Stpm Trial 2009 Maths2 Q&a (n9)

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CONFIDENTIAL*

2

1. A discrete random variable X has a probability distribution as shown in the table below. X= x

0

m

2

3

P(X = x)

0.1

2n

0.3

0.2

(a) Find the value of n. [1 mark]

(b) Given that E ( X ) = 1.6 , find the value of m.

[1 mark] (c) Given that = Y X 1 + X 2 where X 1 and X 2 are independent random variables of X, find E (Y ) .

[2 marks]

2. The mean height of 100 students selected randomly in a college was found to be 150 cm with a standard deviation of 5 cm. (a) Estimate the mean and standard deviation of the mean height of all the students in the college. [3 marks] (b) Estimate the standard error of the mean height of the students. [2 marks]

3. The demand for tiger prawns in Malaysia depends on the price. Price per kg (RM) Sales ( ‘000kg )

27 10

26 12

25 15

24 19

23 27

22 37

21 44

Calculate the Pearson’s correlation coefficient.

20 59 [5 marks]

4. The number of laptop computers that are sold in a week by 16 representatives in a town is as follows: 6

10

9

59

22

14

25

26

11

27

50

27

37

38 19 38

(a) Draw a stemplot to represent the above data. [3 marks] (b) Hence, find the median and the semi-inter quartile range of this distribution. [4 marks]

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CONFIDENTIAL*

3

5. The following table shows the number of digital camera sold in a departmental store for the year 2007 and 2008. Type of digital camera A B C D

2007 2008 Price(RM) Quantity Price(RM) Quantity 350 53 450 81 500 36 550 75 800 35 900 42 1200 60 1500 50

By using 2007 as the base year, (a) calculate the average of relative quantity index of digital cameras A, B, C and D for the year 2008. [3 marks] (b) calculate the Paasche price index for the year 2008 and explain your answer. [3 marks]

6. Events A and B are such that 1 3 , P ( B | A) = , P ( A) = 4 3 Find (a) P ( A  B ) ,

P ( A' B) =

1 4

(b) P ( B ) , Determine whether events A and B are independent. Give reasons for your answer. [7 marks]

7. The relationship between two variables x and y are found to be as the following: x

15

16

17

18

19

20

21

22

y

2.4

2.5

2.6

2.6

3.0

3.5

3.6

3.4

(a) Plot the scatter diagram for the above data and state the relationship between x and y. [3 marks] (b) Find the equation of the regression line of y on x in the form of y = a + bx, where a and b are expressed correct to two decimal places. Draw the graph of the regression line on your scatter diagram. [7 marks]

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CONFIDENTIAL*

4

8. The following table shows the activities for a project and their preceding activities and duration. Activity Preceding Activities Duration (days) 3 A 3 B 7 C 1 D A 2 E D,J 2 F B 1 G C 1 H E,F,G 1 J B (a) Draw an activity network for the project showing the earliest start time and the latest start time for each activity. [6 marks] (b) State the critical path and the minimum completion time. [2 mark]

9. The table below shows the duration in minutes of 160 telephone calls made in one month by a trading company. Duration (minutes) Frequency 0.0 – 2.9 12 3.0 – 5.9 33 6.0 – 8.9 45 9.0-11.9 38 12.0-14.9 19 15.0-17.9 7 18.0-20.9 6 (a) Calculate the mean call duration by the trading company. [2 marks] (b) Plot the cumulative frequency curve for the grouped data above. Hence, estimate the median for the durations of the telephone calls. [4 marks] (c) Describe the skewness of the distribution. [2 marks]

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CONFIDENTIAL*

5

10. The following table shows the quarterly profits (RM’000) of a company. Year 2003 2004 2005

Quarter 1 26 32 34

2 44 56 58

3 100 120 122

4 46 50 52

(a) Calculate the centred four-quarter moving averages for the above data. [4 marks] (b) Calculate the quarterly seasonal variation index using the multiplicative model. [4 marks] (c) Predict the amount of profit for the first quarter of the year 2006. [4 marks] 11. A bakery shop bakes two types of breads, A and B, which are made of three types of ingredients: P, Q and R. One loaf of type A bread needs 3 units of ingredient P, 1 unit of ingredient Q and 3 units of ingredient R. One loaf of type B bread needs 4 units of ingredient P, 2 units of ingredient Q and 8 units of ingredient R. The bakery has 480 units of ingredient P, 180 units of ingredient Q and 640 units of ingredient R. The profit for Type A bread is RM2.00 per loaf, whereas the profit of type B bread is RM3.00 per loaf. (a) Formulate the linear programming problem to obtain maximum profit. [4 marks] (b) Using the graphical method, determine the number of loaves of each type of bread that must be baked to obtain the maximum profit and find this maximum profit. [9 marks]

