Download 1) The function f is defined by:

Page 1 of 8
Herzlia Highlands
Advanced Programme
Mathematics
(Stats & Prob)
Grade 12
Prelims 2016
Notes
1. Calculators in radian mode.
2. Calculators in RADIAN MODE.
3. There is 1 hour to complete 6 questions for 100 marks.
4. Unless otherwise stated, answers to be given to 2 decimal places.
5. You should have 3 pages of formulae plus a Normal Distribution table.
These are the IEB formulae sheets you will get at the end of the year, so
if the formula you are looking for is not here, either you should know
this formula, or you are looking for the wrong one!
Page 2 of 8
1. Based on previous statistics carried out at a certain nut factory in the Kommetjie
area, it was found that the standard deviation for the percentage peanuts placed in
a collection of mixed nuts is about 5%. When buying a bag of 100 of these mixed
nuts, it is found to contain 30 peanuts. Give a 97% confidence interval for the true
percentage of peanuts in a mixed bag of nuts.
[10]
2. The types and numbers of nuts in the bag of 100 nuts chosen in question 1 are as
follows:
30 Peanuts; 25 Almonds; 20 Cashews; 15 Brazil;10 Macadamia
If 20 nuts are randomly taken from this bag, determine the probability that:
2.1. There will be equal amounts of each type?
(4)
2.2. There will be at most 10% peanuts in the sample of 20?
(6)
2.3. There is 10% or less in the sample of 20 more than once if you count the
number of peanuts and then replace the sample of 20 and then repeat this
process 10 times?
(8)
[18]
3. The manufacturer claims that reducing the size of the aperture through which the
nuts are discharged into the bag will increase the percentage of peanuts from the
current setting of 30% and thereby increase their profit margin per bag. The
aperture is reduced and a sample of 50 nuts is taken. The percentage of peanuts in
this sample is found to be 33%. Test the validity of the claim at the 5%
significance level.
[12]
Page 3 of 8
4. The random variable X has the following probability mass function:
 2k

P( X  x)   x
0
4.1. Show that k 
x  1, 2, 3, 4
otherwise
12
.
50
(4)
4.2. Calculate the mean and standard deviation of X.
(15)
4.3. Determine P( X  2) .
(3)
4.4. Determine P( X  3 | X  2) .
(5)
[27]
5. A herd of elephants you are monitoring has 12 individuals in it and you are
interested in how the herd size may change over the coming year. A good
breeding year will occur with a probability of 0,7 and will result in 5 baby
elephants being born, whereas a bad breeding year will occur with probability
0,3 and will result in only 2 baby elephants being born.
There are another two smaller herds of size 4 and 6 respectively. The probability
of no merge is 0,5, of merging with the smallest herd is 0,3 and of merging with
the herd of size 6 is 0,2. Assume that the breeding success has nothing to do with
the merging of the herds.
5.1. Let X be the random variable giving the size of the herd at the end of the
year. Give the probability mass function for X.
(15)
5.2. Given that the herd size is bigger than 15, what is the probability that it is at
most 20?
(5)
[20]
Page 4 of 8
6. With the aim of scaring some rowdy grade 11’s recently, I came up with a
proposed probability density function (pdf) for the depth of the hole they would
make in the ground if I was to throw them out of the classroom window. They
were told that if they did not answer the questions relating to the pdf correctly,
they would be used to test the accuracy of said pdf!
The following is the pdf for the depth of the hole (in feet):
c( x 2  x  2)
0 x2
f ( x)  
elsewhere
0
6.1. Determine the value of c.
(5)
6.2. Determine P(0,5  X  1)
(5)
6.3. Determine the mean, E ( X ) .
(6)
6.4. Find the median of X. (you might need a smidgen of Newton)
(8)
[24]
END OF PAPER!!!
{11 bonus marks}
Page 5 of 8
INFORMATION SHEET
General Formulae
n
n
i 
1 n
i 1
n
i 2 
i 1
x0
 x if
x 
 x if
– b ± b 2 – 4ac
x=
2a
i 1
x0
n(n  1) n 2 n


2
2 2
nn  12n  1 n 3 n 2 n



6
3
2 6
n 2 n  1
n 4 n3 n 2
i 




4
4
2
4
i 1
2
n
3
z  a  bi
z*  a  bi
n A  n B  n  AB
n An  n n A
 A
n A  n B  n  
B
log b x
log a x 
log b a
Calculus
ba n
Area  lim 
  f  xi 
n  n 
i 1
f '( x)  lim
h 0
f ( x  h) – f ( x )
h
 f ' g ( x).g ' ( x) dx 
b
 x n1 
x
dx




 n  1 a
a
b
n
dy dy dt


dx dt dx
f ( g ( x))  c
 f ( x).g ' ( x)dx  f ( x).g ( x)   g ( x). f ' ( x) dx  c
xr 1  xr 
f ( xr )
f ' ( xr )
b
V    y 2 dx
a
Page 6 of 8
Function
xn
sin x
cos x
tan x
cot x
sec x
cosec x
Derivative
nx n 1
cos x
 sin x
sec 2 x
 cosec 2 x
sec x. tan x
 cosec x. cot x
1
arcsin x
1 x2
1
1 x2
arctan x
f ( g ( x))
f ( x). g ( x)
f ( x)
g ( x)
f ' ( g ( x)). g ' ( x)
g ( x). f ' ( x)  f ( x). g ' ( x)
g ( x). f ' ( x)  f ( x).g ' ( x)
g ( x)2
Trigonometry
A
1 2
r
2
s  r
a
b
c
=
=
sin A sin B sin C
In ABC:
a 2  b 2  c 2 – 2bc. cos A
Area 
sin 2 A  cos 2 A  1
1
ab.sin C
2
1  tan 2 A  sec 2 A
sin  A  B  sin A. cos B  cos A sin B
sin 2 A  2 sin A cos A
sin A. cos B 
1
sin( A  B)  sin( A  B)
2
sin A. sin B 
1
cos( A  B)  cos( A  B)
2
cos A. cos B 
1
cos( A  B)  cos( A  B)
2
1  cot 2 A  cosec 2 A
cos A  B  cos A cos B  sin A sin B
cos 2 A  cos 2 A  sin 2 A
Page 7 of 8
Statistics
P( A) =
n( A)
n( s )
P  B | A 
P  B  A
P  A
P( A or B) = P( A) + P( B) – P( A and B)
n
Pr 
n!
 n  r !
n
n
n!
Cr    
 r   n  r !r !
 p  N  p 
 

