ADDITIONAL MATHEMATICS PROJECT

WORK

2012

NAME : NUR IFWATUL FAIQAH KASWADI

CLASS : 5 CAMBRIDGE

MATRIX NO. : 11506

I/C NO. : 9501229-01-5424

N

O.TITLE

PAG

E1 INTRODUCTION 2

2 OBJECTIVES 3

3 HISTORY 4

4 PROJECT TASK 6

5 FURTHER EXPLORATION 16

6 REFLECTION 19

1

First of all, I would like to say Alhamdulillah thank to the God, for giving

me the strength and health to do this project work.

Furthermore, I also want to give my appreciation to my parents for all

their support in financial and moral throughout this project work. Without

them standing with me, I would not be able to finish this project.

Besides, I would like to thank my Additional Mathematics teacher, Mr

Baharom and Madam Azimah for guiding me throughout this project. He

gives a lot of guidance and information about this project. Without his

guidance, I would be lost to do the project since I never done it before.

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Last but not least, I would like to give appreciation to all my friend,

who do this project with me throughout days and nights. Also not forgotten

all my classmates and friends who are willing to share their opinion and

information.

The aims of carrying out this project work are:

i. to apply and adapt a variety of problem-solving strategies to solve

problems;

ii. to improve thinking skills;

iii. to promote effective mathematical communication;

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iv. to develop mathematical knowledge through problem solving in a way

that increases students’ interest and confidence;

v. to use the language of mathematics to express mathematical ideas

precisely;

vi. to provide learning environment that stimulates and enhances

effective learning;

vii. to develop positive attitude towards mathematics.

Since much interest has been evinced in the historical origin of the

statistical theory underlying the methods of this book, and as some

misapprehensions have occasionally gained publicity, ascribing to the

originality of the author methods well known to some previous writers, or

ascribing to his predecessors modern developments of which they were quite

unaware, it is hoped that the following notes on the principal contributors to

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statistical theory will be of value to students who wish to see the modern

work in its historical setting.

Thomas Bayes' celebrated essay published in 1763 is well known as

containing the first attempt to use the theory of probability as an instrument

of inductive reasoning; that is, for arguing from the particular to the general,

or from the sample to the population. It was published posthumously, and we

do not know what views Bayes would have expressed had he lived to publish

on the subject. We do know that the reason for his hesitation to publish was

his dissatisfaction with the postulate required for the celebrated "Bayes'

Theorem." While we must reject this postulate, we should also recognise

Bayes' greatness in perceiving the problem to be solved, in making an

ingenious attempt at its solution, and finally in realising more clearly than

many subsequent writers the underlying weakness of his attempt.

Whereas Bayes excelled in logical penetration, Laplace (1820) was

unrivalled for his mastery of analytic technique. He admitted the principle of

inverse probability, quite uncritically, into the foundations of his exposition.

On the other hand, it is to him we owe the principle that the distribution of a

quantity compounded of independent parts shows a whole series of features

- the mean, variance, and other cumulants - which are simply the sums of

like features of the distributions of the parts. These seem to have been later

discovered independently by Thiele (1889), but mathematically Laplace's

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methods were more powerful than Thiele's and far more influential on the

development of the subject in France and England. A direct result of

Laplace's study of the distribution of the resultant of numerous independent

causes was the recognition of the normal law of error, a law more usually

ascribed, with some reason, to his great contemporary, Gauss.

Gauss, moreover, approached the problem of statistical estimation in

an empirical spirit, raising the question of the estimation not only of

probabilities but of other quantitative parameters. He perceived the aptness

for this purpose of the Method of Maximum Likelihood, although he

attempted to derive and justify this method from the principle of inverse

probability. The method has been attacked on this ground, but it has no real

connection with inverse probability. Gauss, further, perfected the systematic

fitting of regression formulae, simple and multiple, by the method of least

squares, which, in the cases to which it is appropriate, is a particular

example of the method of maximum likelihood.

