Additional Mathematics

ADDITIONAL MATHEMATICS MATHEMATICS Form 5 2013

Name :Mohd Hafiz Bin MohdBasir I/C No : 961207-12-5859 Form : 5 ST 3 School: SMK KuharaTawau Teacher :Khamsiah Seharing 1 Additional Mathematics 2013

No

Contents

Page

1

Acknowledgement

3

2

Objectives

4-5

3

Introduction

6-7

4

Part 1

8-12

5

Part 2

13-29

6

Part 3

30-37

7

Further Exploration

38-39

8

Reflection

40

2 Additional Mathematics 2013

No

Contents

Page

1

Acknowledgement

3

2

Objectives

4-5

3

Introduction

6-7

4

Part 1

8-12

5

Part 2

13-29

6

Part 3

30-37

7

Further Exploration

38-39

8

Reflection

40


First of all, I would like to say thank you to my friends, teachers and parents for givingme their full support in making this project successful. Not forgotten to my family for providing everything, such as money, to buy anything thatare related to this project work and thei r advise, which is the most needed for this project.Internet, books, computers and all that act as my source to complete this project.They alsosupported me and encouraged me m e to complete this task so that I will not procrastinate duringdoing this project work. Next, I would like to thank my teacher, Puankhamsiahseharingfor guiding me and my friendsthroughout this project. We had some difficulties in doing this task, b ut she taught us patientlyand gave me guidance throughout the journey until we knew what to do. She tried her best tohelp us until we understand what we supposed to do with the project work. Besides that, my friends who were doing the same project as mine and shared ourideas.They were helpful that when we combined and discussed together, but b ut however, thistas thistask was done individually.Last but not least, any party which involved either directly or indirect in comp co mple leti ting ng this th ispr proje oject ct work work.. Than Thank k you you ever everyo yone ne


We, as the students taking Additional Mathematics are required to carry ou t a project work while we are in Form 5. 5 . This year the Curriculum Development Division, Ministry of Education has prepared for tasks for us with four choices, and I chose the third task. We are to choose and complete only one task based on our area of interest. This project can be done in groups or individually, but each of us are expected to submit an individually written report. Upon completion of the Additional Mathematics Project Work, we are to gain valuable experiences exp eriences and able to: 1.Apply and adapt a variety of problem solving strategies to solve routine and nonroutineproblems. 2.Experience classroom environments which are challenging, interesting and meaningfuland hence improve their thinking skills. 3.Experience classroom environments where knowledge and skills are applied inmeaningful ways in solving real-life problems. 4.Experience classroom environments where expressing ones mathematical thinking,reasoning and communication are highly encouraged and expected. 5.Experience classroom environments that stimulates and enhances effective learning. 6.Acquire effective mathematical communication through oral and w riting, and to use thelanguage of mathematics to express ex press mathematical ideas correctly and precisely. 7.Enhance acquisition of mathematical knowledge and skills through problem- solving inways that increase interest and confidence.


8.Prepare ourselves for the demand of our future undertakings and in workplace. 9.Realises that mathematics is an important and powerful tool in solving re al-life problemsand hence develop positive attitude towards mathematics. 10.Train ourselves not only to be independent learners but also tocollaborate, to cooperate and to share knowledge in an engaginghealthy environment. 11.Use technology especially the ICT appropriately and effectively. 12.Train ourselves to appreciate the intrinsic values of mathematics and to become morecreative and innovative. 13.Realises the importance and the beauty of mathematics.


