Study "Mathematics / Statistics" Essays 111-162

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William Gosset Term Paper

… William Gosset

William Sealey Gosset was one of the leading statisticians of his time, particularly with his work on the concept of standard deviation in small samples. His theories which were published under the name of "student" are still used today in both the study of statistics and the practical application.

Gosset was born on June 13, 1876, in Canterbury, England to Colonel Frederic Gosset and Agnes Sealy Vidal. Gosset was well educated from the beginning first at Winchester, a prestigious private school, then at New College at Oxford. He received his degree in mathematics in 1897, followed two years later by a degree in chemistry (O'Connor and Robertson). It was the combination of these two fields of study that gave Gosset a career and an opportunity to create his theory.

Upon graduation, Gosset was hired as a chemist by the Arthur Guinness and Son Company in Dublin. Working in the brewery, required Gosset to constantly attempt to find the best varieties of barley for use in the production of Guinness. This was a complicated procedure of taking small samples to determine the best quality product. Gosset continuously played around with the results of various samples of barley in order to find ones of the best quality with the highest yields that were capable of adapting to changes in soil and weather conditions. Much of his work was trial and error both in the laboratory and on the farms, but he also spent time with Karl Pearson, a biometrician, in 1906-07, at his Galton Eugenics Laboratory at University College (O'Connor and Robertson).

Pearson assisted Gosset with the mathematics of the process. Gosset published his findings under the name of "student" because the brewery would not permit him to publish. The brewery feared that trade secrets would get out if information about the brewing process was published. Consequently, Gosset had to assume a pseudonym even though his information would not have impacted the business in the way the brewery was concerned (O'Connor and Robertson).

Gosset published his work in an article called "The Probable Error of a Mean" in a journal operated by Pearson called Biometrika. As a result of Gosset's pseudonym, his contribution to statistics is called the Student t-distribution. Gosset's work caught the attention of Sir Ronald Aylmer Fisher, a statistician and geneticist of the time. Fisher declared that Gosset had developed a "logical revolution" with his findings about small samples and t-distribution (O'Connor and Robertson).

In his work with the barley for the brewery, Gosset was concerned with estimating standard deviation for a small sample. A large sample's standard deviation has a normal distribution. However, Gosset did not have the luxury of working with large samples. He had to find a way to determine the standard deviation for a small sample without having a preliminary sample to make an estimate. Gosset developed the t-test to satisfy this need.…… [read more]

Scores of First Born and Second Term Paper

… ¶ … scores of first born and second born children on the Perceptual Aberration test. Ha: There is a significant difference in mean scores of first born and second born children on the Perceptual Aberration test. The alternative hypothesis is one sided, since the null hypothesis is concerning non-directional data, in that we are not predicting directional information.

The data will be examined using an independent t test. This test is used since the groups in these circumstances are not related. If the samples were correlated, where each individual had two scores under two treatment conditions, the dependant t test would have been used.

The test statistic in this experiment represents the difference between the mean scores of the children tested divided by the standard error of the difference. This result would represent whether there was a significant difference between the means. If so, we would reject the null hypothesis. By using the t test, we are able to determine the ratio of the mean difference in test scores when compared to the error of differences in the means. A large mean difference does not guarantee a large t, hence the use of the standard error of difference.

1d. Following a .05 level of significance, and after calculating the df (92), the critical value needed to reject the null hypothesis is 2.367. Our calculations show that t=.529982. Thus, we would accept the null hypothesis because our calculated value (t = .529982) is less extreme than the critical values (2.367 or -2.367).

1e. The observed mean difference in test scores between first (M=17.2563) and second (M=16.14815) born children was not significantly different, t (92) = .529982, p>.05.

t =


4331.62 +1497.407






t = 1.14815 / SQRT (5829.037/92) * ((1/27) + (1/27))

t =

1.14815 / SQRT (63.3591 *.074074) = 1.14815 / 2.166395 = .529982

2a. H0: There is no difference in the population means of the different scores on recall of words testing following memory techniques. Ha: There is a difference in the population means of the different scores on recall of words testing following memory techniques.

2b. An independent two samples t test is not appropriate because our samples include more than two samples. We are comparing the results of three groups of subjects.