r  n  r 

PR  r 
N
 
n
Z
2
n(  x )  (  x )
2

z
( x  y )   X  Y 
 x2
nx
n ( xy)   x  y
X 
x

n
Z
b
n
n x
P  X  x     p x 1  p 
 x
b

 y2
ny
 xy  nxy
2
2
 x  n( x )
b
 ( x  x )( y  y )
2
 (x  x )
Page 8 of 8
NORMAL DISTRIBUTION TABLE
Areas under the Normal Curve
1 z e  ½ x 2 dx
2  0
H(-z) = H(z), H() = ½
H(z) =
Entries in the table are values of H(z) for z  0.
z
0
0,1
0,2
0,3
0,4
,00
,0000
,0398
,0793
,1179
,1554
,01
,0040
,0438
,0832
,1217
,1591
,02
,0080
,0478
,0871
,1255
,1628
,03
,0120
,0517
,0910
,1293
,1664
,04
,0160
,0557
,0948
,1331
,1700
,05
,0199
,0596
,0987
,1368
,1736
,06
,0239
,0636
,1026
,1406
,1772
,07
,0279
,0675
,1064
,1443
,1808
,08
,0319
,0714
,1103
,1480
,1844
,09
,0359
,0753
,1141
,1517
,1879
0,5
0,6
0,7
0,8
0,9
,1915
,2257
,2580
,2881
,3159
,1950
,2291
,2611
,2910
,3186
,1985
,2324
,2642
,2939
,3212
,2019
,2357
,2673
,2967
,3238
,2054
,2389
,2704
,2995
,3264
,2088
,2422
,2734
,3023
,3289
,2123
,2454
,2764
,3051
,3315
,2157
,2486
,2794
,3078
,3340
,2190
,2517
,2823
,3106
,3365
,2224
,2549
,2852
,3133
,3389
1,0
1,1
1,2
1,3
1,4
,3413
,3643
,3849
,4032
,4192
,3438
,3665
,3869
,4049
,4207
,3461
,3686
,3888
,4066
,4222
,3485
,3708
,3907
,4082
,4236
,3508
,3729
,3925
,4099
,4251
,3531
,3749
,3944
,4115
,4265
,3554
,3770
,3962
,4131
,4279
,3577
,3790
,3980
,4147
,4292
,3599
,3810
,3997
,4162
,4306
,3621
,3830
,4015
,4177
,4319
1,5
1,6
1,7
1,8
1,9
,4332
,4452
,4554
,4641
,4713
,4345
,4463
,4564
,4649
,4719
,4357
,4474
,4573
,4656
,4726
,4370
,4484
,4582
,4664
,4732
,4382
,4495
,4591
,4671
,4738
,4394
,4505
,4599
,4678
,4744
,4406
,4515
,4608
,4686
,4750
,4418
,4525
,4616
,4693
,4756
,4429
,4535
,4625
,4699
,4761
,4441
,4545
,4633
,4706
,4767
2,0
2,1
2,2
2,3
2,4
,4772
,4821
,4861
,48928
,49180
,4778
,4826
,4864
,48956
,49202
,4783
,4830
,4868
,48983
,49224
,4788
,4834
,4871
,49010
,49245
,4793
,4838
,4875
,49036
,49266
,4798
,4842
,4878
,49061
,49286
,4803
,4846
,4881
,49086
,49305
,4808
,4850
,4884
,49111
,49324
,4812
,4854
,4887
,49134
,49343
,4817
,4857
,4890
,49158
,49361
2,5
2,6
2,7
2,8
2,9
,49379
,49534
,49653
,49744
,49813
,49396
,49547
,49664
,49752
,49819
,49413
,49560
,49674
,49760
,49825
,49430
,49573
,49683
,49767
,49831
,49446
,49585
,49693
,49774
,49836
,49461
,49598
,49702
,49781
,49841
,49477
,49609
,49711
,49788
,49846
,49492
,49621
,49720
,49795
,49851
,49506
,49632
,49728
,49801
,49856
,49520
,49643
,49736
,49807
,49861
3,0
3,1
3,2
3,3
3,4
,49865
,49903
,49931
,49952
,49966
,49869
,49906
,49934
,49953
,49968
,49874
,49910
,49936
,49955
,49969
,49878
,49913
,49938
,49957
,49970
,49882
,49916
,49940
,49958
,49971
,49886
,49918
,49942
,49960
,49972
,49889
,49921
,49944
,49961
,49973
,49893
,49924
,49946
,49962
,49974
,49896
,49926
,49948
,49964
,49975
,49900
,49929
,49950
,49965
,49976
3,5
3,6
3,7
3,8
3,9
,49977
,49984
,49989
,49993
,49995
4,0
,49997