The first of the distributions characteristic of modern tests of

significance, though originating with Helmert, was rediscovered by K Pearson

in 1900, for the measure of discrepancy between observation and

hypothesis, known as c2. This, I believe, is the great contribution to statistical

methods by which the unsurpassed energy of Prof Pearson's work will be

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remembered. It supplies an exact and objective measure of the joint

discrepancy from their expectations of a number of normally distributed, and

mutually correlated, variates. In its primary application to frequencies, which

are discontinuous variates, the distribution is necessarily only an

approximate one, but when small frequencies are excluded the

approximation is satisfactory. The distribution is exact for other problems

solved later. With respect to frequencies, the apparent goodness of fit is

often exaggerated by the inclusion of vacant or nearly vacant classes which

contribute little or nothing to the observed c2, but increase its expectation,

and by the neglect of the effect on this expectation of adjusting the

parameters of the population to fit those of the sample. The need for

correction on this score was for long ignored, and later disputed, but is now, I

believe, admitted. The chief cause of error tending to lower the apparent

goodness of fit is the use of inefficient methods of fitting. This limitation

could scarcely have been foreseen in 1900, when the very rudiments of the

theory of estimation were unknown.

The study of the exact sampling distributions of statistics commences

in 1908 with "Student's" paper The Probable Error of a Mean. Once the true

nature of the problem was indicated, a large number of sampling problems

were within reach of mathematical solution. "Student" himself gave in this

and a subsequent paper the correct solutions for three such problems - the

distribution of the estimate of the variance, that of the mean divided by its

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estimated standard deviation, and that of the estimated correlation

coefficient between independent variates. These sufficed to establish the

position of the distributions of c2 and of t in the theory of samples, though

further work was needed to show how many other problems of testing

significance could be reduced to these same two forms, and to the more

inclusive distribution of z. "Student's" work was not quickly appreciated, and

from the first edition it has been one of the chief purposes of this book to

make better known the effect of his researches, and of mathematical work

consequent upon them, on the one hand, in refining the traditional doctrine

of the theory of errors and mathematical statistics, and on the other, in

simplifying the arithmetical processes required in the interpretation of data.

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Recently, the Malaysian government has launched a campaign of 10 000

steps a day to create awareness to the public on healthy lifestyle. At the

school level, all students are required to sit for SEGAK test to determine the

fitness level of students based on a few physical tests. Among the elements

of the test is taking the pulse rate of each student.

Based on the SEGAK test conducted in your school, get the pulse rate of

50 students before and after the step up board activity.

Complete the table below by using the data obtained.

9

Students Pulse rate ( bpm - beats per min)Before After

12...50

Table 1

(a)(i) Find the mean, mode and median of the pulse rate before the step

up board

activity for the 50 students.

(ii) Compare the pulse rate before the step up board activity of

students in your

school with a standard pulse rate. Give your comment.

(b)Find the mean, mode and median of the pulse rate after the step up

board activity for the 50 students.

(c) Construct a frequency distribution table for the pulse rate after the

step up board activity using a suitable class interval.

(i) Represent your data using three different statistical graphs

based on your frequency table.

10

(ii) Determine the mean, mode and median of the pulse rate by

using appropriate method.

(d)Compare the mean, mode and median obtained in part (b) and (c).

Give your comment.

(e)Calculate the standard deviation based on the frequency table by using

three different methods. Draw your conclusion.

ANSWERS: Pulse Rate of 50 Students Before and After The Step Up Board Activity.

Students Pulse rate (bpm – beats per min)Before After

1 82 1152 57 1283 73 1234 75 1055 64 1176 72 1147 119 1418 69 1149 79 13010 83 11711 75 11612 86 13013 56 9614 69 15015 87 12216 64 9117 81 8618 96 11019 56 8120 80 12021 66 9122 82 110

11

23 66 8024 75 12025 82 10226 75 10427 77 12028 79 10929 82 11230 77 9331 80 11032 72 9733 85 11734 75 10735 69 9136 83 12237 84 12138 91 13039 68 8840 73 10341 75 11042 82 11243 66 9944 70 10145 72 10546 76 10847 92 14148 68 9149 75 9850 81 105

(a)(i) Find the mean, mode and median of the pulse rate before the step up

board activity for

the 50 students.

Mean :

(56+56+57+64+64+66+66+66+68+68+69+69+69+70+72+72

+72+73+73+75+

75+75+75+75+75+75+76+77+77+79+79+80+80+81+81+82

+82+82+82+

12

82+83+83+84+85+86+87+91+92+96+119)

50

= 76.42

Mode : 75

Median :

56,56,57,64,64,66,66,66,68,68,69,69,69,70,72,72,72,73,73,75,75

,75,75,75,75,75,76,

77,77,79,79,80,80,81,81,82,82,82,82,82,83,83,84,85,86,87,91,92

,96,119.