2c. The between groups ANOVA test statistic will measure if there is a difference…… [read more]

Historic Mathematicians Term Paper

… Historic Mathematicians

Born on January 29, 1700 Daniel Bernoulli was a famous Swiss Mathematician. His father -- Johan Bernoulli was the head of mathematics at Groningen University in the Netherlands. His father planned his future so that Daniel would become… [read more]

Chaos Theory Has Filtered Down Book Review

… Chaos Theory has filtered down to the public through such short discussions of the issue as are found in films like Jurassic Park or on television documentaries. The issue are more complex than can be indicated in such media depictions,… [read more]

Probability Statistics Term Paper

… Probability: Its Use in Business Statistics

Business, one might say, is an exercise in probability. No one knows exactly what the market will do in the future, not even the most skilled analysts and prognosticators. One can only make educated guesses, and the use of probability models and statistics enables the professional to make such guesses, even though, no consumer behaves perfectly according to mathematical economic metric models. If used correctly, statistical analysis can be important guides that enable one to purse intelligent business practices and function as aids in the decision making process, even though they are only, ultimately projected 'guesses' as to how the economic environment will evolve, given a variety of variable factors.

Probability, in its most ideal mathematical form, attempts to make use of various concepts to determine what is likely to occur, given a particular set of variable circumstances. One of the most important uses of probability in business is to determine what a particular consumer market's spending habits are likely to be, given a particular set of events. For instance, if the Federal Reserve lowers interest rates yet again, and consumer spending is likely to increase, what is the most desirable course of action, in terms of production of a business that manufactures durable goods, if all other market aspects remain relatively unchanged? Probability theory can also be used to assess what to do if a new and potentially variable competitor advances into a market, pricing comparable goods competitively against one's own product line. What will consumers do, and how will the market behave, given these circumstances?

Probability theory thus deals with what is variable and also with what is unknown in projected circumstances or futures. One must know certain fixed attributes about the circumstances, such as certain fixed production costs, but the use of probability theory allows for the introduction of a set of uncertain or variable factors.

Thus, the use statistical probability attempts to project a variety of foreseeable futures, so the businessperson can prepare for the possible negative aspects of these foreseeable futures. These unknowns are represented, in equations, as variables or unknowns. Various scenarios can be plugged into these placeholders, represented as 'xy' in integral calculus functions.…… [read more]

Guillaume Francois Antoine De L'hopital Term Paper

… Apparently, out of respect to the mathematician who made much of his fame possible, L'Hopital abandoned the project.

'L'Hopital was a major figure in the early development of the calculus on the continent of Europe" (Robinson 2002). During this time of scientific and mathematic enlightenment in Europe, and particularly in France, L'Hopital established himself as one of the world's premier mathematicians and book writers. It is noteworthy that many of the accomplishments L'Hopital is credited with have come into question over the years. Most obvious among these include the rule that is named after him, which every calculus student has been forced to memorize for the past three hundred years. Despite these questions, perhaps the most telling thing about L'Hopital is that he was widely accepted and respected by his peers. He became the third man on continental Europe to learn calculus simply because he impressed the man who later became his tutor. "According to the testimony of his contemporaries, L'Hopital possessed a very attractive personality, being, among other things, modest and generous, two things which were not widespread among the mathematicians of his time." (Robinson 2002). He died on the second of February, 1704, in Paris; the city of his birth.

Works Cited

1. Addison and Wesley. Calculus: Graphical, Numerical, Algebraic. New York: Addison-Wesley Publishing, 1994.

2. Feinberg, Joel and Russ Shafer-Landau. Reason and Responsibility. Boston: Wadsworth Publishing, 1999.

3. Goggin, J. And R. Burkes. Traveling Concepts II: Frame, Meaning and Metaphor. Amsterdam: ASCA Press, 2002.

4. Greenberg, Michael D. Advance Engineering Mathematics: Second Edition. Delaware: University of Delaware, 1998.

5. O'Connor, J.J. And EF Robertson. "Blaise Pascal." JOC/EFR. December 1996. School of Mathematics and Statistics, University of St. Andrews, Scotland.…… [read more]

Statistical Analysis Reported in Two Term Paper

… First, no such mention was ever made in the beginning of the study with respect to gender differences. Second, logistic regression analysis and/or techniques have no earthly association with differences. Had the authors wanted to determine whether or not differences… [read more]

Mathematician - Maria Gaetana Agnesi Term Paper

… She had written 2 volumes of mathematical books, the Institutioni analytiche ad uso della gioventu italiana (Analytical Institutions), that covers elementary and advanced mathematics which she started to develop when she was teaching mathematics to her younger brothers. Her books aim to present a complete lecture of algebra and mathematical analysis.

Maria Gaetana Agnesi was well-known for her "The Witch of Agnesi," which, actually, should be called "The Curve of Agnesi." The Italian term "versiera," or plane curve, was mistakenly translated by John Colson into the word "witch" (Parente, 2003). Thus, "The Curve of Agnesi" was also known as "The Witch of Agnesi." Elif Unlu describes "The Witch of Agnesi" by stating the following.

Agnesi wrote the equation of this curve in the form y = a*sqrt (a*x-x*x)/x because she considered the x-axis to be the vertical axis and the y-axis to be the horizontal axis [Kennedy]. Reference frames today use x horizontal and y vertical, so the modern form of the curve is given by the Cartesian equation y*x^2=a^2(a-y) or y = a^3/(x^2 + a^2). It is a versed sine curve, originally studied by Fermat.

