Additional Mathematics Project Work 2015 Contents B i l Title Page 1 Title :”Probability distribution and its use in solving problem in real life situations” 2 2 Appreciation 3 3 Part 1 Introductions 4-6 4 Part 2 4.1Result of the survey 4.2Frequency distribution table: Representation Mean and standard deviation for the weight Ogive 4.3Assuming the weight is normally distributed 7-14 5 Part 3: 5.1BMI for each student 5.2Mean and standard deviation for the BMI Ogive 5.3Assuming the BMI normally distributed 15-19 1
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Additional Mathematics Project Work 2015
ContentsBil Title Page1 Title :”Probability distribution and
its use in solving problem in real life situations”
2
2 Appreciation 33 Part 1
Introductions4-6
4 Part 24.1Result of the survey 4.2Frequency distribution table: Representation Mean and standard
deviation for the weight
Ogive4.3Assuming the weight is normally distributed
7-14
5 Part 3:5.1BMI for each student5.2Mean and standard deviation for the BMI
Ogive5.3Assuming the BMI normally distributed
15-19
6 Further Exploration 207 Reflection 21
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Additional Mathematics Project Work 2015
2
Title
Additional Mathematics Project Work 2015
Appreciation:
I would like to thank my Additional Mathematics teacher Puan Nor Liza Binti Mohammad for explaining to us and guide us throughout this project work.Without her , I wouldn’t have been able to do this project so smoothly.
I would also like to take this opportunity to thank my friends Teo Feng Tian ,Eileen Toh for all cooperating in searching for the information ,obtaining the data , sharing their ideas and helping out in writing this report . I would like to thank them for their full commitment in doing this project work .
Lastly I would like to thank my family members. Especially my parents for their moral and financial support.
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Appreciation
Additional Mathematics Project Work 2015
Go to internet to explore “Probability distribution , binomial distribution , normal distribution “
Decisions made by government , business , education department , sporting bodies ,etc , are often made after careful consideration of statistical evidence . Statistics play a vital role in many areas of our society . Statistics are a tool for helping to made rational decisions about variables described by data sets.
Amongst other things , governments use statistics to help formulate future policies . Businesses often use statistics to aid decision making ,for example , whether or not to enter the market with an alternative to a product when there are already several of them on the market . Statistical information about sport has increased dramatically in recent years. We only need to watch a ‘one day international ‘ cricket match to observe the many statistical graph and tables used to help make the viewer more informed.
Following are some examples of the types of problems we may face , and where statistical methods may help to answer them:
A younger executive of a hotel chain claimed that lowering the room tariff by 10% would increase the patronage by 20%. Would this be true ?
A manufacturer of AAA batteries claimed that her batteries outlasted all other leading brands by at least 100 hour . Is she correct?
Does the unemployment rate affect the crime rate fir that city? An employer claims that younger employees (<30 years) have on
average twice as many sick days as older ones(≥30 years) Which drug for helping to quit smoking has the greatest chance of
success.
The binomial probability model:
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Introduction
Additional Mathematics Project Work 2015
The binomial probability model is a sequence of trials, each of which consist of repetition of a single experiment .We assume the trials to be independent . Furthermore we assume that there are only two possible outcome for each trial that is success or failure . Many real-world situation have the characteristics of the binomial probability model.
Examples of binomial distribution being applied to real life problems:1. A machine produce light bulbs to meet certain specification ,and 80% of the
bulbs produced meet there specification . A sample of 6 bulbs is taken from the machine’s production .What is the probability that 3 or more of them fail to meet specification?
2. Suppose that 60% of the voters intend to vote for a conservative candidate . What is the probability that a survey poiling 8 people reveals that 3 or fewer intend to vote for a conservative candidate?
3. In a 15 item true-false examination , what is the probability that a student who guesses on each question will get at least 10 correct answer ? If another student has 0.8 probability of correctly answering each question , what is the probability that this student will answer at least 12 questions correctly ?
The normal distribution model :
Frequency distributions can assume almost any shape or form , depending on the data . However , the data obtained from many experiments often follow a common pattern which has been thoroughly investigated . For example , heights of people , weight of people , test scores and coin tossing all lead to data which have the same kind of frequency distribution is referred to as normal distribution or the Gaussian distribution. The graph of the normal distribution , called the normal distribution , called the normal curve , is the bell-shaped curve shown below:
Examples of normal distribution being applied to real life problems:
A company manufactures 60000 pencils each day . Quality control studies have shown that on the average, 4% of the pencils are defective . A random sample of 500 pencils is selected from each day’s production .What is the probability that in the sample there are at least 12 and no more 24 defective pencils ?
