## Statistics on the Number of Students Who Play Cricket, Hockey, and Volleyball

QUESTION 1 a) Among a group of students, 50 played cricket, 50 played hockey and 40 played volley ball. 5 played both cricket and hockey, 10 played both hockey and volley ball, 5 played cricket and volley ball and 10 played all three. i) Draw a Venn diagram base on the information given. ii) How many number of students if every student played at least one game? iii) How many students played cricket only? iv) How many students played hockey and volley ball, but not cricket?

b) In a survey involving 170 respondents on their interest of Astro Channel : Astro Prima (P), Astro Ria (R) and Astro Mustika (M), the following information is obtained. 72 respondents like Astro Mustika 78 respondents like Astro Prima 78 respondents like Astro Ria 20 respondents like Astro Ria and Astro Mustika 15 respondents like Astro Prima and Astro Mustika 29 respondents like Astro Prima and Astro Ria

i) Draw a Venn diagram to represent all the above information. ii) How many respondents like Astro Mustika only? iii) How many respondents at least like two astro channels? iv) How many respondents like Asto Prima and Astro Mustika, but not Astro Ria? (Total mark : 20 marks)

QUESTION 2 (a) The data shows the sample of the number of coconut trees planted by 50 farmers. 60 77 70 81 73 69 79 69 75 67 65 71 62 66 78 76 64 71 73 79 70 66 64 89 81 61 73 78 73 68 68 77 74 63 71 65 87 67 63 74 74 80 70 72 75 82 76 81 68 74 i) Construct a frequency table using the class width of 5 and the first class interval is set as 55 – 59. The frequency table must also include the information for class boundary, midpoint, and cumulative frequency. ii) Based on the frequency table in part (i), calculate the values of median and standard deviation. iii) By using a scale of 2 cm to represent 5 trees on the x-axis and 2 cm to represent 5 farmers on the y-axis, draw an ogive graph for the data above

(b) Table below shows the marks obtained from a sample of a group of students in a Science examination. Marks 40 60 75 83 88 Number of Students 4 6 10 6 x i) If the mean of the students’ marks is 75, find the value of x. ii) State the minimum value of x if the mode is 88. (Total mark : 20 marks)

## Calculation of Means

QUESTION 3 a) Three machines, X, Y and Z, manufacture a component. Machine X manufactures 35% of the components, machine B manufacture 40% of the components and machine C makes the rest. A component is either acceptable or not acceptable: 7% of components made by machine A are not acceptable, 12% of components made by machine B are not acceptable, and 2% of those made by C are also not acceptable. i) Find the probability that a component is made by either machine A or machine B. ii) Two components are picked at random. What is the probability that they are both made by machine B? iii) Three components are picked at random. What is the probability that they are each made by a different machine? iv) A component is picked at random. What is the probability that it is not acceptable? v) A component is picked at random. It is not acceptable. What is the probability it was made by machine B? vi) A component is picked at random and is acceptable. What is the probability it was made by either machine A or machine B? (Total mark : 20 marks)

QUESTION 4 a) Evaluate the following integrals:

b) Given the two curves 2 x y 1 2( 2) and x y 6 7 . By sketching the graph, find the area of the region bounded by the given curves by using two methods: i) integrating with respect to x ii) integrating with respect to y

QUESTION 5 a) Given u ( 1,5,6), v ( 2, 3, 5) . Find the following vector operations. i) 2u v ii) 3 2 u v b) If 1 2 1 0 1 2 A , 1 2 3 5 0 0 1 2 1 B Verify A BC AB C ( ) ( ) . c) Johny, Sara and Chong went to the craft store to purchase supplies for making decorations. Johny purchased three sheets of craft papers, four boxes of markers and five glue sticks. His bill was RM24.40. Sara spent RM30.40 when she bought six sheets of craft papers, five boxes of markers and two glue sticks. Chong, purchases total RM13.40 when he bought three sheets of craft papers, two boxes of markers and one glue stick. i) Obtain a system of linear equations to represent the given information. ii) Use Cramer’s rule to solve the system of linear equations. Hence, state the unit cost for craft paper, box of markers and glue stick.

## Statistics on the Number of Students Who Play Cricket, Hockey, and Volleyball

- For ease with the calculations let’s use letters to represent the games.

That is Cricket

H for Hockey V for Volley

- Drawing the Venn diagram to represent the scenario

The number of students who play:

Cricket 50

Hockey 50

Volley 40

Cricket and Hockey 5

Hockey and Volley 10

Cricket and Volley 5

- If every student play at least one game this means either a student play one, two or three games but there is no single student who does not participate in games.

The number of students will therefore be

This gives the total number of students to be 100

- The number of students who play cricket only

Total playing cricket is 50, all three games are played by 10 students.