When Agnesi first wrote her 2 volumes of Analytical Institutions, she used her genius in mathematics to teach her younger brothers, and the young Italians as well. Her prowess in mathematics was shared when, after the success of her book, she became a professor of mathematics in the University of Bologna.


Crowley, Paul. Maria Gaetana Agnesi.

New Advent. 08 Dec 2003.

Unlu, Elif. Maria Gaetana Agnesi.

1995. Agnes Scott College. 08 Dec 2003.

Parente, Anthony. I Wrote the First Surviving Mathematical Work by a Woman.

2003. 08 Dec 2003.… [read more]

Pascal's Triangle Who Really Invented Term Paper

… In fact, the understanding of probabilities the triangle helped mathematicians understand has led to the development of "average gain" or "probable gain" formulas that are still used extensively in business and industry (Borel, 1963, p. 20).

The basic formula for the triangle is simple, as one expert notes.

If we assume a fictitious row of noughts prolonging each of these lines to right and left, it is possible to lay down the following rule: each number in any one of these lines is equal to the sum of whatever number lies immediately above it in the preceding line, and whatever number lies immediately to the left of that number. Thus the third number in the fifth line is 10 = 6 + 4; the fourth number in this same line is 10 = 4 + 6; the fifth number is 5 = 1 + 4 (Borel, 1963, p. 18).

There is one problem with Pascal's formula, however. Unfortunately, as the numbers increase, the triangle takes much longer to solve, and the formula becomes ungainly. This created problems with the formula initially, but mathematicians have learned to cope with the formula and have created alternates that let them work with the numbers more effectively, as this expert notes. "Mathematicians have established certain formulas that allow them to work out the numbers which appear in Pascal's Triangle, as well as the sums of whole rows of these numbers included between fixed limits" (Borel, 1963, p. 18). Thus, Pascal's triangular theory was not perfect, but the formula has lasted through time, been improved, and still makes the study of probabilities cognitive.

However, this simple formula has made quite a difference in mathematics circles for centuries for a number of reasons. First, his treatise on these binomial coefficients later helped contribute to Sir Isaac Newton's eventual invention of the general binomial theorem for fractional and negative powers. In addition, Pascal carried on a long correspondence with Pierre de Fermat, and in 1654, this correspondence helped contribute to the development of the foundation of the theory of probability, which is one of our most important mathematical developments even today.

Interestingly enough, Pascal devoted the last eight years of his short life to philosophy and religion, and gave up his studies in the sciences and mathematics. One must wonder what he could have accomplished had he continued his studies, and indeed, what improvements he could have made to his triangle had he given it even more time and effort. His discoveries and inventions live on today, along with his name, as one of the greatest minds of all time, and he contributed greatly to our lives today, from a clearer understanding of probabilities to measuring the weather, dispensing medications, and ultimately computing our calculations quickly and efficiently.

In conclusion, Blaise Pascal died in 1662 at the age of thirty-nine - two years before the significance of his triangle would be known to those outside his academic circle, and the final formula would be published. Today, mathematicians… [read more]

Proof, a Nova Episode Aired Term Paper

… This is another way of looking at solving complex problems. The show made the problem seem all encompassing (which it was to Wiles), and used a variety of experts to explain just what Wiles was attempting to prove, and why it was so important to the mathematical community. They took a topic which could have been boring and nearly incomprehensible, and made it interesting enough to keep the viewer watching. In fact, NOVA managed to get the viewer behind Wiles, and by the end of the show, when it seemed like he might not prove his theory, it was almost as if I was rooting for him to continue and not give up. To end the program, NOVA said, "Andrew Wiles is probably one of the few people on earth who had the audacity to dream that you could actually go and prove this conjecture" (NOVA). Therefore, this story is as much about dreams and goals as it is about pursuing something complex throughout your life to fruition. Andrew Wiles dared to dream, and in the end, his most complex "proof" may have been that sometimes dreams come true - with hard work, determination, and thinking "outside the box," - or in this case, the theorem.

This video is also quite important in what it shows about how people learn to do mathematics, and it was somewhat how I learned to do mathematics. Wiles broke down an extremely complex problem into bits and pieces, but he also had to look at it in unaccepted and untried ways. This is often how new truths are learned in any area. He also said that he suddenly had some kind of understanding that had not been there before. "I had this incredible revelation. [...] It was the most -- the most important moment of my working life. It was so indescribably beautiful; it was so simple and so elegant, and I just stared in disbelief for twenty minutes" (NOVA). While I have not attempted to solve complex problems such as Wiles', I had a hard time "getting" algebra at first, and it seemed like it took me years and years of study to understand even the most simple equation. Then suddenly, one day in class, I looked at an equation, and it suddenly just "made sense," and I could see the solution without struggle. I finally "got" it, and I know just how Wiles felt when the solution suddenly came to him. It was an incredible feeling, and once I had "gotten" it, not only was mathematics simpler, it was not so frightening or frustrating.

