The following presentation is an introduction to the Algebraic Methods – part one for level 4 Mathematics. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
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Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
Chapter 5 part2- Sampling Distributions for Counts and Proportions (Binomial ...nszakir
Mathematics, Statistics, Sampling Distributions for Counts and Proportions, Binomial Distributions for Sample Counts,
Binomial Distributions in Statistical Sampling, Binomial Mean and Standard Deviation, Sample Proportions, Normal Approximation for Counts and Proportions, Binomial Formula
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
Chapter 5 part2- Sampling Distributions for Counts and Proportions (Binomial ...nszakir
Mathematics, Statistics, Sampling Distributions for Counts and Proportions, Binomial Distributions for Sample Counts,
Binomial Distributions in Statistical Sampling, Binomial Mean and Standard Deviation, Sample Proportions, Normal Approximation for Counts and Proportions, Binomial Formula
An Integrated Approach to Ensure Viral Vector and Gene Therapy Commercial Rea...Merck Life Sciences
Come learn more about our integrated approach to ensure viral vector and gene therapy commercial readiness. We will discuss topics relating to process development for viral vector manufacturing, biosafety testing and commercial readiness.
Significant progress has been made for the use of viral vectors for gene therapy. Promising clinical trial results as well as recent FDA approval for CAR-T cell therapy to treat certain children and young adults with B-cell lymphoblastic leukemia have signaled advancements in the field. This marks a historic action, providing opportunities for new viral vector technologies to transform medicine and the way patients are treated and even cured. The need for process development for viral vector manufacturing to improve yield to meet patient demand, biosafety testing for product characterization, potency and safety and commercial readiness to accelerate therapy to-market are critically important. Here, we emphasis an integrated approach that allows our customers solutions to ensure viral vector and gene therapy commercial readiness to meet the growing market need.
In this webinar, you will learn:
● Process development advances for production scale-up of viral vectors for gene therapy
● Methods specific for viral gene therapy product characterization, purity, potency, safety and release testing
● Commercial readiness through our US and UK Centers of Excellence for viral product manufacturing
Detail Description about Probability Distribution for Dummies. The contents are about random variables, its types(Discrete and Continuous) , it's distribution (Discrete probability distribution and probability density function), Expected value, Binomial, Poisson and Normal Distribution usage and solved example for each topic.
Accounting for uncertainty is a crucial component in decision making (e.g., classification) because of ambiguity in our measurements.
Probability theory is the proper mechanism for accounting for uncertainty.
The following presentation is a part of the level 4 module -- Electrical and Electronic Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
The following presentation is a part of the level 4 module -- Electrical and Electronic Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
The following presentation is a part of the level 4 module -- Digital Logic and Signal Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
The following presentation is a part of the level 4 module -- Digital Logic and Signal Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
The following presentation is a part of the level 4 module -- Digital Logic and Signal Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
The following presentation is a part of the level 4 module -- Digital Logic and Signal Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
The following presentation is a part of the level 4 module -- Digital Logic and Signal Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
The following presentation is a part of the level 4 module -- Digital Logic and Signal Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
The following presentation is a part of the level 4 module -- Digital Logic and Signal Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
The following presentation is a part of the level 4 module -- Digital Logic and Signal Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
The following presentation is a part of the level 5 module -- Electronic Engineering. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
The following presentation is a part of the level 5 module -- Electronic Engineering. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
The following presentation is a part of the level 4 module -- Digital Logic and Signal Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
The following presentation is an introduction to the Algebraic Methods – part one for level 4 Mathematics. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
