how to solve normal distribution problems

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From Wikipedia

How to Solve It

How to Solve It (1945) is a small volume by mathematician George Pólya describing methods of problem solving.

Four principles

How to Solve It suggests the following steps when solving a mathematical problem:

  1. First, you have to understand the problem.
  2. After understanding, then make a plan.
  3. Carry out the plan.
  4. Look backon your work.How could it be better?

If this technique fails, Pólya advises: "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"

First principle: Understand the problem

"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:

  • What are you asked to find or show?
  • Can you restate the problem in your own words?
  • Can you think of a picture or a diagram that might help you understand the problem?
  • Is there enough information to enable you to find a solution?
  • Do you understand all the words used in stating the problem?
  • Do you need to ask a question to get the answer?

The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.

Second principle: Devise a plan

Pólya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

  • Guess and check
  • Make an orderly list
  • Eliminate possibilities
  • Use symmetry
  • Consider special cases
  • Use direct reasoning
  • Solve an equation

Also suggested:

  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backward
  • Use a formula
  • Be creative
  • Use your head/noggin

Third principle: Carry out the plan

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals.

Fourth principle: Review/extend

Pólya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.

The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:

The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.

The book has achieved "classic" status because of its considerable influence (see the next section).

Other books on problem solving are often related to more creative and less concrete techniques. See lateral thinking, mind mapping, brainstorming, and creative problem solving.

Influence

  • It has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication.
  • Marvin Minsky said in his influential paper Steps Toward Artificial Intelligence that "everyone should know the work of George Pólya on how to solve problems."
  • Pólya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education.
  • Russian physicistZhores I. Alfyorov, (Nobel laureate in 2000) praised it, saying he was very pleased with Pólya's famous book.
  • Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels Pólya's work.


From Yahoo Answers

Question:A factory has a machine designed to produce 1 kg bags of sugar. It is found that the average weight of sugar in the bags is 1.02 kg. Assuming that the weights of the bags are normally distributed, find the standard deviation if 1.7% of the bags weigh below 1 kg. Give your answer correct to the nearest 0.1 gram. I know how to find the percentage by using mean and standard deviation, but I do not understand how to get standard deviation from just the mean and percentage. Please help, it would be greatly appreciated.

Answers:Okay, let's vision a normal bell curve with X and Z scores beneath it. At the mean, the X score is 1.02 and the Z score is 0 (because the mean on the Z scale is always zero). Now, when the area under the curve to the left is equal to 0.017, the X score is 1.0 (given by the problem) and the Z score is -2.12 (found by doing an inverse normal distribution look up using any statistical program, Z table, or TI83 calculator). We know the X to Z transformation is: Z = (X - mean)/std deviation This means that the standard deviation is equal to (X - mean)/Z Substituting in the numbers we know: Standard deviation = (1.0 - 1.02)/(-2.12) = 0.0094

Question:PartyPalace balloon sellers determined that the diameter of their small sized balloons are normally distributed with a mean of 4.2 inches and a standard deviation of .03 inches. If a balloon is selected at random, what is the probability that its diameter will be between 4.6 inches and 5.02 inches? I would appreciate seeing how the problem is solved and not just an answer.

Answers:You need a statistical table of normal distribution probabilities to solve this. If you wanted, you could use the NORMSDIST function in Microsoft Excel. To see how this works, read this article: http://www.ehow.com/how_5260332_use-normsdist-function-microsoft-excel.html Draw the normal distribution bell curve. Draw a solid vertical line down the middle, dividing the curve into two areas, each with an area of .5000. Now, we need to calculate two z-scores. Recall that the formula for z-score is: z=(event-mean)/standard deviation The first z-score is: z=(4.6-4.2)/.03=13.33 to two decimal places (this seems a bit high; are you sure you wrote the problem correctly?) The second z-score is: z=(5.02-4.2)/.03=27.33 to two decimal places (again, are you sure you typed this correctly? Maybe you mean .3 and not .03??) Basically what you do is calculate the z-scores, plot them on the graph, and calculate the area between them.

Question:The lifetimes of a particular brand of DVD player are normally distributed with a mean of 8 years and a standard deviation of six months. Find each of the following probabilities where x denotes the lifetime in years. In each case, sketch the probability. a.P(7 x 9) d. P(x 8) g. P(x 10) h. P(x>10) I having problems solving the following problem, any help would be wonderful and my textbook is not really helping. Thanks!