The Proof" is an elegant look at a complex subject, and it not only made mathematics more human, it made it clear how the best problem solving approach is one that takes a complex problem, breaks it down into more solvable areas, and then looks at every angle of the problem to find a solution. That solution might be, in the end, simple, but it needed alternate thinking… [read more]

Low Math Term Paper

… In the book, Ma provides an example of a Chinese teacher who has this profound understanding.

This teacher prepares for their lesson by considering what they will teach and what it means. They link the lesson that will be taught… [read more]

Sine, Cosine, and Tangent Term Paper

… Because of trigonometry, it was now possible to determine the approximate volume of a star simply by finding its diameter. When it was first discovered, people used simple right-angle trigonometry to find heights of mountains and tall buildings.

It was soon discovered that the entire wave spectrum could be described in terms of frequency and amplitude, and graphed by trigonometric functions, such as sine, cosine and tangent.

The Babylonian measure of 360° formed the study of chords. With this information, sine and cosine were loosely defined as =1. Another Greek mathematician, Menelaus, wrote six books on chords. Ptolemy subsequently created a complete chord table. His new discovery included a variety of different theorems such as a quad inscribed inside a circle has the property that the product of the diagonals = sum of products of opposite sides; the half angle theorem; the sum and difference formulae; the inverse trigonometry functions; and more sine and cosine rules.

How Sine, Cosine and Tangent are Used Today

Today, sine, cosine and tangent are still used for astronomy and for geography, as well as in navigation and mapmaking. The trio is also used in physics with the study of visible light and fluid motion. Engineers today use trigonometric functions for military engineers and conveyors.

Trigonometric functions are the functions of an angle. These functions are important when studying triangles and modeling periodic phenomena. The trigonometric functions may be accurately defined as ratios of two sides of a right triangle containing the angle, or as ratios of coordinates of points on the unit circle.

Of the six trigonometric functions, sine, cosine and tangent are the most important. Sine, cosine, and tangent are used when you know an angle and a length of one of the sides of a right triangle, and you want to know the length of another side. For these functions, the angle is in radians, not degrees

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. (Moyer) The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

Without sine, cosine and tangent, the mathematical tables on our computer screens would only show blank pages, and scientific calculators would not react to punching in numbers. Draftsmen would make serious errors when designing buildings, geologists would have inaccuracies of measurement, and so on.

Trigonometry has even been used in analyzing motor vehicle collisions. (Kaye) Geometry is used to determine curve radii for use in circular motion calculations while sine, cosine and tangent are used in momentum, vaults and road grade determinations.

Trigonometric functions were originally developed for astronomy and geography, but scientists are now using them for other purposes, too. Besides other fields of mathematics, trigonometry is used in physics, engineering, and chemistry.

Within mathematics, trigonometry is used primarily… [read more]

Theory on Plate Tectonics Term Paper

… Tragedy was to strike again, only a year after taking up this post, when in 1808 his father dies, and then in 1809, whilst in childbirth, his wife dies, and the second son, who she was giving birth to, was also to die soon after. However, is work does not appear to have suffered in the long-term, but the short-term saw him take time off of work and devote himself to his three children (Schaaf, 1964).

In 1810 he remarried, there were another three children, but this is generally though to have been a marriage of convenience rather than a love match (Schaaf, 1964).

Some of his major works included work on how to calculate the orbit of the planets. In his work Theoria Motus Corporum Coelestium he examined and discussed the use of differential equations, conic sections and the elliptic orbits, and then in the next volume of this work he then showed how the orbit of a planet could be estimated and then the estmate could be further refined (Rassias, 1991). By 1817 he had made is contributions to astronomy, and despite continuing observations he did not add more to the theoretical framework of astronomy (Schaaf, 1964).

Gauss did look to other subjects, publishing a total of one hundred and fifty papers over his career, he contributed to many other areas. Papers included Methodus nova integralium valores per approximationem inveniendi which was a practical essay that concerned the use of approximate integration, a discussion of statistical estimators in Bestimmung der Genauigkeit der Beobachtungen and geodesic problems in Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractate (Schaaf, 1964).

During the 1820's the work of Gauss appeared to start taking him more in the direction of geodesy. This may have started when, in 1818, he was requested to undertake a geodesic survey of Hanover, to link up to the Danish grid that was already in existence. He took total charge, and made the measurements during the day, and in the evenings he would reduce them to the calculations. It was during this survey, and as a result of the survey needs, that he invented the heliotrope (Rassias, 1991). Unfortunately, in the survey there were erroneous base lines used (Rassias, 1991).