A Brief History of Statistics By the 18th century, the term "statistics" designated the systematic collection of demographicand economic data b y states. In the early 19th century, the meaning of "statistics" broadened,then including the discipline concerned with the collection, summary, and analysis of data.Today statistics is widely employed in government, business, and all the sciences. Electronic computers have expedited statistical computation, and have allowed statisticians to develop"computer-intensive" methods. The term "mathematical statistics" designates the mathematical theories of probability and statistical inference, which are used in statistical practice. The relationbetween statistics and probability theory developed rather late, however. In the 19th century,statistics increasingly used probability theory, whose initial results were found in the17th and18th centuries, particularly in the analysis of games of chance (gambling). B y 1800, astronomyused probability models and statistical theories, particularly the method of l east squares, whichwas invented by Legendre an d Gauss. Early probability theory and statistics was systematizedand extended by Laplace; following Laplace, probability and statistics have been in continualdevelopment. In the19th century, social scientists used statistical reasoning and probabilitymodels to advance the new sciences of experimental psychology and sociology; physicalscientists used statistical reasoning and probability models to advance the new sciences of thermodynamics and statistical mechanics. The development of statistical reasoning was closelyassociated with the development of inductive logic and the scientific method. Statistics is not afield of mathematics but an autonomous mathematical science, like computer science oroperations research. Unlike mathematics, statistics had its origins in public administration andmaintains a special concern with demography and economics. Being concerned with thescientific method and inductive logic, statistical theory has close association with the philosophyof science; with its emphasis on learning from data and making best predictions, statistics hasgreat overlap with the decision science and microeconomics. With its concerns with data,statistics has overlap with information science and computer science.


Statistics Today During the 20th century, the creation of precise instruments for agricultural research, publichealth concerns (epidemiology, biostatistics, etc.), industrial quality control, and economic andsocial purposes (unemployment rate, econometric, etc.) necessitated substantial advances instatistical practices. Today the use of statistics has broadened far beyond its origins. Individualsand organizations use statistics to understand data and make informed decisions throughout thenatural and social sciences, medicine, business, and other areas. Statistics is generally regardednot as a subfield of mathematics but rather as a distinct, albeit allied, field. Many universitiesmaintain separate mathematics and statistics departments. Statistics is also taught in departmentsas diverse as psychology, education, and public health.


1.Importance of data analysis in daily life. There are many benefits of data analysis however , the most important ones is it helps structuring the findings from different sources of datacollection like survey research. It is again very helpful in breaking a macro problem into micro parts. Data analysis acts like a filter when it comes to acquiring meaningful insights out of hu ge data-set. Every researcher has sort out huge pile of data that he/she has collected, before reaching to a conclusion of the research question. Mere data collection is of no use to the researcher. Data analysis proves to be crucial in this process. It provides a meaningful base to critical decisions. It helps to create a complete dissertation proposal. One of the most important uses of data analysis is that it helps in keeping human bias away from research conclusion with the help of proper statistical treatment. With the help of data analysis a researcher can filter both qualitative and quantitative data for anassignment writing projects. Thus, it can be said that d ata analysis is of utmost importance for both the research and the researcher. Or to put it in another words data analysis is as important to a researcher as it is important for a doctor to diagno se the problem of the patient before giving him any treatment.


2.A.1.Type of measure of central tendency and of measure of dispersion.

Central tendency gets at the typical score on the variable while dispersion gets at how much variety there is in the scores. Both the central tendency and the dispersion are customary on report, and it is desirable when the scores are in single variable.

I.Mean. The mean is what in everyday conversation is called the average. It is calculated by adding the values of all the valid cases together and dividing by the number of the valid cases.

x  

OR

 fx   f

The mean is an interval/ratio measure of central tend ency. Its require that the attributes of variable are represent numeric scale.

II.Mode Mode is attribute often in data set. For ungrouped data , the mode is found by finding the modal class and the two classes adjacent to the modal class .Crossed the two lines from the adjacent and we will find the intersection. The modal intersection value is known as median.


III.Median. The median is a measure of central tendency. It identifies the value of the middle case when the cases have been placed in order or in linefrom low to high. The middle of the line is as far from being extreme as you get .

   F    C m  L   2  f m      There are as many cases in line in front of the middle case as behind the middle case.The median is the attribute used by the middle class. When you know the value of the median , you know that at least half the cases had that value or a higher value, while atleast half the cases had that value or a lower value.


2.A.2.Types of measure of dispersion While measures of central tendency are used to estimate “ normal” values of data set, measures of dispersion are important for describing the spread of the data, or its variation around central value.Two distinct samples may havethe same mean or median, but completely different levels of variability, or vice versa.A proper description of a set ofdata should haveinclude both of the characteristics.There are various methods that can be used to measurethe dispersion of a data set, such as standard variation and variance.

