pyMaxima-Sitzung

Thursday September 27, 2007 06:22 AM


Lösung zu S. 106/ Nr. 4

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"]])$

Funktionen-Plot
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!"$

Funktionen-Plot

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!"$