Other work included may theories that were also discovered independently of Gauss by other mathematicians which have gained the recognition. For example, he had formed the ideas for non-Euclidean geometry, claiming to have discovered it fifty four years before Lobachevsky, but he still praised it. The fifty four-year framework may not be correct, but there are certainly some vague references to it in some of his work (Schaaf, 1964).

It was in 1832 when Gauss started to work with Weber, regarding terrestrial magnetism, many ideas were mentioned, and Dirichlet's principle was also included, but with a proof. They also proved there could only be two poles with Allgemeine Theorie (Schaaf, 1964).

The papers and theories have outlasted the name and reputation of their founder. However, the long-term impact of… [read more]

Exploring the Correlation Between Age and Cell Phone Use Chapter

… Computer Lab: Hypothesis Testing Correlations

The following null hypothesis is applicable for testing the correlation between the two variables "Age" and "Q18: "On an average day, about how many phone calls do you make and receive on your cell phone?"

Ho = The age of the cell phone user is not related to the average number of cell phone calls made or received by the cell phone user.

The Pearson correlation coefficient is .055 (p = 0.05, 2-tailed). In addition, Spearman's rho (-.244) and Kendall's tau (-.340) both show the correlation as significant at the 0.01 level (2-tailed). The null hypothesis is rejected at 0.01. There is a positive relationship between the age of the cell phone user and the average number of cell phone calls made or received by the cell phone user.

The number of observations for the "Age" variable was 1917 and the number of observations for the "Q18: "On an average day, about how many phone calls do you make and receive on your cell phone?" variable was 2252. User-defined missing values were treated as missing. Statistics for each pair of variables was based on all the cases that had valid data for that pair.

The assumptions about the data, including normality and linearity, were tested by examining the descriptive statistics and tests for skewness and kurtosis as shown in the output table directly below. The data are assumed to be normally distributed and independent. Some outliers are present in the data, which is to be expected since not all frequent users of cell phones are young. Variables such as the type of work or employment in which cell phone users engage can strongly influence the frequency of the cell phone calls made and received. These considerations encourage additional statistical analysis of other variables and perhaps additional research.


Q18. On an average day, about how many phone calls do you make and receive on your cell phone?

AGE. What is your age?









Std. Error of Mean









Std. Deviation





Std. Error of Skewness






Std. Error of Kurtosis…… [read more]

Math Problems and Concepts in Teaching Research Paper

… Mathematics of Mathematic Puzzles

Using mathematics puzzles is a frequently-used deployed pedagogical device to teach critical concepts to math students of all ages. "Understanding in mathematics is born not only from formulas, definitions and theorems but, and even more so, from those networks of related problems… Mathematicians seek knowledge. In search of knowledge, they enjoy themselves tremendously inventing and solving new problems" (Bogomolny 2015). The mathematics of puzzles is also designed to challenge the student's instinctive conceptions of how the world works. For example consider the question posed 1702 of how long would a rope encircling the earth have to be "for it to be one foot off the ground all the way around the equator -- this can be used to illustrate the geometric concept of finding the circumference (Pickover 2010). The student's intuitive instinct would be that the rope must be extremely long but in fact the answer is only 2pi. "If r is the radius of the Earth, and 1 + r is the radius in feet of the enlarged circle, we can compare the rope circumference before (2pir) and after (2pi (1 + r))" (Pickover 2010).

Other examples of these teaching devices are the multiplying 'wheat on a chessboard' dilemma. In this problem, the grains of wheat, beginning with one, are doubled every square leading to such a proliferation of grains that it would be impossible to provide that much wheat in reality, thus illustrating the concept of geometric growth (Pickover 2010). There is also the barber's paradox "which involves a town with one male barber who, every day, shaves every man who doesn't shave himself, and no one else" which is used to illustrate the concept of set theory (Pickover 2010).

The applicability of apparently pointless mathematical queries to vital foundational concepts in math has also come to light in the evolving discipline of game theory, frequently used by economists to illustrate how people make choices. For example, in the classical case of the Prisoner's Dilemma, two separately-imprisoned individuals are placed in a cell. If neither confess, both men go free, if both confess, both get a reduced sentence but still do jail time, while if only…… [read more]

History and Math Puzzles Research Paper

… Mathematical puzzles are not simply fun and interesting topics on which to muse: they have great significance within the history of the discipline, often connecting many generations of mathematicians who strive to unravel such riddles. "What is the value of such puzzles and enigmas? One important value is the fact that a math puzzle has an answer. We spend much of our time puzzling over problems that do not appear to have easy answers, if they have answers at all, and math puzzles offer a simplified way of solving problems that lead to a satisfying conclusion" (Bright 2010). Math problems are frequently-deployed mental exercises to help individuals understand certain concepts better. But some math problems have been so widely regarded as unsolvable they have become notorious.

