.TL Weekly past paper 2 (stats) .AU Lucas Standen .AI QMC .AB .CD .I I thought for all our sanities, we might appreciate me trying to type this :) .AE .PP 1)a) .EQ 22 - 16 over sigma = 1.28155 .EN .EQ sigma = 22 - 16 over 1.28155 .EN .EQ sigma = 4.68 .EN .PP 1)b) .EQ P(L < 16) = P(Z < 13-16 over 4.68 ) .EN .EQ P(L < 16) = 26.1% .EN .PP 1)c) .EQ P(S < 8) = 0.1 .EN .EQ {8 - mu} over sigma = -1.2816 .EN .EQ {17 - mu} over sigma = 0.8416 .EN .EQ sigma = 4.238 .EN .EQ mu = 13.432 .EN .PP 1)d) .EQ 13.4 > 16 .EN .EQ "this supports thier belief" .EN .PP 2)a) .EQ W = " The number of cards out of the 20 winning ones" .EN .EQ W " ~ " B(20, 0.45) .EN .EQ S = " The number of stores with 12 or more winning cards" .EN .EQ S " ~ " B(8, p) .EN .EQ p = P(w >= 12) = 0.1308 .EN .EQ P(S >= 2) = 0.2818 .EN .PP 2)b) .EQ "There is a large number of trials, so the rate should be close 0.5" .EN .PP 2)c) .EQ X " ~ " N(135, 74.25) .EN .EQ P(X < 122.5) .EN .EQ P(Z < {122.5 - 135} over sqrt 74.25 ) = 0.0734 .EN .PP 2)d) .EQ "The probability is higher than 2.5% therefore there is not enough evidence to suggest that the proportion is not 45%" .EN .PP 3)a)i) .EQ X " ~ " N(30, 2 sup 2 ) .EN .EQ P(X = 31) = 0 .EN .PP 3)a)ii) .EQ P(X > 31) = 0.3085 .EN .PP 3)b)i) .EQ 0.0668 times (1 - 0.0668) sup 4 = 0.0507 .EN .PP 3)b)ii) .EQ P(Y > 1) = 1 - P(Y <= 1) .EN .EQ 1 - 0.9610 = 0.039 .EN .PP 3)c) .EQ H " ~ " N( mu , 1.5 sup 2 ) .EN .EQ P(H > 42) = 0.0005 .EN .EQ z = {42 - mu} over 1.5 = 3.29 .EN .EQ mu = 37.06 .EN .PP 4)a) .EQ "mean "= 1680 over 60 = 28 " mins " .EN .EQ S sub xx = 47654.4 - 1680 sup 2 over 60 = 614.4 .EN .EQ sigma = sqrt {S sub xx over 60} = 3.2 .EN .PP 4)b) .EQ H sub 0 : mu = 27.5 .EN .EQ H sub 1 : mu > 27.5 .EN .EQ z = 28 - 27.5 over {3 over sqrt 60} = 1.2909 .EN .EQ z = 1.64485 .EN .EQ "no sufficent evidence to support her belief" .EN .PP 4)c)i) .EQ "assuming that you will always have a p of 0.2 is not likely" .EN .PP 4)c)ii) .EQ "moddel the fastest and slowest people as 2 distributions" .EN