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.
2
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;
3
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
4
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
5
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
6
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
7
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.
8
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
24
(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.
25
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
26
(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.
27
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)
28
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
29
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.
30
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.
31
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