12. The mass of a type of pill produced by a pharmacy store has a normal distribution with mean µ g and standard deviation 0.2 g (a) Given µ = 7 g, find the probability that the mean mass of a random sample of 16 pills exceed 7.05g. [4 marks] (b) If the mean mass of the random sample of the 16 pills is 7.2 g, find a 96% confidence interval for the population mean, µ. State with reason whether the manager’s claim that µ = 8 g is true or false. [6 marks] (c) Determine the minimum sample size needed so that the difference between sample mean and true mean µ is less than 0.1 g at a 90% confidence interval. [5 marks]

CONFIDENTIAL*

1. (a)

0.1 + 2n +0.3+0.2 = 1 n = 0.2

(b)

1 Mark Scheme

0(0.1) + m(0.4) + 2(0.3) + 3(0.2) = 1.6 m=1

(c)

E(X 1 ) = E(X 2 ) = 1.6 = E (Y ) E ( X 1 ) + E ( X 2 ) = 1.6 + 1.6 = 3.2

2. (a) Estimate of population mean = sample mean = 150 cm Estimate of population standard deviation =

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B1 B1 M1 A1

B1

ns 2 n −1

100 ( 5 ) = 99 = 5.025 or 5.03 2

M1 A1

(b) Estimate of the standard error of the mean = =

s2 n

( 5)

2

100 = 0.5

M1 A1

∑ xy −

3. = S xy

= 4952 −

∑ x∑ y n (188)( 223)

8

= −288.5 = S xx

∑x

2

( ∑ x) −

= 42

∑y

= S yy

2

B1 - Any three of the summation correct

2

M1 – method to calculate any of the Sxy, Sxx, Syy

n

= 4460 −

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2

CONFIDENTIAL*

(188)

2

A1 – All the values correct

8

(∑ y) −

2

n

= 8345 −

( 223)

= 2128.875

2

8

∴Coefficient of correlation, r = =

S xy S xx S yy −288.5

( 42 )( 2128.875)

M1 A1

= −0.9648

or

= = , ∑ y 223 = , ∑ xy 4952 = , ∑ x 2 4460 = , ∑ y 2 8345 ∑ x 188

r=

= −

n∑ xy − ∑ x ∑ y

[n∑ x 2 − (∑ x) 2 ][n∑ y 2 − (∑ y ) 2 ] 8(4952) − 188(223) 336 × 17031

= − 0.9648

(his value)

B1-Any three correct

M1 – calculating any of the Sxy, Sxx, Syy A1 – all values correct M1 A1

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3

CONFIDENTIAL* 4(a) 0 6 1 0 2 2 3 7 4 5 0 Key: 2|5 = 25

9 1 5 8

4 6 8

M1 – uniform scale, Key

9 7

7

A1 – all correct entries B1 – Stem for 4

9

(b) Median = 25.5

B1

Q 1 = 12.5 , Q 3 = 37.5

(both)

B1

1 (37.5 - 12.5) 2 = 12.5

Semi-inter quartile range =

5(a) Type of digital camera A B C D

152.83 + 208.33 + 120.00 + 83.33 4

∑I

B1 – correct values for Iq q

n

= 141.13 (b) Paasche price index = =

A1

Sales Quantity qn 2007 (q o ) 2008 (q n ) I q = q × 100 o 53 81 152.83 36 75 208.33 35 42 120.00 60 50 83.33

Average Relative Quantity Index = =

M1

M1 (his value) A1

∑p ∑p

q

2006 2006 2005 q2006

×100

450 ( 81) + 550 ( 75 ) + 900 ( 42 ) + 1500 ( 50 ) ×100 350 ( 81) + 500 ( 75 ) + 800 ( 42 ) + 1200 ( 50 )

= 119.473 The price of the items have increased by 19.47%

M1 (strict) A1 B1

CONFIDENTIAL* 6 (a)

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4

3 4 P ( B  A) 3 = 4 P ( A) P ( B | A) =

M1

3 × P ( A) 4 3 1 P ( A  B )= × 4 3 1 = 4 1 (b) P ( A' B) = 4 1 P ( B) − P ( A B) = 4 1 1 P ( B) − = 4 4 1 P ( B) = 2 Since P ( A  B ) ≠ P ( A ) P ( B ) or P ( B | A ) ≠ P ( B ) thus A and B are not independent. 7(a) Uniform scale and at least 5 correct points All correct Positive linear relationship (b) y= a + bx B1 = = , ∑ y 23= .6 , ∑ xy 444= .4 , ∑ x 2 2780 ∑ x 148 P ( B  A )=

1 ∑ xy − x y b= n 1 ∑ x2 − x 2 n 1 148 23.6  ( 444.4 ) −     8  8  8  = 2 1  148  ( 2780 ) −   8  8  = 0.186 a= y − bx

M1 A1

M1 A1 B1 – deduction B1 - reason D1 D1 B1

M1 – usage for the formula (his value) A1

23.6  148  − 0.186   8  8  = −0.486 The equation of the regression line: y = −0.486 + 0.186 x ( x , y ) = (18.5, 2.95)

=

Straight line passing through his ( x , y )

B1 √ (his value of b) A1 D1 D1

8(a)