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Additional Mathematics Project Work 2015
The average height of 2000 woman in a random sample is 158 cm . The standard deviation is 6 cm. The heights have a normal distribution . How many woman are between 155 cm and 162 cm?
Records shows that the average life expectancy of a pair of shoes is 2.2 years with a standard deviation of 1.7 years . A manufacturer guarantees that shoes lasting less than a year are replaced free. For every 1000 shoes sold , how many shoes should the manufacturer expect to replace free?
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Part 2
Additional Mathematics Project Work 2015
1. A survey on the distribution of height and weight of 50 students in my school are shown in the table below:
From the ogive , percentage of students with weight more than 60 kg
=18.550
×100=37 %
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Additional Mathematics Project Work 2015
3. The weight of the students are normally distributed with mean 56.46 and standard deviation =10.381 that is x-N( 56.46,10.381)
i. Percentage of students with weight more than 60 kg
P(x>60)=P(z>60−56.46
10.381 )=P(z>0.3410)=0.3666
36.66% of the students weight more than 60 kg ii. Percentage of students with weight less than 45 kg
P(x<45)=P(z<45−56.46
10.381 )=P(z<-1.1039)=0.1348
13.48% of students weight less than 45 kgiii. The value of m if 90% students has weight more than m kg
P(x>m)=0.9
P(z>m−56.46
10.381 )=0.1
P(z>-m−56.46
10.381 )=0.1
-m−56.46
10.381 =1.281
m=43.16kg
90%of the students weight more than 43.16kg
4. The conclusion we obtained regarding the percentage of students weighting more than 60 kg using ogive is 37% and using normal distribution is 36.67%.Therefore they are the the same using two different methods.
5. P(x=3)=10∁ 3 (0.3667 )(0.6333)=0.2418
If 10 students are picked, the probability that 3 students have weight more than 60 kg is 0.2418.
6 My school has 800 students . The number of students with weight more than 60 kg is estimated to be
Here are more tips from Dawn Jackson Blatner, RD, author of The Flexitarian
Diet : Eat vegetables to help you feel full. Drink plenty of water. Get tempting foods out of your home. Stay busy -- you don't want to eat just because you're bored. Eat only from a plate, while seated at a table. No grazing in front of the 'fridge. Don't skip meals.
Keeping a food journal -- writing down everything you eat -- can also help you
stay on track.
We can also adopt a healthier lifestyle by following these easy step:
1. Think positive and focus on gratitude
2. Eat your vegetables
3. Set a “5-meal ideal”
4. Exercise daily
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Additional Mathematics Project Work 2015
5. Get at good night's sleep
One famous mathematician who contributed to the development of normal distribution .
Kari Frederik Gauss examined errors in measurement and found an appropriate error measurement function . It was still is agreed that the characteristic of a measurement error function are :
The measurement errors of the same size but of opposite sign are equally likely
Small errors are more likely than larger ones , and extremely larger errors are very unlikely,
The total area under the function is 1 The most likely measurement is the mean of the distribution.
Gauss was able to prove that the normal distribution is the only error function that satisfies all four conditions.
The normal distribution is fundamental in the study of probability and statistics , partly because it is the model for errors in measurement .Because of the significance of Gauss’ fundamental discovery, this distribution is also called the Gaussian distribution.
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Further Exploration
Additional Mathematics Project Work 2015
From doing this project , we have learn the following skills and values:
(a) Use what we learn in the class to solve real life problemWe learned from the chapter 6 and 7 and 8 ,it helps us to calculate the question by using the formulae.
(b) Importance of teamwork.We work like a team to search for the information. Our friendship is being strengthened
(c) Creative in presenting the resultsBy using the table, bar chart ,line graph to present the results is more innovating. The data is shown more clearly and easy to be understand .
(d) Using technology to help us made our work easier.For example , we calculate the answers by setting the formulae in the Microsoft Word. Then, the scientific calculator also helps us a lot in doing this project. So ,we are to get the answer more accurately.