= 75+75

2

= 75

(ii) Compare the pulse rate before the step up board activity of students

in your

school with a standard pulse rate. Give your comment.

13

= For teenager and adults, the current standard for a normal pulse is 60 to 100

beats per minute. Your pulse rate will be faster when you exercise or under stress or

having fever. When you're resting, your pulse rate will be slower. To have a pulse below

60 beats per minute is to have insufficient beating of the heart and weakness in the

body. Sometimes, a low heart rate is brought on by vascular heart disease or immunity

problems. A pulse over 100 beats per minute is not healthy unless you are a newborn.

(b)Find the mean, mode and median of the pulse rate after the step up board

activity for the 50 students.

1) Mean =

80+81+86+88+91+91+91+91+93+96+97+98+99+101+102+103+10

4+105+105+105+

107+108+109+110+110+110+110+112+112+114+114+115+116+11

7+117+117+120+

120+120+121+122+122+123+128+130+130+130+141+141+150

14

50

= 110.06

2) Mode = 110

3) Median

80,81,86,88,91,91,91,91,93,96,97,98,99,101,102,103,104,105,105,105,107,

108,109,110,

110,110,110,112,112,114,114,115,116,117,117,117,120,120,120,121,12

2,122,123,128,

130,130,130,141,141,150

= 110+110

2

= 110

(c) Construct a frequency distribution table for the pulse rate after the step up

board activity using a suitable class interval.

PULSE RATE (bpm) FREQUENCY71 – 80 181 – 90 3

91 – 100 9101 – 110 14

15

111 – 120 12121 – 130 8131 – 140 2141 – 150 1

(i) Represent your data using three different statistical graphs

based on your frequency table.

Bar Chart

16

FREQU

ENCY

71 – 80 81 – 90 91 – 100 101 – 110

111 – 120

121 – 130

131 – 140

141 – 150

0

2

4

6

8

10

12

14

16

Histogram

75.5 85.5 95.5 105.5 115.5 125.5 135.5 145.50

2

4

6

8

10

12

14

16

PULSE RATE (bpm)

FR

EQ

UEN

CY

Frequency polygon

17

Frequ

ency

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

65.5 155.5145.5135.5125.5115.5105.595.585.575.5

Ogive

0 1 2 3 4 5 6 7 8 90

10

20

30

40

50

60

CULMULATIVE FREQUENCY

CULMULATIVE FREQUENCY

110.5100.590.580.570.5 150.5140.5130.5120.5

CUMULATIVE FREQUENCY

PULSE RATE

18

PULSE RATE (bpm)

(ii) Determine the mean, mode and median of the pulse rate by

using appropriate method.

Mean = x =

∑ fx

∑ f

Mean = 75.5(1)+85.5(3)+95.5(9)+105.5(14)+115.5(12)+125.5(8)+

135.5(2)+145.5(1)

50

= 109.5

Mode = 107

19

Median =

L + ( N2 − F

f m)c

=90.5 + ( 502

− 13

9 )10

= 103.83

(d)Compare the mean, mode and median obtained in part (b) and (c).

Give your comment.

= the mean, mode and median in group data is mpore accurate

than in ungroup data.

Mean,

mode and

median

20

Measure of Central

TendencyUngrouped data Grouped data

Mean 110.06 109.5

Mode 110 107

Median 110 103.83

obtained in (b) is more accurate compared to (c). All the values are taken into

consideration while calculating mean, mode and median in part (b), whereas, in

part (c) values are calculated based on class interval or midpoint.

(e)Calculate the standard deviation based on the frequency table by using

three different methods. Draw your conclusion.

METHOD 1 : Using calculator

σ = 66.323

σ2 = 4398.81

METHOD 2 : Using formula 1

Pulse

rate

Frequency

, f

Midpoi

nt , x

x2 fx2 Fx

21

71 – 80 1 75.5 5700

.25

5700.

25

75.5

81 – 90 3 85.5 7310

.25

2193

0.75

342

91 – 100 9 95.5 9120

.25

8208

2.25

895.5

101 –

110

14 105.5 1113

0.25

1558

23.50

1477

111 –

120

12 115.5 1334

0.25

1600

83.00

1386

121 –

130

8 125.5 1575

0.25

1260

02.00

1004

131 –

140

2 135.5 1836

0.25

3672

0.50

271

141 –

150

1 145.5 2117

0.25

2117

0.25

154.5

22

= 200.0055

METHOD 3 : Using formula 2

Pulse

rate

Frequency

, f

Midpoi

nt , x

x2 fx2 Fx

71 – 80 1 75.5 5700

.25

5700.