Amazingly, many of the earliest puzzles were only solved in the modern era. For example, the Greek Archimedes constructed a puzzle asking how many ways there were to divide a square into 14 pieces. Only in the 21st century did a Cornell professor of mathematics discover that there "are a total of 17,152 solutions, but some can be considered equivalent if a rotation or a reflection are performed" (Pitici 2008). In another famous mathematical puzzle pertaining to spatial relations called the bridges of Konigsberg, the central problem arose when "seven bridges were built so that the people of the city could get from one part to another;" people began to speculate how to walk over the various bridges to go straight across the city while crossing each bridge only once ("The beginnings of topology," Math Forum). The problem of the bridges was one of the first questions ever posed in the evolving discipline of topology in mathematics, which reflects the fact that the "properties of the shapes remain the same" regardless of whether they are "stretched or compressed" ("The beginnings of topology," 2015).

The concept of paradoxes was also highly significant to the discipline of mathematics. Zeno's paradox about the "infinite divisibility" of space asked the question of how motion was possible ("Zeno's paradox," 1998). Obviously, motion is possible from an observable perspective but in theory motion is impossible because to reach one point, first we need…… [read more]

Easy Explanation of Analysis of Variance Essay

… ¶ … goals of statistics is to create an easy way to compare two or more factors in a meaningful way. Basic comparison tests, such as the t-test, allow for the comparison of two groups. Analysis of Variance (ANOVA) allows for the comparison of three or more different groups. While the concept of ANOVA may seem complex, it is simply a way of describing how the various items that belong within a single large group vary from one another. This can be very important because there are two main types of variance in a group. One type of variance is the type of variance that can be found naturally within a group; this type of variance is frequently referred to as random variance and may also be described as in-group variance. The other type of variance is the type of variance that is caused by the impact of independent variables on the dependent variable and may also be described as between-group variance. Without understanding the underlying variance in a group, one could overestimate or underestimate the impact that an independent variable was having on the dependent variable. An ANOVA test does not provide conclusive data; instead, it provides analysts with a key to understanding other test results, so that they can understand the significance of those results (Investopedia, 2015).

An ANOVA test can help explain whether there are any significant differences between the average results for three or more different groups. In an experimental context, each of the three groups will be as similar as possible at the beginning and the in-group variance can help explain how much variation is simply the result of random chance. Then, once the independent variable is applied to the group, the changes are examined to determine what type of impact the independent variable has on the dependent variables. The goal of the ANOVA is to test the null hypothesis, which is a hypothesis developed for the purpose of research, which states that the independent variable will have no impact on the dependent variable.…… [read more]

Hypothesis Testing Essay

… Alpha level determines the confidence with which the researcher can decide to reject the null hypothesis. When researchers establish an alpha level prior to performing the research they are essentially deciding that if the null hypothesis is true then the probability of getting significant results in this study by either sampling error or by chance is whatever the alpha level they have set is (conventionally set at 0.05; Runyon, Coleman, & Pittenger, 2000). The choice of the alpha level is arbitrary, in other words researchers decide where to set it before their analysis. It is been a convention in research to use the 0.05 level; however, this is certainly not set in stone and recently this convention has come under some sharp criticisms (e.g., see Ioannidis, 2005). There are times when researchers will decide to use a more liberal or conservative alpha level in the research.

For example, suppose that a researcher is developing a way to diagnose a very serious and debilitating disease that typically cannot be diagnosed until the infected individual is nearly terminal. Also let us assume that if people with this disorder can be identified before it becomes terminal the cure is relatively safe. In this case the cost of making false positive errors (Type I errors) is low whereas the cost of making a false negative error (Type II error) is considered high and researchers are often motivated to increase their decision -- making criteria by raising their alpha level (Runyon et al., 2000). Also when researchers are investigating areas where there is very little known about the potential outcome and they are trying to develop theoretical models they may also decide to use a more liberal alpha level in order to identify potential significant constructs that will be scrutinized more conservatively in the future.

There also times when researchers will opt for a more conservative alpha level. In fact, there is one specific situation where researchers should adopt a more conservative alpha level but often do not. This occurs when researchers are making multiple statistical comparisons on the same data set (e.g., when the comparisons are not independent of one another; Runyon et al., 2000). The Type I error rate for any single statistical comparison is set by the researcher prior to their analysis and this is the…… [read more]

Limit Your Summary Term Paper

… Low working memory individuals did not suffer appreciably under pressure, showing that working memory is indeed crucial to success and also sensitive to pressure constraints such as social evaluation.