Answers:For any normal random variable X with mean and standard deviation , X ~ Normal( , ), (note that in most textbooks and literature the notation is with the variance, i.e., X ~ Normal( , ). Most software denotes the normal with just the standard deviation.) You can translate into standard normal units by: Z = ( X - ) / Where Z ~ Normal( = 0, = 1). You can then use the standard normal cdf tables to get probabilities. If you are looking at the mean of a sample, then remember that for any sample with a large enough sample size the mean will be normally distributed. This is called the Central Limit Theorem. If a sample of size is is drawn from a population with mean and standard deviation then the sample average xBar is normally distributed with mean and standard deviation / (n) Also, since the normal is a continuous random variable: P(X x) = P( X < x) because integration over a point is zero. An applet for finding the values http://www-stat.stanford.edu/~naras/jsm/FindProbability.html calculator http://stattrek.com/Tables/normal.aspx how to read the tables http://rlbroderson.tripod.com/statistics/norm_prob_dist_ed9.html In this question we have X ~ Normal( x = 8 , x = 0.25 ) X ~ Normal( x = 8 , x = 0.5 ) Find P( 7 < X < 9 ) = P( ( 7 - 8 ) / 0.5 < ( X - ) / < ( 9 - 8 ) / 0.5 ) = P( -2 < Z < 2 ) = P( Z < 2 ) - P( Z < -2 ) = 0.9772499 - 0.02275013 = 0.9544997 ===== Find P( X > 8 ) P( ( X - ) / > ( 8 - 8 ) / 0.5 ) = P( Z > 0 ) = P( Z < 0 ) = 0.5 This is easy because the normal is symmetric about it's mean, i.e., for any normal random variable X with mean P( X < ) = P( X > ) = 0.50 == == == Find P( X < 10 ) P( ( X - ) / < ( 10 - 8 ) / 0.5 ) = P( Z < 4 ) = 0.9999683 == == == == == == == == == Find P( X > 10 ) P( ( X - ) / > ( 10 - 8 ) / 0.5 ) = P( Z > 4 ) = P( Z < -4 ) = 3.167124e-05

Question:Guys can you help me solve these problems. Its kinda complicated. 1) In a study about life expectancy of 200 thirty-year-old males, the mean is 65 and the standard deviation is 10. What proportion of the males could live past 75? 2) Assume that the people's ages at the onset of a certain disease have a normal distribution with a mean of 60 years and a standard deviation of 5 years. What is the probability that an individual will develop the disease before the age of 50? thank you guys for your help in advance. im still working/ solving on 30 problems.

Answers:1) Let X be the age of a male. For any normal random variable X with mean and standard deviation , X ~ Normal( , ), (note that in most textbooks and literature the notation is with the variance, i.e., X ~ Normal( , ). Most software denotes the normal with just the standard deviation. You can translate into standard normal units by: Z = ( X - ) / Moving from the standard normal back to the original distribution using: X = + Z * Where Z ~ Normal( = 0, = 1). You can then use the standard normal cdf tables to get probabilities. An applet for finding the values: http://www-stat.stanford.edu/~naras/jsm/FindProbability.html calculator: http://stattrek.com/Tables/normal.aspx how to read the tables: http://rlbroderson.tripod.com/statistics/norm_prob_dist_ed9.html In this question we have X ~ Normal( = 65 , = 100 ) X ~ Normal( = 65 , = 10 ) Find P( X > 75 ) P( ( X - ) / > ( 75 - 65 ) / 10 ) = P( Z > 1 ) = P( Z < -1 ) = 0.1586553 2) In this question we have X ~ Normal( = 60 , = 25 ) X ~ Normal( = 60 , = 5 ) Find P( X < 50 ) P( ( X - ) / < ( 50 - 60 ) / 5 ) = P( Z < -2 ) = 0.02275013

From Youtube

ck12.org Normal Distribution Problems: Qualitative sense of normal distributions :Discussion of how "normal" a distribution might be

MATH 110 Normal Distribution Problems Solved: Part B :MATH 110 Normal Distribution Problems Solved: Part B