2 3

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5

CONFIDENTIAL*

5

D 5

4

1

6

A 3 1 0

0

2

J B 3

3 3

1 5

C 7

6

F 2

8

7

H 8

1

9

G 1

4 7

E

7

D1 – correct sequence and flow of the network diagram D1 – correct activities and duration B1 – correct values of EST B1 – correct values of EFT B1– correct values of LST B1– correct values of LFT (b) B1 - Critical Path : C-G-H (strict) B1 - Minimum completion Time = 9 days

1384 160 = 8.65

9)(a) Mean = (b)

M1 A1

Less than Cumulative Frequency 2.95 12 5.95 45 8.95 90 11.95 128 14.95 147 17.95 154 20.95 160 B1- correct boundaries and cumulative frequencies Graph D1 – uniform scale and correct points (his values) D1 – all correct B1 - median = 8.4 ± 0.2 (c) X = 8.65,median = 8.0 The distribution is skewed to the right or positively skewed because median < mean

B1 B1

9

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6

CONFIDENTIAL*

Cumulative Frequency

160 −

X X X

140 − X 120 −

100 − X 80 −

60 −

X

40 −

20 − X 0X

‫׀‬ 3

‫׀‬ 6

‫׀‬ 9

‫׀‬ 12

‫׀‬ 15

‫׀‬ 18

‫׀‬ 21

Marks

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7

CONFIDENTIAL* 10 (a) Year

1

Original Data, Y 26

2

44

3

100

4

46

1

32

2

56

3

120

4

50

1

34

2

58

3

122

4

52

Quarter

2003

2004

2005

Moving Average

54.0 55.5 58.5 63.5 64.5 65.0 65.5 66.0 66.5

M1

Trend, T

Y T

54.75

1.83

57.00

0.81

61.00

0.52

64.00

0.88

64.75

1.85

65.25

0.77

65.75

0.52

66.25

0.88

A1

M1

A1

(b) Year

Quarter

1 2003 2004 0.52 2005 0.52 Total 1.04 Average 0.52 Correction factor 0.9926 Seasonal Variation 0.52 M1: Average

2 0.88 0.88 1.76 0.88 0.9926 0.87

M1 adjusted seasonal variation A1 all correct

3 1.83 1.85

4 0.81 0.77

3.68 1.58 1.84 0.79 0.9926 0.9926 1.83 0.78 B1: Correction factor

CONFIDENTIAL*

(c) Range of trend values = 66.25 – 54.75 = 11.5 11.5 Average change in each period = 7 11.5 ) = 71.18 Thus, the projected trend value = 66.25 + 3( 7 Forecast value for the first quarter of 2006 = 71.18 × 0.52 = 37.01 Forecast profit for the first quarter of 2006 is RM 37 010 11(a)

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8

M1 , A1 M1 A1

x = number of loaf of type A bread. y = number of loaf of type B bread 3x + 4y ≤ 480 x + 2y ≤ 180 3x + 8y ≤ 640

B1 B1 B1

x ≥ 0, y ≥ 0 Maximizing, P = 2x + 3y

B1

(b) 5 lines (including axis and search line) 1 mark for every correct line D5 Correct region B1 Number of loaf of Type A bread = 120 Number of loaf of Type B bread = 30 Maximum profit = 2 (120 ) + 3 ( 30 ) = RM 330.00 OR All 4 extreme points (0,80) , (80,50) , (120,30) (160,0) Substitute in obj. function x = 120, y =30 (both) RM 310.00

B1 dependent on search line M1 (substitution) A1 M1 M1 A1 A1

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9

CONFIDENTIAL*

y

120 −

100 −

80 X −

60 − X

3x + 4y = 480

40 − X

2x + 3y = 90

3x + 8y = 640

20 −

0

‫׀‬ 20

‫׀‬ 40

‫׀‬ 60

‫׀‬ 80

‫׀‬ 100

‫׀‬ 120

‫׀‬ 140

X‫׀‬ 160

x + 2y = 180 ‫׀‬ 180

x

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10

CONFIDENTIAL* 12. X ~ N(µ, 0.22) (a) µ = 7 , n = 16  0.22  X ~ N  7,   16      7.05 − 7  P ( X > 7.05 ) = P  Z >  0.22    16   = P ( Z > 1) = 0.1587

= ( 7.097 , 7.303) At the level of confidence of 96%, the manager’s claim that µ = 8 is false because the interval (7.097,7.303) does not contain 8 g Zα 2

1.645

σ2 n

B1 for X > 7.05 M1

A1

(b) Z 0.02 = 2.054 96% confidence interval for the population mean µ, is  0.22  =  7.2 ± 2.054    16   = ( 7.2 ± 0.1027 )

(c)

B1

≤ 0.1

B1 M1 for 7.2 ± K M1 for the correct term K A1 B1 False, B1 Reason

M1 for ≤ 0.1

0.22 ≤ 0.1 n 2

2  1.645  n≥  ( 0.2 )  0.1  n ≥ 10.82 The minimum sample size is 11

M1 - substitution B1 z-value 1.645 M1 A1