I.Standard Deviation. The standard deviation tells you the approximate average distance of cases from the mean. This is easier to comprehend than the squared distance of cases from the mean. The standard deviation is directly related to the variance. If you know the value of the variance, you can easily figure out the value of the standard deviation. The reverse is also true. If you know the value of the standard deviation, you can easily calculate the value of the variance. The standard deviation is the square root of the variance

  fx      x      f  2

2

II.Variance The mean of the squared deviation scores about the mean of destribution Variance is defined as the average of the squared deviations from the mean. To calculate the variance, you first subtract the mean from each number and then square the results to find the squared differences. You then find the ave rage of those squared differences. The result is the variance.


2.B Type of measures of central tendency stated in (A.1) which is mode , median and mean their examples in daily life. I. Mean Mean or the average, is a important statistic used in sports.Coaches use averages to determine how well a player is performing.General managers ma y use averages to determine how good a player is and how much money that the player is worth.

II.Median The median is used in economics. For example, the U.S census bureau finds the median household income.According to the U.S Census Bureau,” median household income” is defined as “ the amount which devides the income distribution into an equal groups,half having income above that amount , and half having income below that amount.

III.Mode The mode may be beneficial for a managers of a shoe store.For example, you would see size 17 shoes stocked on the floor.Why? Because very few people have a same shoe size.Therefore,store managers may look at data and determine which shoe sold the most. Managers would want to stock the floor with best selling shoe size.



Monthly test marks subject: Bahasa Malaysia No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Students Name Farrah AfifahFilzahBintiFauzee Muhammad DzharulIdham Bin Talip FirdausLiau Junior Jonathan V Joannes JumariaBintiBahari Muhammad KhairuddinSanuddin HafizulIkmal Bin Ma”mun Muhammad FareezAsyraf Bin Rudy @ Rudy Ariffin Kartika Sari DewiBinti Abdul Samad Ahmad AfiqAmsyar Bin Matam AlfieraAyuNurulA”inBintiIszianto Muhammad Mazlan Bin Ngatiman AdiyIdhamAbdillah Mohd Hafiz Bin MohdBasir UmmuKhairatulFarhaniBintiBurhan HarreyadiAdha Bin Sadimin NorazilahBinti Jim Nursayfayani Foo BintiMohdSaiful Muhammad Faruq Bin Pramadansha Asbuddin Bin Asdi RabiatulAdawiyahBintiDarwis SufianaMohdSufyan Asmin Bin Mandja CalveianiBintiAkim MohdFarid Bin Muslia N urulAmyliaMohd Rata NurAzwaniBinti Ismail ZainAzhar Bin Sapar MohdApfis Bin Sayuman AkmalHazwam Bin Wahyono MohdHamizee Bin MazlanSufree SalwahBintiHatibe NurulAinBintiDulmi HananBintiHussin Suriani Abdullah MahranRafhanahBintiMohdSiri MohdNurIqmalSamsudin AirisyaHirnanishaBintiHirwan MarshenaBintiJunie Mas AizaBinti Musa


Mark 13 15 18 25 35 37 38 38 39 40 40 40 41 42 42 45 46 46 47 48 48 48 50 50 52 52 53 53 55 57 57 58 59 63 64 66 67 71 71 71

Marks 1-20 21-40 41-60 61-80 81-100

Number of Students 3 9 21 7 0


No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40



x

Mark, 13 15 18 25 35 37 38 38 39 40 40 40 41 42 42 45 46 46 47 48 48 48 50 50 52 52 53 53 55 57 57 58 59 63 64 66 67 71 71 71

 x  1900

Mark, x 169 225 324 625 1225 1369 1444 1444 1521 1600 1600 1600 1681 1764 1764 2025 2116 2116 2209 2304 2304 2304 2500 2500 2704 2704 2809 2809 3025 3249 3249 3364 3481 3969 4096 4356 4489 5041 5041 5041



x

2



2

98160

2.a) Mean a. Mean, x 

x 

 x  1900   40 x 

1900 40

x  47.5

2.b)Median arranging the marks obtained by students in order of magnitude 13,15,18,25,35,37,38,39,40,40,40,41,42,42,45,46,46,47,48,48,48,50,50,52,52,53,53,55,57,57,58, 59,63,64,66,67,71,71,71