Cricket and hockey 5 students and finally cricket and volley 5 students.

Hence cricket only will be.

- The number of students who are playing hockey and volleyball only, but no cricket are 10 students.

This number can be obtained directly from the Venn diagram by checking the intersection of H and V.

- The survey involves 170 respondents on their interests in Astro Channels. Using parameters

Astro prima be P

Astro Ria be R

Astro Mustika be M

P and M

Total 170 respondents

- Assuming no respondent like all the 3 channels that means the value of x is zero. Then the number of respondents who like Astro Musika only will be

Total who like Astro Musika

Then

And

M only will therefore be

- The number of respondents who like at least 2 channels.

Total will be

- Respondents who like , this value can be observed directly from the Venn diagram.

Question 2

- The frequency tables

- Value of the median

- Drawing the ogive graph

- The mean of the data is 75.

Mean is calculated using the formula Simplifying this equation gives

- The mode is the marks with the highest frequency. If the mode of the data is 88 then the value of x should be at a minimum 11.

Question 3

- Probability a component is made by machine A or B

- The probability that all the two components are made by machine B

ways on which the components can be arranged will have

as the definitive answer.

- The probability of A and B and C

This gives 0.035

Since there are nine ways of arranging the products depending on which machine produced them then the answer is raised to power 3 to give

- The probability of not acceptable

- The probability of B given not acceptable

- Probability of acceptable

Question 4

143.25

- Sketching the graphs

- Integrating with respect to x

The equation

When integrated with respect to x gives

On the other hand, when integrated with respect to x gives 1. With thus the area under the two curves when obtained using integration with respect to x will be 1Integrating with respect to y

Question 5

- You given and Then Is On the other hand Therefore, the value of which the value of as it is positive hence the absolute value will be the same.

The is to verify that To begin we computer This means Thereafter we computer This equals (AB) which is Since the final solution of (AB)C then we have successfully proven that

- To initiate the solution lets first replace the items bought by a set of letters

Sheets of crats papers be S

Boxes of markers be B

Glue sticks be G

- For Johny For Sara nd for Chong

- Using Cramer’s rule to solve the system of linear equations

The Matrix computed is

From here we write down the main matrix. This is We then find the determinant of this matrix. Which will be Then the 1^{st} column of the main matrix is replaced by the solution vector and the determinant of the resultant matrix obtained The determinant will be given by e next step will be to replace the 2^{nd} column of the main matrix with the solution vector and obtain the determinant The resultant matrix will be The determinant will be Thereafter we replace the 3^{rd} column of the matrix with the solution vector and determine the resultant determinant.

The matrix is The determinant will be Now the value of the items will be obtained by From the solution obtained the unit costs will be as follows

For Craft paper Box of Markers Glue sticks

Class interval |
Class boundary |
Mid points (x) |
Frequency (f) |
Cumulative Frequency |
fx |
x^2 |
fx^2 |

55-59 |
54.5-59.5 |
57 |
0 |
0 |
0 |
3249 |
0 |

60-64 |
59.5-64.5 |
62 |
7 |
7 |
434 |
3844 |
26908 |

65-69 |
64.5-69.5 |
67 |
11 |
18 |
737 |
4489 |
49379 |

70-74 |
69.5-74.5 |
72 |
15 |
33 |
1080 |
5184 |
77760 |

75-79 |
74.5-79.5 |
77 |
10 |
43 |
770 |
5929 |
59290 |

80-84 |
79.5-84.5 |
82 |
5 |
48 |
410 |
6724 |
33620 |

85-89 |
84.5-89.5 |
87 |
2 |
50 |
174 |
7569 |
15138 |

Sum |
50 |
3605 |
262095 |
||||

x |
y |
||||||

1 |
22-23.6 |
0 |
|||||

2 |
24-25.6 |
2.8 |
|||||

3 |
26-27.6 |
7.2 |
|||||

4 |
28-29-6 |
13.2 |
|||||

5 |
30-31.6 |
17.2 |
|||||

6 |
32-33.6 |
19.2 |
|||||

7 |
34-35.6 |
20 |

References

Freedman, D. (2005). Statistical Models: Theory and Practice. Cambridge University Press.

Gut, A. (2005). Probability: A Graduate Course. Springer-Verlag.

Katz, V. J. (2008). A history of mathematics . Boston: Addison-Wesley.

Merriam-Webster. (2017). Integral Calculus - Definition of Integral calculus.

Ruskey, F., Savage, C. D., & Wagon, S. (2006). The Search for Simple Symmetric Venn Diagrams. Notices of the AMS, 1304–11.

Sandifer, E. (2003). How Euler Did It. The Mathematical Association of America.

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