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Boundary value problem

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator.

To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.

Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle.

Initial value problem

A more mathematical way to picture the difference between an initial value problem and a boundary value problem is that an initial value problem has all of the conditions specified at the same value of the independent variable in the equation (and that value is at the lower boundary of the domain, thus the term "initial" value). On the other hand, a boundary value problem has conditions specified at the extremes of the independent variable. For example, if the independent variable is time over the domain [0,1], an initial value problem would specify a value of y(t) and y'(t) at time t=0, while a boundary value problem would specify values for y(t) at both t=0 and t=1.

If the problem is dependent on both space and time, then instead of specifying the value of the problem at a given point for all time the data could be given at a given time for all space. For example, the temperature of an iron bar with one end kept at absolute zero and the other end at the freezing point of water would be a boundary value problem.

Concretely, an example of a boundary value (in one spatial dimension) is the problem

y(x)+y(x)=0 \,

to be solved for the unknown function y(x) with the boundary conditions

y(0)=0, \ y(\pi/2)=2.

Without the boundary conditions, the general solution to this equation is

y(x) = A \sin(x) + B \cos(x).\,

From the boundary condition y(0)=0 one obtains

0 = A \cdot 0 + B \cdot 1

which implies that B=0. From the boundary condition y(\pi/2)=2 one finds

2 = A \cdot 1

and so A=2. One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case is

y(x)=2\sin(x). \,

Types of boundary value problems

If the boundary gives a value to the normal derivative of the problem then it is a Neumann boundary condition. For example, if there is a heater at one end of an iron rod, then energy would be added at a constant rate but the actual temperature would not be known.

If the boundary gives a value to the problem then it is a Dirichlet boundary condition. For example, if one end of an iron rod is held at absolute zero, then the value of the problem would be known at that point in space.

If the boundary has the form of a curve or surface that gives a value to the normal derivative and the problem itself then it is a Cauchy boundary condition.

Aside from the boundary condition, boundary value problems are also classified according to the type of differential operator involved. For an elliptic operator, one discusses elliptic boundary value problems. For an hyperbolic operator, one discusses hyperbolic boundary value problems. These categories are further subdivided into linear and various nonlinear types.