8. How are these findings relevant to everyday life? Provide at least two examples. At least one of these examples should be an original example. That is, it should not be mentioned by the author(s) of the research article.

These findings are highly relevant for a number of real life scenarios. One example is in school children or in students at university level. Students who do well on their homework and low-pressure class exercises might choke under pressure, such as when they are a leader of a team, when there are time constraints on them, or when they are being watched closely. Another example is in the workplace. Employees who depend on their working memory might choke in high-pressure scenarios, such as public speaking engagements or competing with a colleague for an important contract. The implications are that persons who have a high capacity for success should develop their skills in coping with anxiety.

9. Describe two ways in which this research expands upon the theories and concepts discussed in class? You may wish to consult your textbook and lecture notes. Please cite appropriately (in APA format) when you discuss information from your text (e.g., Goldstein, 2011) and from your class notes (e.g., Trammell, personal communication, 2013). Please limit your response to 8 sentences.

The theories and concepts discussed in class have to do more with general cognitive psychology. Short-term memory is discussed, but not in terms of working memory in the context of performance on cognitively demanding tasks like mathematics. We have not learned that working memory is like a flash drive, in that the brain is able to process material effectively and rapidly by relying on short-term memory space. Yet anxiety also takes up this short-term memory space. Invading working memory, anxiety therefore inhibits performance. This would seem to be true for everyone, but it is especially true for those who rely most on short-term or working memory. It appears that the most successful or most capable persons are those who rely strongly on their working memory.

10. What research questions remain unanswered? That is, describe at least two areas for future research. Please limit your response to 8 sentences.

This study raises several questions for future research. One is related to the different effects of different types of pressure. It would be helpful to know what types of pressure affect what types of people. For instance, some people might not find financial incentives stressful and would therefore not have any problem being distracted by that stimulus when solving math problems. Other people might find that social pressures are less important. It would also be helpful to know if time constraints and other factors are relevant. Furthermore, it would be interesting to know if there are gender differences between high capacity persons. Another question for future research is how… [read more]

And Standard Deviation Case Study

… 87, which is outside of the confidence interval.

The null hypothesis is therefore rejected -- the bottles contain a mean fill that is lower than the fill allowable within the 95% confidence interval. The complaint is correct -- the bottles are not sufficiently filled.

There are a couple of reasons why the bottles are not being filled properly. Most likely this is a calibration issue with the filling machine. The machine is chronically underfilling, which is most likely explained by calibration. The actual volatility of the fills (standard deviation) is high but does not explain the strong deviation from the expected mean of 16.0.

To solve this problem, I would check the calibration of the filler. Normally, the filler will fill to a level around the mean, usually at the 95% confidence interval -- or better if it is a more expensive filler. So if the machine is doing that here, it clearly thinks that 14.9 ounces is 16 ounces, or somebody changed the target fill level to 14.9 from 16. So the machine either needs to have its fill level target…… [read more]

ANOVA: Fobt, Tukey's HSD, and Effect Size Calculations Essay

… Statistics and Probability

A researcher investigated the number of viral infections people contract as a function of the amount of stress they experienced during a 6-month period. The obtained data:

Amount of Stress

Negligible Stress

Minimal Stress

Moderate Stress

Severe… [read more]

Multivariate Analysis Is Appropriate Essay

… discrete data), or make an unlimited number of comparisons. Like any research tool multivariate statistics have their limitations. Different types of multivariate tests allow a researcher to answer different types of questions, but a researcher's ability to make inferences is always limited by the type of data collected in the methodology used to collect it.

I am interested in how gender rates differ in cases of human trafficking into the United States. Currently there is a study that proposes looking at the differences in gender in human trafficking cases based on the data from The National Human Trafficking Resource Center. This particular to the research could be adequately performed using a simple t-test or one-way ANOVA. However, other variables could be added to the methodology and multivariate analyses could offer much more information regarding the differences in males and females involved in human trafficking. Using other variables such as geographical area trafficked to (e.g., in the United States this could be divided into Midwest, North, South, etc.), the type of context into which human victims are placed (e.g., hard labor, domestic work, prostitution, etc.), ethnic background of the trafficked individual, and others allows for a richer analysis. In this case there would be multiple independent variables that can contribute to the differences in the number of males and females that are trafficked into the United States and multivariate techniques such as factorial ANOVA (or its counterpart multiple regression depending on the type of question) is more appropriate. In this type of study we would expect that the particular context would lead to differences in the rates of males and females trafficked into the country. For example, we would expect that people put into hard labor jobs such as… [read more]

Meyer Et Al. Meyer, Wang Essay

… In the data. Different methodologies designate the types of conclusions that can be made from the analyses. For example in the current study Meyer et al. (2009) employ a correlational design, therefore they are unable to make causal inferences but can describe the relationships, their relative strengths, and in some cases direction given the analyses.