More from School of Design Engineering Fashion & Technology (DEFT), University of Wales, Newport (14)
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
3. The Probability Distribution Before discussing the Distribution namely: The Binomial Distribution The Poisson Distribution The Normal Distribution We need first to discuss elementary probability and how to write down probability statements and how to evaluate them. Probability
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6. Definition of Probability When all the equally possible occurrences have been enumerated, the probability Pr of an event happening is the ratio of the number of ways in which the particular event may occur to the total number of possible occurrences. Probability Statements. Suppose the Pr of a machine producing a defective item is known then the Pr that in a random sample of eight items produced by the machine, the Pr of not more than 2 being defective will be given by the probability statement: Pr(not more than 2 being defective) = P(≤2) = P(0) + P(1) + P(2) i.e. P(no defects) + P(1 defect) + P(2 defects)
7. Again the chance or probability of say at least 3 items being defective would be given by: P(≥3) = P(3) + P(4) + . . . + P(8) - - - - (A) Or since the total Pr = 1 we could write (A) as: P(≥3) = 1 – [P(0) + P(1) + P(2)] Note: This is a very useful way of writing (A) and one we will be using regularly in our work. Probability
8. The Binomial Distribution Before discussing the binomial distribution we need to look at the binomial theorem for a positive integer as a power. (a + b) 2 = (a + b)(a + b) = a 2 + ab + ba + b 2 = a 2 +2ab + b 2 (a + b) 3 = (a + b)(a 2 +2ab = b 2 ) = a 3 + 3a 2 b + 3ab 2 + b 3 This can be extended and the multipliers form Pascal’s triangle as below Each number is the sum of the two numbers above it. Probability 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 (a+b) 0 (a+b) 1 (a+b) 2 (a+b) 3 ……
9. The Binomial Distribution In general the Binomial Expansion of when n is a positive integer is given by: (binomial theorem) Consider the expansion of ( ¾ + ¼) 5 here a = ¾, b = ¼ and n = 5 Exercise Write down the first 3 terms of the Binomial Expansion of ( ⅞ + ⅛ ) 6 and evaluate their values by the use of a calculator. [ 0.4488, 0.3847, 0.1374] Probability
10. We can now consider the Binomial Distribution Coin tossing and dice rolling have been used as a means of introducing probability. The consideration of how a coin falls can conveniently illustrate some fundamental probability theory. We must assume that we are dealing with “fair coins” i.e. not two heads or coins that are in any way biased – there is one chance in two or a Pr of 0.5 of obtaining a head and a Pr of 0.5 of obtaining a tail. The development of the Binomial Distribution cab be illustrated by the following simple example of coin tossing. Consider three coins being tossed simultaneously – There are eight possible head (H) and tail (T) combinations: Probability
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12. Now consider the binomial expansion of ( ½ + ½) Where ( ½ + ½ ) Probability of tails Probability of heads i.e. Pr of Failure i.e. Pr of Success when the coin is tossed Pr (success) + Pr (failure) = 1 = ⅛ + ⅜ + ⅜ + ⅛ The Pr’s agree with those found above, namely ⅛, ⅜, ⅜ and ⅛ of obtaining no heads, 1 H, 2 H and 3 H respectively when three coins are tossed simultaneously. Probability
13. We could repeat this for the tossing of four coins. It is clear that with such probability distributions (known as binomial distributions) that the probability results can be obtained by using the binomial distribution. General Binomial Distribution In the binomial distribution of Where p = Pr of the event happening or success q = Pr of the event not happening or failure n = Number of tests Where p + q = 1 Then the terms of the binomial expansion namely: Give the respective Pr’s of obtaining 0, 1, 2, 3, etc events happening or successes. Note the order Probability
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16. 2. The probability of passing an examination is 0.7. Determine the Pr that out of 8 students (a) just 2 (b) more than two will pass the examination. p = Pr of the event happening or success [0.7] q = Pr of the event not happening or failure [0.3] n = Number of tests [8] Pr (>2 pass) = 1 – [Pr (0 pass) + Pr (1 pass) + Pr (2 pass)] Pr (>2 pass) = 1 – [0.00006561 + 0.00122472 + 0.01000188] Pr (>2 pass) = 1 – 0.01129 = 0.98871 = 98.87% Probability
17. 3. The incidence of occupational disease in an Industry is such that the workmen have a 25% chance of suffering from it. What is the probability that in a random sample of 7 workmen (a) no one (b) not more than 2 (c) at lease 3 will contract the disease. p = Pr of the event happening or success [0.25] q = Pr of the event not happening or failure [0.75] n = Number of tests [7] Pr (not 2+ dis) = Pr (0 dis) + Pr (1 dis) + Pr (2 dis) Pr (≤2 inf) = 0.1335 + 0.3115 + 0.3115 = 0.7565 = 75.65% Pr (≥3 inf) = 1 - Pr (≤2 inf) = 1 – 0.7565 = 0.2435 = 24.35% Probability