Algebraic statistics

Algebraic statistics is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing. Traditionally, algebraic statistics has been associated with the design of experiments and multivariate analysis (especially time series). In recent years, the term "algebraic statistics" has been sometimes restricted, sometimes being used to label the use of algebraic geometry and commutative algebra in statistics. The tradition of algebraic statistics In the past, statisticians have used algebra to advance research in statistics. Some algebraic statistics led to the development of new topics in algebra and combinatorics, such as association schemes. Design of experiments For example, Ronald A. Fisher, Henry B. Mann, and Rosemary A. Bailey applied Abelian groups to the design of experiments. Experimental designs were also studied with affine geometry over finite fields and then with the introduction of association schemes by R. C. Bose. Orthogonal arrays were introduced by C. R. Rao also for experimental designs. Algebraic analysis and abstract statistical inference Invariant measures on locally compact groups have long been used in statistical theory, particularly in multivariate analysis. Beurling's factorization theorem and much of the work on (abstract) harmonic analysis sought better understanding of the Wold decomposition of stationary stochastic processes, which is important in time series statistics. Encompassing previous results on probability theory on algebraic structures, Ulf Grenander developed a theory of "abstract inference". Grenander's abstract inference and his theory of patterns are useful for spatial statistics and image analysis; these theories rely on lattice theory. Partially ordered sets and lattices Partially ordered vector spaces and vector lattices are used throughout statistical theory. Garrett Birkhoff metrized the positive cone using Hilbert's projective metric and proved Jentsch's theorem using the contraction mapping theorem. Birkhoff's results have been used for maximum entropy estimation (which can be viewed as linear programming in infinite dimensions) by Jonathan Borwein and colleagues. Vector lattices and conical measures were introduced into statistical decision theory by Lucien Le Cam. Recent work using commutative algebra and algebraic geometry In recent years, the term "algebraic statistics" has been used more restrictively, to label the use of algebraic geometry and commutative algebra to study problems related to discrete random variables with finite state spaces. Commutative algebra and algebraic geometry have applications in statistics because many commonly used classes of discrete random variables can be viewed as algebraic varieties. Introductory example Consider a random variable X which can take on the values 0, 1, 2. Such a variable is completely characterized by the three probabilities p_i=\mathrm{Pr}(X=i),\quad i=0,1,2 and these numbers clearly satisfy \sum_{i=0}^2 p_i = 1 \quad \mbox{and}\quad 0\leq p_i \leq 1. Conversely, any three such numbers unambiguously specify a random variable, so we can identify the random variable X with the tuple (p0,p1,p2)∈R3. Now suppose X is a Binomial random variable with parameter p = 1 − q and n = 2, i.e. X represents the number of successes when repeating a certain experiment two times, where each experiment has an individual success probability of q. Then p_i=\mathrm{Pr}(X=i)={2 \choose i}q^i (1-q)^{2-i} and it is not hard to show that the tuples (p0,p1,p2) which arise in this way are precisely the ones satisfying 4 p_0 p_2-p_1^2=0.\ The latter is a polynomial equation defining an algebraic variety (or surface) in R3, and this variety, when intersected with the simplex given by \sum_{i=0}^2 p_i = 1 \quad \mbox{and}\quad 0\leq p_i \leq 1, yields a piece of an algebraic curve which may be identified with the set of all 3-state Bernoulli variables. Determining the parameter q amounts to locating one point on this curve; testing the hypothesis that a given variable X is Bernoulli amounts to testing whether a certain point lies on that curve or not.

From Yahoo Answers

Question:The time required in minutes for each of the 50 students to read 20 pages of a book is recorded below: 43 52 47 49 36 48 41 47 50 32 45 48 40 43 48 36 51 44 49 53 37 34 42 47 45 47 44 50 31 48 43 45 44 36 51 51 43 53 46 39 50 42 42 47 38 49 46 40 38 45 With the information, i) Prepare a frequency distribution table. ii) Draw a histogram. iii) Draw a frequency polygon plotting points please up load it to www.ripway.com then give me the link to the file.... thx a bunch ^_^ HERE SOME SAMPLE CLICK LINK BELOW http://h1.ripway.com/bookworms/SAMPLE.docx

Answers:From a previous answer you have the answer to part i) of your question, namely the frequency distribution table: 31-35 3 36-40 9 41-45 15 46-50 17 50-55 6 55-60 0 The data was grouped into six bins (classes). The first class is from 31 to 35 minutes and there are 3 students in that bin. Therefore the frequency is 3. It appears that the data were rounded to the nearest minute and so a time 30.5 minutes would be rounded to 31 minutes. Similarly, any time less than 35.5 minutes but equal to or greater than 34.5 minutes would be rounded to 35 minutes. Therefore that first class really includes values from 30.5 to 35.5 (called the class interval). The next class is really from 35.5 to 40.5, and so on. In part (ii) you draw a histogram with class boundaries 30.5, 35.5, 40.5, 45.5 and so on. See the source below for the explanation. In part (iii) you plot a graph (a frequency polygon) by plotting the frequencies at the midpoint of each class. For example, the first point is plotted at frequency = 3 on the y-axis and time = 33 minutes on the x-axis.