Median =

48  48 2

= 48 48 is the median mark


2.c)Mode

13,15,18,25,35,37,38,39,40,40,40,41,42,42,45,46,46,47,48,48,48,50,50,52,52,53,53,55,57,57,58, 59,63,64,66,67,71,71,71

From the data above The score that appeared with the most frequency are 40,48 and 71 Mode are 40,48,71

2.d) Standard Deviation   x Standard Deviation ,     

2

    x  

 x  1900   40 x  47.5



 98160  2     47.5  40 

 2454  2256.25 

197 .75


2

Marks 1-20 21-40 41-60 61-80 81-100



3.A) From table one , find the

I)

Mark

Frequency

Midpoint,x

Fx

1-20

3

10.5

31.5

21-40

9

30.5

274.5

41-60

21

50.5

1060.5

61-80

7

70.5

493.5

81-100

0

90.5

0

 fx Mean, x   f  fx  1860 x 

 f  40



 fx  f

1860 40

= 46.5


II)

Mode The class with highest frequency is 41-60 Modal class: 41-60 Mode =

41  60 2

= 50.5

III) Median a. Method one using Formulae  1     F  C Median , m = L   2  fm      L = Lower boundary N = Total Frequency F = Frequency before class median fm = Frequency of class median C = Size class 1

 N

m =

2

 1     F  C L   2  fm     

 12  20  21 

20  40  40.5   

1

2

= 20

 48.12

N = 40 F = 12 fm = 21 C = 20 L = 40.5 21 Additional Mathematics 2013

b. Median : Method Two Using Graph


IV) Standard Deviation a. Method one  f  x  x     f

2





40 1900  47.5

 f  1900 x  40

x  47.5






2

40





40 3431756 .25 40

= 1852.5



b. Method two

Mark

Frequency

Midpoint,x

fx 2

1-20

3

10.5

330.75

21-40

9

30.5

8372.25

41-60

21

50.5

53555.25

61-80

7

70.5

34791.75

81-100

0

90.5

0

 fx  97050

 f  40

 fx   x     f

2

2

2 



97050  40

 47.5

 fx  97050



2426 .25  2256 .25

f  40 



170

x  47.5

 13.04

2


2

V)

Interquartile range a. Method one using formulae Mark

Frequency

1-20

3

Cummulative frequency 3

21-40

9

12

41-60

21

33

61-80

7

40

81-100

0

40

Interquartile = Q3 - Q1 1

Qq   4



1 4

40

= 10

3

Q3   4



3 4

40

= 30

N = 40

N = 40

F=3

F = 12

fm = 9

fm = 21

C = 20

C = 20

L = 40.5

L = 40.5


= 3 = 30

 1     F  C Q  L   4 fm       1

 10  3   40.5   20  9 

 3     F  C Q  L   4 fm       3

 30  12   40.5   20  21 

 40.5  15.55 = 40.5 + 17.14 = 56.06

= 57.64

Hence, interquartile range

 Q Q 3

1

= 57.64 – 56.06 = 1.58


b. Method two using graph


3.B) The most appropriate measure of central tendency . The most appropriate measure of central of tenden cy that reflect the performance of my class are the median method.The median have an accurate value and distance of the median are bigger than the mean and the mode.

3.C) Advantages of using standard deviation compared to interquartile range . The standard deviation is the better measure of dispersion and brings a lot of advantages compared to interquartile range because the value of standard deviation is small and more accurate compared to interquartile range .


4)

Ungrouped Data Mean Standard Deviation 4. a)

47.5 1852.5

Grouped Data 46.5 13.04

Grouped data gives a more accurate representation becauseit have small value in

standard deviation but bigger mean value compared to the ungrouped data. In mathematics theory, it tells that it is better if the value of standard deviation is small and the mean value is bigger.

4. b)

Ungrouped data is suitable be used when the value is small .Example from the

value 1 to 10, and the grouped data is more suitable when the value is bigger , such as 1 to 50.