25

75.5

81 – 90 3 85.5 7310

.25

2193

0.75

342

91 – 100 9 95.5 9120

.25

8208

2.25

895.5

101 –

110

14 105.5 1113

0.25

1558

23.50

1477

111 – 12 115.5 1334 1600 1386

23

120 0.25 83.00

121 –

130

8 125.5 1575

0.25

1260

02.00

1004

131 –

140

2 135.5 1836

0.25

3672

0.50

271

141 –

150

1 145.5 2117

0.25

2117

0.25

154.5

= 746.67

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(a) Resting Heart Rate

Resting Heart Rate (RHR) is the number of beats for 60 seconds which is

done during the morning ( after getting up from sleep) before doing any

exercise.

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My Resting Heart Rate is 60. This is suit for an adult due to the normal

resting heart rate ranges for adults from 60 to 100 beats per minute.

(b) Maximum Heart Rate

Maximum Heart Rate (MHR) = 220 – age.

Target Heart Rate (THR) = (MHR – RHR) x 0.6 + RHR - lower limit

Target Heart Rate (THR) = (MHR – RHR) x 0.8 + RHR - upper limit

MHR = 220 – 17

= 203

THR1 = (203 – 60) x 0.6 + 60

= 145.8

THR2 = (203 – 60) x 0.8 + 60

= 174.4

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(c) Pulse rate for another persons.

Nu

mber

Person RHR MHR THR

Upper

limit

Lower

limit

1 Mother 62 173 150.8 128.6

2 Father 65 185 161 137

3 Teacher1 61 175 152.2 129.4

4 Teacher2 60 177 153.6 130.2

5 Athlete 43 203 171 139

6 Non-athlete 69 203 176.2 149.4

(d) Conclusion about the level of fitness and lifestyle.

Pulse rates vary from person to person. The pulse is lower when the

person is at rest and increases when the person is doing exercise because

more oxygen-rich blood is needed by the body when in exercise.

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Many things can cause changes in the normal heart rate, including age,

activity level, and the time of day.

The target heart rate can guide people how hard he should exercise so he

can get the most aerobic benefit from his workout.

The pulse rate can be used to check overall heart health and fitness level.

Generally lower pulse rate is better.

Keep in mind that many factors can influence heart rate, including:

Activity level

Fitness level

Air temperature

Body position

Emotions

Body size

Medication use

Age

etc

Although there's a wide range of normal heart rate, an unusually high or

low heart rate may indicate an underlying problem. Consult any doctor if the

resting heart rate is consistently above 100 beats per minute (tachycardia)

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or below 60 beats per minute (bradycardia); especially if a person having

other signs or symptoms, such as fainting, dizziness or shortness of breath.

Some people gain the most benefits and lessen the risks when they

exercise in the target heart rate zone. Usually this is when their exercise

heart rate (pulse) is 60 percent to 80 percent of their maximum heart rate.

To find out if a person exercised in their target zone which is between 60

percent and 80 percent of their maximum heart rate, stop exercising and

check their 10-second pulse. If their pulse is below the target zone, increase

the rate of exercise. If their pulse is above the target zone, decrease the rate

of exercise.

Last Year, in order to sit for Addmath paper, my friends and I must

complete a project. I don't know about my friends, but I chose to do Project 1

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together with friends. Project 1 is the easiest as it involving SEGAK Test, the

others were mind blowing stuff. In every project, we are required to create a

piece of art that has connection with AddMath. Either poster, symbols,

stories, or a poem, which I have chosen to do...

Just now, I was tidying up my papers, I found the draft of the poem. So

I would like to share it with all of you.

Additional Mathematics,

Are u as easy as a click,

Do u become easier as we speak,

You are the one i seek,

you are the one i need.

Since i ever heard of u,

i become afraid of u

but when i know u,

u attract me out of the blue.

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with u, although it hard to be right,

i try my best not to be out of sight,

to show the light,

and practices at night

now I shall see the light.

and it is so bright...

Don't laugh at my piece of work... but this is how I truly feel about

additional mathemathics. I really love the subject because it felt so good

when we solve the question correctly.

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