Question:Maybe someone can explain this better to me. My calculations are not coming out right. I need to create a frequency histogram and relative frequency histogram, however before I can construct those graphs I need to get my class frequencies right. For some reason, my answers keep coming out wrong. I asked my instructor about it and he also told me that I am wrong and the book is right, but he didn't elaborate (online class). Here are the math exercises from my book: In Exercises 3 and 4, use the following data set.The data represent the actual liquid volume (in ounces) in 24 twelve-ounce cans. 11.95 11.91 11.86 11.94 12.00 11.93 12.00 11.94 12.10 11.95 11.99 11.94 11.89 12.01 11.99 11.94 11.92 11.98 11.88 11.94 11.98 11.92 11.95 11.93 3. Make a frequency histogram using seven classes. 4. Make a relative frequency histogram of the data set using seven classes. ---------------------------------- I determined that the class range was .04. (Since the upper limit is 12.10 and the lower limit is 11.86. So 12.10 - 11.86 = .24 and 24/7 = .034 but you round up to .04) Now when creating my classes I got the following (Sorry about the underscore, it was the only way I could space them out): Class__________MyFrequencies______Book's Answer 11.86 - 11.89_________3_______________2 11.90 - 11.93_________5_______________4 11.94 - 11.97_________8______________10 11.98 - 12.01_________7_______________6 12.02 - 12.05_________0_______________1 12.06 - 12.09_________0_______________0 12.10 - 12.14_________1_______________1 So here's my problem, the Book reads the frequencies totally different. Why? According to the data set, the book says the first class' frequency is 2. But that doesn't make sense because there are three items that can be included in that class: 11.86 11.88 11.89 So, can someone please explain to me what it is that I am doing wrong. I just don't understand why 3 is not correct. But if I lowered the LCL wouldn't that make inaccurate frequencies? The thing is, I have had no problem determining frequencies. It was just this problem that was giving me problems. So I figured maybe it was an error. In your personal opinion, is the book really right then? If so, can you explain a little better why? My class width is correct. I mean feel free to double check me if you want. But we were taught to round up unless it was a whole number (like 4.00). Thanks Ohno. Sorry if I burdened you. I just wasn't understanding it quite right.

Answers:When you have your class sets ... your highest and lowest classes have your highest and lowest numbers as your LCL.... (i.e. class one LCL = 11.86, and your class seven LCL = 12.1) try different ways of setting up the class boundaries so that each item gets spaced in..... EX. (your classes are set up like this) 11.86 - 11.89 11.90 - 11.93 11.94 - 11.97 ... and so on.... TRY messing with your class sets, since your class seven has 3 extra numbers that aren't even being used... 11.85 - 11.88 11.89 - 11.92 11.93 - 11.96 so on and so forth until you see that your frequencies match up with the book's ______________________________________________ Edit: I just spent time to do this for setting up class lists with the numbers you listed. If you lower each LCL and UCL by .01, then your answers will match up with the book. 11.85 - 11.88 11.89 - 11.92 11.93 - 11.96 11.97 - 12.00 12.01 - 12.04 12.05 - 12.08 12.08 - 12.11 If you set up those as your class lists, your frequencies will match up with the book.

