1

SMK Sacred Heart, Sibu, Sarawak.

950/2 STPM Mathematics M ( Management Maths ) Paper 2

1(a) Axis : [ M 1 ]

Histogram : [ GA 2 ]

1(b) Top five percentage P95 = x95% = x76 → P95 class is 79.5 – 99.5

95

9580 72

10079.5 99.5 79.5

8P

= 89.5 :. The minimum mark is 89.5 [ B1M1A1]

1(c) mean , 4580

57.2580

x , standard deviation, 4580

57.2580

x ,

7

mod 49.5 10 55.867 4

e

,

Pearson coefficient of skewness = mod 57.25 55.86

0.075318.45

mean e

[ B1A1 ]

:. The data is almost symmetry abaou the mean. [ B1 ]

2. P ( delegates that contain at least one boy and at least one girl )

= 1 – P ( all boys ) – P ( all girls )

=

4 5

4 4

9 9

4 4

1C C

C C =

1 5 201

126 126 21 [ B1M2 A1 ]

2

3.(a) 3

3

3 1k x dx

32

3

3 12

xk x

9 9

9 9 12 2

k

[ M1 ]

18 k = 1

1

18k [ A1 ]

3(b) E[X] = 33 3 2

3 3

1 1 33

18 18 3 2

x xx x dx

=

1 27 279 9 1

18 2 2

[ M1 ]

E[X2 ]=

33 4 32

3 3

1 1 33

18 18 4 3

x xx x dx

=

1 81 8127 27 3

18 4 4

[ M1 ]

Var[X] = E[X2] – (E[X] )

2

= 3 - 12 [ M1 ]

= 2 [ A1 ]

3(c)

3

1( )

4

1 13

18 4

q

p X q

x dx

[ M1 ]

2 9 13 9

2 2 4

qq

22 12 17 0q q [ M1 ]

q = - 3.7 or q = - 2.29

( No solution - 3 < q < 3 ) :. q = - 2.29 [ A1 ]

4. 2 24926, 2218180, 410, 17964, 194008, 12x x y y xy n

Coefficient of determination,

2

2

2 2

12 194008 4926 410

12 2218180 4926 12 17964 410r

[ B1

M1]

= 0.85185 [ M1 ]

r = 0.9230 [ A1 ]

:. There is a very strong positive correlation between x and y . [ B1 ]

3

5(a) Laspeyres price index,

5 5 10 3 14 2 13 2 12 2100

4 5 8 3 10 2 12 2 11 2LP

[ M1 ]

= 120.9 [ A1 ]

:. The housewife expenditure in January 2013 has increased 20.9 % cpompared with Jan 2012.

[ B1 ]

5(b) Paasche quantity index,

6 5 2 10 1 14 2 13 1 12100

5 5 3 10 2 14 2 13 2 12PQ

[ M1 ]

= 76.69 [ A1 ]

:. Due to reduction in comsumption of fish in 2013, the housewife expenditure decreased by

23.31 %. [ B1 ]

6(a) Axis : [ M1 ]

Points & Graph: [ GA2 ]

The data is increasing linearly

showing an upward trend and has

maximum value at third quarter and

minimum value at first quarter.

[ B1 ]

6(b) (c ) Year Quarter Sales, y 4 points mov. Ave. Centred 4 points m.a. Y Seasonal variation, y -Y

2010

1 23 - - -

2 32 - - -

3 68 39 39.50 28.5

4 33 40 41.00 -8

2011

1 27 42 42.625 -15.625

2 40 43.25 43.75 3.75

3 73 44.25 44.625 28.375

4 37 45 45.375 -8.375

2012

1 30 45.75 46.25 -16.25

2 43 46.75 47.50 -4.5

3 77 48.25 - -

4 43 - - -

[ M3A1]

4

Year / Quarter 1st 2nd 3rd 4th

2010 - - 28.5 -8

2011 -15.625 -3.75 28.375 -8.375

2012 -16.25 -4.5 - -

Average seaonal

variation

-15.9375 -4.125 28.4375 -8.1875

Adjusting factor 0.046875 0.046875 0.046875 0.046875

Seasonal variation -15.984375 -4.171875 28.390625 -8.234375

The seasonal variation index is as follow ( 4 s.f. )

Seasonal variation

Index

-15.98 -4.172 28.39 -8.234

[ M3A1 ]

Section B

7(a) X : Life span of YES battery.

X~N( 2, 0.42 )

P ( X < 1 ) = 1 2

2.5 1 2.5 0.0620.4

P Z P Z

[ B1M1A1 ]

7(b) X~ B ( 1000 , 0.0062 )

N > 50 , np = 1000 x 0.0062 = 6.2 ( > 5 ) npq = 1000 x 0.0062 x 0.9938 = 6.16156 ( >5 )

: . Normal distribution is taken as suitable approximation. X ~ N ( 6.2 , 6.16156 ) [ B1 ]

10.5 6.2

10 . . 10.5 1.7323 0.95846.16156

P X c c P X P Z p Z

[ M3A1]

7(c) X ~ N ( 2 , 0.42 )

2 2

2 0 0.50.4

P X P Z P Z

[ M1 ]

Y : number of battery have life span exceeding two years.

Y ~ B ( 10 , 0.5 ) [ B1 ]

0 10 1 9 2 8 3 710 10 10 10

0 1 2 3

4 1 4

1 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.8281

P Y P Y

C C C C

[ M2A1 ]

5

8(a) [ GA2 ]

8(b) 2 28, 425, 22679, 1353, 229047, 72013n x x y y xy

2 2

8 72013 425 1353

8 22679 425 8 229047 1353

0.9036

r

[ B1M2A1]

8(c) r = 0.9036, it shows there is a strong positive linear correlation between the mass (x) and the

height (y) of students. [ B1 ]

8(d) y = a + b x

2

8 72013 425 13531.337

8 22679 425b

1353 4251.337 98.10

8 8a

: . The equation of least square regression line is y = 98.10 + 1.337 x [ B1M1A2]

8(e) When x = 45, y = 98.10 + 1.337 ( 45 ) = 158.27 cm ( 2d.p. )

: . The estimate height of the a student when the mass is 45 kg is 158.27 cm. This estimation is not

valid as the mass, 45 kg is not in the range of data collected. [ M1B1 ]

8(f)

Students A B C D E F G H

x 4 6 3 7.5 7.5 5 2 1

y 2 6 4.5 7 8 4.5 3 1

d2 4 0 2.25 0.25 0.25 0.25 1 0

2 8d

Spearman rank correlation coefficient,

2

6 81 0.9048

8 8 1sr

[ M1A1 ]