# Section 3; Question 1: # c( pnorm( 1.33 ) - pnorm( -0.44 ), pnorm( 0.94 ), 1 - pnorm( -1.48 ), pnorm( -4.32 ) ) # Section 3; Question 2: # c( pnorm( 2.07 ) - pnorm( 0 ), pnorm( -0.11 ) - pnorm( -0.64 ), 1 - pnorm( -1.06 ), pnorm( -2.33 ), 1 - pnorm( 4.61 ) ) # Section 3; Question 4: # c( sqrt(2*pi) * ( pnorm( 1.24 ) - pnorm( 0 ) ), ( 6/sqrt(2*pi) ) ) # Section 3; Question 5: # qnorm( 0.33 ) qnorm( 1-0.2236 ) qnorm( 0.5004 + pnorm(-1) ) qnorm( ( 1 - 0.8 )/2 ) qnorm( pnorm( 2.03 ) - 0.15 ) # Section 3; Question 6: # qnorm( 1 - 0.25 ) - qnorm( 1 - 0.75 ) # Section 3; Question 7: # n = 74806 p = 1/12 t = 306000/50 - 1 + 0.5 pnorm( ( t - n*p )/sqrt( n*p*(1-p) ) ) # Section 3; Question 8: # n = 1600 p = 0.8 m = n*p s = sqrt( n*p*(1-p) ) pnorm( ( 1310.5 - m )/s ) - pnorm( ( 1259.5 - m )/s ) # Section 3; Question 9: # n = 400 p = 0.55 dbinom( 200, n, p ) m = n*p s = sqrt( n*p*(1-p) ) #pnorm( ( 200.5 - m )/s ) - pnorm( ( 199.5 - m )/s ) # normal approximation pnorm( ( 199.5 - m )/s ) # Section 3; Question 10: # n = 100 p = 0.7 m = n*p s = sqrt( n*p*(1-p) ) pnorm( ( 80.5 - m )/s ) - pnorm( ( 74.5 - m )/s ) # Section 3; Question 11: # n = 747 p = 3/12 m = n*p s = sqrt( n*p*(1-p) ) 1 - pnorm( (343.5-m)/s ) # Section 3; Question 12: # n = 15*100 p = 1/5 m = n*p s = sqrt( n*p*(1-p) ) 1 - pnorm( (325.5-m)/s ) # Section 3; Question 13: # n = 10 p = 0.7 print( c( 9*p/(1-p), 9*(1-p)/p ) ) exact = sum( dbinom( 4:8, n, p ) ) m = n*p s = sqrt( n*p*(1-p) ) approx = pnorm( (8.5 - m)/s ) - pnorm( ( 3.5 - m)/s ) print( sprintf('exact= %.4f; approx= %.4f', exact, approx ) ) # Section 3; Question 14: n = 42200 p = 0.38 m = n*p s = sqrt( n*p*(1-p) ) m + s * qnorm(1-0.2) # Section 3; Question 15: n = 200 p = 1/2 m = n*p s = sqrt( n*p*(1-p) ) print( sprintf("central limit prob= %f", pnorm( 5/s ) - pnorm( -5/s )) ) # Section 3; Question 16: # ex = sum( 1:6 )/6 ex2 = sum( (1:6)^2 )/6 varx = ex2 - ex^2 n = 100 z = ( 369.5 - n*ex )/sqrt( n*varx ) print( z ) print( pnorm( z ) ) # Section 3; Question 17: # ex = 5 * ( 18 / 38 ) - 5 * ( 20 / 38 ) ex2 = 5^2 * ( 18 / 38 ) + 5^2 * ( 20 / 38 ) varx = ex2 - ex^2 n = 100 z = ( -50 - n*ex )/sqrt( n*varx ) print( z ) print( 1 - pnorm( z ) ) # Section 3; Question 19: # lambda = 50 m = lambda s = sqrt( lambda ) p = 1 - ( pnorm( (60.5-m)/s ) - pnorm( (-0.5-m)/s ) ) print( sprintf( "central limit prob= %f", p ) ) # Section 3; Question 20: # lambda = 3 exact = 1 - sum( dpois( 0:7, lambda ) ) m = lambda s = sqrt( lambda ) p = 1 - ( pnorm( (7.5-m)/s ) - pnorm( (-0.5-m)/s ) ) print( sprintf( "exact prob= %f central limit prob= %f", exact, p ) ) # Section 3; Question 21: # m = 30000 s = 5000 print( 1 - pnorm( (25000-m)/s ) ) # Section 3; Question 22: m = 100 s = 16 print( 1 - ( pnorm( (135-m)/s ) - pnorm( (80-m)/s ) ) ) # Section 3; Question 23: m = 20000 s = 5000 print( 1 - pnorm( (30000-m)/s ) ) # Section 3; Question 24: # m = 266 s = 16 print( 1 - pnorm( (305-m)/s ) ) # Section 3; Question 25: # p_1 = 1 - ( pnorm( (75-60)/10 ) - pnorm( (0-60)/10 ) ) p_2 = pnorm( (75-80)/5 ) - pnorm( (0-80)/5 ) print( sprintf("p(D|D^c) = %f; p(D^c|D)= %f", p_1, p_2) ) # Section 3; Question 26: # m = 12.5 s = 0.2 p = 1 - ( pnorm( (13-m)/s ) - pnorm( (12-m)/s ) ) print( c( p, 1000*p ) ) # Section 3; Question 27: # m = 565 s = 75 p = 1 - pnorm( (660-m)/s ) print( c( p, 4250*p ) ) # Section 3; Question 28: # t_AB = qnorm( 1 - 0.2 ) t_CB = qnorm( 1 - 0.2 - 0.26 ) t_DC = qnorm( 1 - 0.2 - 0.26 - 0.36 ) t_FD = qnorm( 1 - 0.2 - 0.26 - 0.36 - 0.12 ) z_breaks = c( t_FD, t_DC, t_CB, t_AB ) print( z_breaks ) print( 12 * z_breaks + 70 ) # Section 3; Question 29: # t = -qnorm( 0.25 ) # this is 20/sigma sigma = 20/t print( sigma ) # Section 3; Question 30: # m = ( 103.5 + 144.5 )/2 print( 144.5 - m ) minus_t = qnorm( ( 1-0.8 )/2 ) print( -20.5 / minus_t ) # Section 3; Question 31: # print( pnorm( (0.08 - 0.09)/0.004 ) ) # Section 3; Question 32: # # men (Michael), women (Laura) print( c( (75-62)/7.6, (92-76.3)/10.8 ) ) # Section 3; Question 33: # n = 9 m = 100 s = 16 print( 1 - pnorm( (103-100)/(s/sqrt(n)) ) ) p = 1 - pnorm( (103-100)/s ) print( p ) print( dbinom( 3, n, p ) ) # Section 3; Question 34: # t = -qnorm( ( 1 - 0.99 )/2 ) print( (2*t/0.1)^2 ) # Section 3; Question 35: m = 3*6 s = sqrt( 3 * 0.3^2 ) print( 1 - pnorm( ( 19-m)/s ) ) t = qnorm( 1-0.005 ) print( 1/(sqrt(3)*t) ) # Section 3; Question 36: m = 40.5 - 41.5 s = sqrt( 0.3^2 + 0.4^2 ) print( 1 - pnorm( (0-m)/s ) ) # Section 3; Question 38: # m = 2 - 1 s = sqrt( 2^2/9 + 1^2/9 ) print( 1 - pnorm( (0-m)/s ) )