Question:PLease if anyone can help me with this assignment- email me or reply to this post. The bottom of the assignment has charts that you need to see, but this will not let me post the charts. I am willing to $$$ for the help with this assignment as long as it is reasonable of coure. Please help. I have done some/ most of it, but I am lost at the charts. Please email and I will reply with the actual assignment. 1. Place these variables in the following classification tables. For each table, summarize your observations and evaluate if the results are generally true. For example, salary is reported as a continuous quantitative variable. It is also a continuous ratio scaled variable. a.Salary- b.Gender- c.Sales volume of MP3 players- d.Soft drink preference- e.Temperature- f.SAT scores- g.Student rank in class- h.Rating of a finance professor- i.Number of home computers- 2. What is the level of measurement for each of the following variables? a.Student IQ ratings. b.Distance students travel to class. c.Student scores on the first statistics test. d.A classification of students by state of birth. e.A ranking of students as freshman, sophomore, junior, and senior. f.Number of hours students study per week. 3. What is the level of measurement for these items related to the newspaper business? a.The number of papers sold each Sunday during 2006. b.The departments, such as editorial, advertising, sports, etc. c.A summary of the number of papers sold by county. d.The number of years with the paper for each employee. 4. For each of the following, determine whether the group is a sample or a population. a.The participants in a study of a new cholesterol drug. b.The drivers who received a speeding ticket in Kansas City last month. c.Those on welfare in Cook County (Chicago), Illinois. d.The 30 stocks reported as a part of the Dow Jones Industrial Average. 5. Alexandra Damonte will be building a new resort in Myrtle Beach, South Carolina. She must decide how to design the resort based on the type of activities that the resort will offer to its customers. A recent poll of 300 potential customers showed the following results about customers preferences for planned resort activities: What is the table called? Basic Table Draw a bar chart to portray the survey results. Bar 1- Like planned activities Bar 2- Do not like planned activities Bar 3- Not sure Bar 4- No answer Professor Salazar- I do not know how to use excel or anything to make charts or pie graphs and I could not get a hold of anyone to help me so this is all I could do. I would rather submit this than nothing. I don t know what else I can do. The same for the pie chart. I was able to get the percentages and the correct distance between each color, but no labels. Draw a pie chart for the survey results. Vacation Activities 1.(Purple)- Like planned activities (21%) 2.(Burgundy or red)- Do not like planned activities (45%) 3.(Yellow)- Not sure (26%) 4. (Aqua)- No answer (8%) 6The following chart summarizes the selling price of homes sold last month in the Sarasota, Florida, area. a.What is the chart called? Line chart b.How many homes were sold during the last month? c.What is the class interval? d.About 75 percent of the houses sold for less than what amount? e.One hundred seventy-five of the homes sold for less than what amount? 7.A chain of sport shops catering to beginning skiers, headquartered in Aspen, Colorado, plans to conduct a study of how much a beginning skier spends on his or her initial purchase of equipment and supplies. Based on these figures, it wants to explore the possibility of offering combinations, such as a pair of boots and a pair of skis, to induce customers to buy more. A sample of cash register receipts revealed these initial purchases: a. How many classes would you recommend? b. What class width would you choose? c. Construct a Frequency Distribution table d. Construct a Histogram e. Construct a Frequency Polygon f. Construct a Cumulative Frequency Curve

Answers:b. Gender- Nominal, discrete c. Sales volume of MP3 players- discrete, ratio d.Soft drink preference- ordinal, discrete e.Temperature- continuous , interval f. SAT scores- ratio , discrete (I am assuming the scores are integers (rounded if necessary) g.Student rank in class- ordinal, discrete h. ratio [ ratings are scaled from 1 to 10 or some other number?] ordinal [ ratings are poor, moderate, good, excellent etc] i.Number of home computers- ratio, discrete 2) a. Student IQ ratings --ratio. b. Distance students travel to class--.continuous c. Student scores on the first statistics test.---discrete d. A classification of students by state of birth.-- nominal e. A ranking of students as freshman, sophomore, junior, and senior. -- ordinal f. Number of hours students study per week. --- ratio 3. What is the level of measurement for these items related to the newspaper business? a. The number of papers sold each Sunday during 2006. -- ratio b. The departments, such as editorial, advertising, sports, etc. -- nominal c. A summary of the number of papers sold by county. --- ratio d. The number of years with the paper for each employee. -- ratio 4. For each of the following, determine whether the group is a sample or a population. a. The participants in a study of a new cholesterol drug. -- sample b. The drivers who received a speeding ticket in Kansas City last month. -- population c. Those on welfare in Cook County (Chicago), Illinois. -- population d. The 30 stocks reported as a part of the Dow Jones Industrial Average. -- sample

Question:For my statistics class, I am supposed to include an original example of a Type 2 Error. I have figured out the Type 1 Error, but 2 is very tricky! Any help would save this poor girl from doing homework all night <3

Answers:It is difficult to know what "an original example" means. They will all have occurred, or been used before. However, the one which always springs to mind as an example of type 2 error is someone being given a blood test for a particular disease, say, and being told that the result is negative when they do, in fact, actually have the disease. Not very original, but I don't think there are any original ones to be had !