ADDITIONAL MATHEMATICS
PROJECT WORK
2012
NAME
: NUR IFWATUL FAIQAH KASWADI
CLASS
: 5 CAMBRIDGE
MATRIX NO. : 11506
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N
O.
TITLE
PAG
E
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INTRODUCTION
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2
OBJECTIVES
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3
HISTORY
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4
PROJECT TASK
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5
FURTHER EXPLORATION
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6
REFLECTION
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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.
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.
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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;
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;
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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 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
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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 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
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be 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 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
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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.
Students Pulse rate ( bpm - beats per min)
Before After 1 2 . . . 50 Table 1
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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.
(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.
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ANSWERS: Pulse Rate of 50 Students Before and After The Step Up Board Activity.
Students Pulse rate (bpm – beats per min) Before After 1 82 115 2 57 128 3 73 123 4 75 105 5 64 117 6 72 114 7 119 141 8 69 114 9 79 130 10 83 117 11 75 116 12 86 130 13 56 96 14 69 150 15 87 122 16 64 91 17 81 86 18 96 110 19 56 81 20 80 120 21 66 91 22 82 110 23 66 80 24 75 120 25 82 102 26 75 104 27 77 120 28 79 109 29 82 112 30 77 93 31 80 110 32 72 97 33 85 117 34 75 107 35 69 91 36 83 122 37 84 121 38 91 130 39 68 88 40 73 103 41 75 110 42 82 112 43 66 99 44 70 101 45 72 105 46 76 108 47 92 141 48 68 91 49 75 98 50 81 105
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(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+ 82+83+83+84+85+86+87+91+92+96+119) 50 = 76.42 Mode : 75 Median : 56,56,57,64,64,66,66,66,68,68,69,69,69,70,72,72,72,73,73,75,75,75,75,75,75,75,7 6, 77,77,79,79,80,80,81,81,82,82,82,82,82,83,83,84,85,86,87,91,92,96,119.
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=
= 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.
= 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.
13 1) Mean = 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+122+122+123+128+130+130+130+141+141+150 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, 110,110,110,112,112,114,114,115,116,117,117,117,120,120,120,121,122,122,123,128, 130,130,130,141,141,150 =
= 110
(c) Construct a frequency distribution table for the pulse rate after the step up board activity using a suitable class interval.
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PULSE RATE (bpm) FREQUENCY
71 – 80 1 81 – 90 3 91 – 100 9 101 – 110 14 111 – 120 12 121 – 130 8 131 – 140 2 141 – 150 1
(i) Represent your data using three different statistical graphs based on your frequency table. Bar Chart FREQUE NCY 0 2 4 6 8 10 12 14 16 71 – 80 81 – 90 91 – 100 101 – 110 111 – 120 121 – 130 131 – 140 141 – 150
15 Histogram Frequency polygon Ogive 0 2 4 6 8 10 12 14 16 75.5 85.5 95.5 105.5 115.5 125.5 135.5 145.5 FR EQU EN C Y PULSE RATE (bpm) 0 2 4 6 8 10 12 14 16 1 65.5 2 75.5 3 85.5 495.5 5 105.5 6115.5 7 125.5 8135.5 9145.5 10155.5 PULSE RATE (bpm) Freque ncy
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(ii) Determine the mean, mode and median of the pulse rate by using appropriate method. Mean = x =
f f x 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 0 10 20 30 40 50 60 0 2 4 6 8 10CULMULATIVE FREQUENCY
CULMULATIVE FREQUENCY 110.5 100.5 90.5 80.5 70.5 120.5 130.5 140.5 150.5 CUMULATIVE FREQUENCY PULSE RATE17 = 109.5 Mode = 107 Median = c f F N L m 2 10 9 13 2 50 5 . 90 = 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.
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Mean, mode and median 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
Measure of Central Tendency
Ungrouped data Grouped data
Mean 110.06 109.5
Mode 110 107
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METHOD 2 : Using formula 1
Pulse rate Frequency , f Midpoint
, x x2 fx2 Fx 71 – 80 1 75.5 5700. 25 5700.2 5 75.5 81 – 90 3 85.5 7310. 25 21930. 75 342 91 – 100 9 95.5 9120. 25 82082. 25 895.5 101 – 110 14 105.5 11130 .25 155823 .50 1477 111 – 120 12 115.5 13340 .25 160083 .00 1386 121 – 130 8 125.5 15750 .25 126002 .00 1004
20 131 – 140 2 135.5 18360 .25 36720. 50 271 141 – 150 1 145.5 21170 .25 21170. 25 154.5 2 2 2 ( ) fx x f = 200.0055
METHOD 3 : Using formula 2
Pulse rate Frequency , f Midpoint
, x x2 fx2 Fx 71 – 80 1 75.5 5700. 25 5700.2 5 75.5
21 81 – 90 3 85.5 7310. 25 21930. 75 342 91 – 100 9 95.5 9120. 25 82082. 25 895.5 101 – 110 14 105.5 11130 .25 155823 .50 1477 111 – 120 12 115.5 13340 .25 160083 .00 1386 121 – 130 8 125.5 15750 .25 126002 .00 1004 131 – 140 2 135.5 18360 .25 36720. 50 271 141 – 150 1 145.5 21170 .25 21170. 25 154.5 fx x f
22 2 2 f x( x) f = 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.
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
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THR1 = (203 – 60) x 0.6 + 60
= 145.8
THR2 = (203 – 60) x 0.8 + 60
= 174.4
(c) Pulse rate for another persons.
Num ber 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
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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.
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
26 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) 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.
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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 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,
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Since i ever heard of u,
i become afraid of u
but when i know u,
u attract me out of the blue.
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.
