maths/hw/wpp2/wpp2.ms


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.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