18. The Mean and Standard Deviation of the Binomial Distribution The mean or average of a Binomial Distribution λ = n x p The Standard Deviation SD = Distribution about the mean value Example The probability of obtaining a defective resistor is given by 1/10 In a random sample of 9 resistors what is the mean number of defective resistors you would expect and what is the standard deviation? Mean = n x p = 9 x 0.1 = 0.9 SD = √(9x0.1x0.9) = √0.81 = 0.9 Probability
21. Since the Poisson Law has only one constant, namely λ it is easier to tabulate than the Binomial Law which has two independent constants, namely n and p. For the Poisson Law the mean value λ like the Binomial Law is n p i.e. λ = n p Also the Standard Deviation S.D. σ = √(n p q), since q is nearly 1 because p is very small σ = √(n p), = √ ( λ ) In using the Poisson Law you only have to find one thing, namely the average value λ before you can use it. Note (i) Either you will be told the value of the mean λ in a given question. Or (ii) You can calculate it by using the result λ = n p Probability
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24. p = Pr of the event happening or success [0.03] q = Pr of the event not happening or failure [0.97] n = Number of tests [100] Pr (>2 def) = 1 – [Pr (0 pass) + Pr (1 pass) + Pr (2 pass)] Pr (>2 def) = 1 – [0.04755 + 0.14707 + 0.22515] Pr (>2 pass) = 1 – 0.41977 = 0.5802 = 58.02% Probability
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28. The Normal Distribution This was first introduced in 1750 by the mathematician De Moivre, through probability theory. Let us consider a simple example of his work. Suppose we toss 10 coins simultaneously, then the expected probability of obtaining 0, 1, 2, 3, … 10 heads is given by the Binomial Expansion of ( ½ + ½) 10 . On expansion we have: Pr(0H) Pr(1H) Pr(2H) etc If we now plot the distribution of the throws of the 10 coins in the form of a frequency polygon, we obtain the diagram on the next slide.
29. Bell Shaped Curve The distribution is symmetrical about the Mean Value of 5T 5H. If the number of coins were increased to a very large number the distribution approximates to the smooth curve shown on the next slide.
30. De Moivre called this the Normal Curve Since Head and Tail arrangements are discrete i.e. you cannot have fractional values only whole numbers, like 1H or 2H – you cannot have 2 ¾H, the curve is not truly continuous. However 50 years later Gauss obtained a truly continuous Normal Curve. Probability
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35. STATISTICAL TABLES Area under the Normal Curve For a normal variable x, having a mean μ and a standard deviation σ , the values tabulated give the area under the curve between the mean μ and any multiple of the standard deviation above the mean. The results are given for values of (x – μ )/ σ , the Standard Unit Variable (z), at intervals of 0.01. The corresponding areas for deviations below the mean can be found by symmetry. 0 z Probability
37. Use of the Normal Curve and Normal Probability Tables Suppose a variable is normally distributed about a mean value of 10 with a standard deviation of 0.5. Suppose we want to find the probability of a variable in the distribution having a value between 10.7 and 11.2 To do this we need to find the corresponding area under the normal curve. (As shown) This can be done by standardising the given values of 10.7 and 11.2 – we evaluate each as a Standard Unit Variable z where Probability
38. When x = 10.7 When x = 11.2 We can now refer to the Table using the values of z of 1.40 and 2.40. The area between 10.7 and 11.2 will be the area up to 11.2 minus the area up to 10.7 Area = z 2.40 – z 1.40 = 0.4918 – 0.4192 = 0.0726 7.26% of all the variables in this Normal Distribution lie between x = 10.7 and x = 11.2 Probability
39. Since all the probability questions in a Normal Distribution are equivalent to areas under the Normal Curve (which can be obtained from the table) let us now look at a few examples in evaluating areas under the Normal Curve. Example 1. Find the area under the normal curve between z = -1.36 and z = 0.85 or z -1.36 and z 0.85 A B Putting in the Normal Curve gives us: Area A 0 -1.36 A = 0.4131 Area B 0 +0.85 B = 0.3023 Total Area = 0.7154
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41. Practical Problems on the Normal Distribution 1. The mean of the mass of a set of components is 151Kg and the standard deviation is 15Kg. Assuming that the masses are normally distributed about the mean, find the probability of a single component having a mass of (a) between 120 and 155Kg and (b) greater than 186Kg. (a) μ = 151Kg and σ = 15Kg – we need to calculate z for each value at the extremes of the range. When x = 120Kg z 1 = (120 – 151)/15 = -2.07 When x = 155Kg z 2 = (155 – 151)/15 = 0.27 Probability
42. Pr mass between 120Kg and 155Kg is the sum of the areas to the left and the right of the centre. Area = 0.4808 + 0.1064 Area = Pr = 0.5872 (b) To find Pr mass >185Kg z = (185 – 151)/15 = 2.27 Pr of having a mass >185 is the shaded area. This is 0.5 – z 2.27 Area = 0.5 – 0.4884 Area = 0.0116 Probability