No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40



Mark 16 18 21 28 38 40 41 41 42 43 43 43 44 45 45 48 49 49 50 51 51 51 53 53 55 55 56 56 58 60 60 61 62 65 67 69 70 74 74 74

Part 3 : number 1 Marks 1-20 21-40 41-60 61-80 81-100



A)

2

Mean, x Marks 1-20 21-40 41-60 61-80 81-100

Frequency,f 2 4 25 9 0 f  40 

Mean, x  fx   f 

2

2040 40

=51


Midpoint,x 10.5 30.5 50.5 70.5 90.5

fx 21 115 1262.5 634.5 0  fx  2040

B)

Mode Mode

 d  C  L    d d   1

1

2

L = lower boundary of the modal class

d 1 = Difference between frequency of modal class an d class before H

d 2 = Difference between frequency of modal class and class after H C = Width of modal class / size of class

Frequency

C

d 1

d 2

mode

 d  C  L    d d   1

1

= 40.5 +

2

 

20

= 40.5 + 11.35 = 51.85


C)

Median

MARKS

FREQUENCY

1 – 20 21 – 40 41 – 60 61 – 80 81 – 100

2 4 25 9 0

 1     F  C Median , m = L   2  f m     

= 40.5 +

 

20

= 40.5 + 11.2 = 51


CUMULATIVE FREQUENCY 2 6 31 40 40

LOWER BOUNDARY 0.5 20.5 40.5 60.5 80.5

D)

Interquartile Range

MARKS

FREQUENCY

1 – 20 21 – 40 41 – 60 61 – 80 81 – 100

2 4 25 9 0

 1     F  C Q q  L   4  f m      = 40.5 +

   

CUMULATIVE FREQUENCY 2 6 31 40 40

LOWER BOUNDARY 0.5 20.5 40.5 60.5 80.5

 3     F  C Q  L   4  f m      3

20

= 40.5 +

 

= 40.5 + 3.2

= 40.5 + 19.2

= 43.7

= 59.7

Hence , Interquartile range = 59.7 – 43.7 = 16


20

E)

Standard Deviation

MARKS 1 – 20 21 – 40 41 – 60 61 – 80 81 – 100

Mean, x 

=

FREQUENCY 2 4 25 9 0 = 40

MIDPOINT, x 10.5 30.5 50.5 70.5 90.5

 fx  f

2040 40

= 51 Standard Deviation =

  fx      x      f  2

 

112430 40

 51

2

2

2811 .5  2601

 14.51


fx 21 122 1262.5 634.5 0 = 2040

fx² 220.5 3721 63756.25 44732.25 0 =112430

Part 3: Number 2 MARKS

FREQUENCY

1 – 20 21 – 40 41 – 60 61 – 80 81 – 100

2 4 25 9 1  f  41

Mean, x



MIDPOINT, x 10.5 30.5 50.5 70.5 90.5

 fx  f

fx

fx²

21 122 1262.5 634.5 90.5  fx  2130.5

220.5 3721 63756.25 44732.25 8190.25  fx =120620.25 2

  fx     x      f  2

2



Standard Deviation,

120620 .25

 fx  2130.5



 f  41

= 15.56



2130.5 41

= 51.96


41

 51.96 

2

Further Exploration 1.Calculating Method . Arranging the marks obtained by 40 students in magnitude

13 40 48 57

15 40 48 58

18 41 50 59

25 42 50 63

35 42 52 64

37 45 52 66

38 46 53 67

38 46 53 71

39 47 55 71

40 48 57 71

Top 20% students 20 100

 40  8

The top 8 students that has the higher mark get the reward. That is 57, 58, 59, 63, 64, 66, 67, 71, 71, 71. =The lowest mark for this group is 57


2. My Class

Mr.Ma’s Class

Mean,

47.5

76.79

Standard Deviation

13.65

10.36

Based on the table above, My class have the smaller value of Mean (47.5) than Mr.Ma’s class (76.79). Especially, the value of Standard Deviation Mr.Ma’s class is smaller (10.36) than My class (13.65). So, We can see Mr.Ma’s class is better than My class. It is because in mathematics theory, it tells that it is better if the value of Stand ard Variation is small and the mean value is bigger.


Additional Mathematics

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