0 <= X <= 0 )"$
(%i21) sumBin(5,0,0,0.166666667)$
P( 0 <= X <= 0 ) = 0.40187757121271
(%i22) 0.40187757121271 + 0.59812242878729;
(%o22) 1.0
Teil b (1) - (3)
(%i23) "P(k1 <= X <= k2) = P(0 <= X <= 21 )"$
(%i24) sumBin(100,0,21,0.166666667)$
P( 0 <= X <= 21 ) = 0.89981652772556
(%i25) "P(X = k) = P(X = 25 )"$
(%i26) sumBin(100,25,25,0.166666667)$
P( 25 <= X <= 25 ) = 0.0098258820675571
(%i27) "P(k1 <= X <= k2) = P(21 <= X <= 100 )"$
(%i28) sumBin(100,21,100,0.166666667)$
P( 21 <= X <= 100 ) = 0.15188785007575
(%i29) "P(k1 <= X <= k2) = P(0 <= X <= 20 )"$
(%i30) sumBin(100,0,20,0.166666667)$
P( 0 <= X <= 20 ) = 0.84811214992425
(%i31) 0.15188785007575 + 0.84811214992425;
(%o31) 1.0
Teil d
(%i32) "Parameter der Binomialverteilung"$
(%i33) bin_n : 300$
(%i34) bin_p : 0.1666666667$
(%i35) bin_q : 1-bin_p;
(%o35) 0.8333333333
(%i36) "Erwartungswert ew"$
(%i37) bin_ew : bin_n * bin_p;
(%o37) 50.00000001
(%i38) "Standardabweichung sigma"$
(%i39) bin_sigma : ev(sqrt(bin_n * bin_p * bin_q),numer);
(%o39) 6.454972244195426
(%i40) "P(ew-sigma <= X <= ew+sigma) (1-Sigma-Umgebung)"$
(%i41) sumBin(bin_n,ceiling(bin_ew-bin_sigma), floor(bin_ew+bin_sigma), bin_p)$
P( 44 <= X <= 56 ) = 0.68627306915098
(%i42) "P(ew-2*sigma <= X <= ew+2*sigma) (2-Sigma-Umgebung)"$
(%i43) sumBin(bin_n,ceiling(bin_ew-2*bin_sigma), floor(bin_ew+2*bin_sigma), bin_p)$
P( 38 <= X <= 62 ) = 0.94769997967356
(%i44) "P(ew-3*sigma <= X <= ew+3*sigma) (3-Sigma-Umgebung)"$
(%i45) sumBin(bin_n,ceiling(bin_ew-3*bin_sigma), floor(bin_ew+3*bin_sigma), bin_p)$
P( 31 <= X <= 69 ) = 0.99745127445438
(%i46) "Aufgabenstellung: k = 70 mal die 6 => außerhalb der 3-Sigma-Umgebung!"$
Teil e (1)/(3)
(%i47) "rechtsseitiger Hypothesentest (Binomial verteilte Zufallsvaribale X)"$
(%i48) "Stichprobenumfang n = 100"$
(%i49) "Irrtumswahrscheinlichkeit alpha = 0.05"$
(%i50) "Nullhypthese H0: p <= 0.1"$
(%i51) "Prüfvariable X ist B(100;0.10)-verteilt"$
(%i52) "Konstruktion des Ablehnungsbereiches K = {gr;...;100}"$
(%i53) "K = {16;...;100}"$
(%i54) "=> Annahmebereich Kquer = {0;...;15}"$
(%i55) "Der Würfel zeigt 20-mal die Sechs => Nullhypothese H0 verwerfen!"$