The type of data also influences the type of analyses one can do. There are a number of different levels of measurement in the current study due to a large number of variables ranging from nominal (e.g., diagnoses or patient employment status), ordinal (e.g., employment status coded as full-time, part-time, or casual), to ratio level variables (e.g., years of nursing work experience). Some of the variables are categorical such as employment status and some variables are continuous variables such as age.

Meyer et al. (2009) used multiple data collection methods that included collecting hospital records, daily unit data, surveys, and patient data forms. In order to ensure that different data sources and collection methods were consistent they calculated the inter-rater reliability for all measures (which they claim was at 90% throughout the study). The use of surveys in the study was extremely important as surveys allow the collection of anonymous data (no identification on the part of the person that takes a survey so they are free to answer candidly) and questions can be asked and rated on scales that allow the researchers to calculate such important internal constructs as a nurse or patient's attitude, opinions, and also allow to collect hard external data such as the number of hours worked, education levels, etc. Surveys remain an important staple in all areas of correlational research as the data can be coded and easily subject to statistical analyses; however, survey data typically does not allow the researcher to determine cause-and-effect associations (Tabachnick & Fidell, 2012). Thus, survey data can be extremely useful, but has limitations.


Jackson, S.L. (2012). Research methods and statistics: A critical thinking approach (4th

ed).Belmont, CA: Wadsworth.

Meyer, R.M., Wang, S., Li, X., Thomson, D., & O'Brien-Pallas, L. (2009). Evaluation of a patient care delivery model: Patient outcomes in acute cardiac care. Journal of Nursing Scholarship, 41(4), 399-410.

Tabachnick, B.G., & Fidell, L.S. (2012). Using multivariate statistics (6th…… [read more]

Experimental Research Research Proposal

… Experimental Research

One of the many important decisions a researcher must make during his or her work is which design to use for the research. Two main experimental designs include within-subjects and between-subjects design. Each has its own merits and drawbacks, and researchers tend to choose these according to the purpose and nature of the experiments or surveys to be conducted. For the experiment in question, where three different types of survey invitations are offered online, either the within- or between-subjects design can be used.

According to MacKenzie (2013), most empirical evaluations of input devices or interaction techniques, like the one to be conducted here, will be comparative. In this specific experiment, three types of online survey invitations are explored; the first containing only a link, the second offering to donate $10 to charity as a reward for participation, and finally, a chance to win $1,000 as a potential reward for participation. The comparative aspect lies in why people would be moved to respond to each of the invitations, and which invitation would receive the most participants. While some cases require a between-subjects design and others a within-subjects design, this particular survey could be studied by using either.

A within-subject design means that there is one group in which each participant is tested for all the conditions. Two major advantages of this design is that fewer participants are required, which means that recruiting, scheduling, briefing, demonstrating, and all the other aspects of the research procedure would be somewhat easier and take less time. Second, there is less variance resulting from participant disposition. Certain dispositions will occur consistently for certain participants, which makes it easier to make concessions for the particular behavior, as far as it influences the experiment results. Further, differences among measurements will then be due to differences in the conditions being measured rather than to disposition or behavior differences among participants (MacKenzie, 2013).

In the Zikmund experiment, a within-subject design would mean a single group of people would be tested for each type of survey invitation. In practical terms, this would mean that the group would be exposed to all three invitations and asked to choose the one they would most likely participate in. They could also be asked to give reasons for their participation. This would include both quantitative and qualitative effects in the experiment results. Quantitatively, the results would then reveal the invitation that is most enticing to participants, while the reasons can be investigated for their consistency with the results and among each other.

Although within-subject designs have distinct advantages, it is also true that there could be interference between the conditions imposed. When exposed to all three choices, for example, some participants may experience some difficulty choosing among them, which could compromise the reliability of the results. For this reason, researchers sometimes choose to opt for a between-subjects design instead (MacKenzie, 2013).

According to Shuttleworth (2013), a between-subjects design refers to an experiment that involves more than…… [read more]

Statistical Research II Measuring Research Paper

… The Median is the number in a set that equates to the middle, if every figure in the distribution were to be listed in ascending or descending order. One of the most attractive aspects of the median from a statistical analysis standpoint comes from the fact that "the median is often used instead of the mean for asymmetric data because it is closer to the mode and is insensitive to extreme values in the sample" (Bickel, 2003), and this ability to resist the effects of variant data makes the median an extremely effective diagnostic tool. Median can also be conceptualized as the exact point within a number set where perfect separation into two halves occurs, with exactly 50% of the values in the distribution falling to one side or another, which is why statisticians often refer to the median as the visual center of a numerical distribution.

Crude Range is the result of subtracting the lowest number in a distribution from the highest. The Range equates to the amount of distinct values which are present between these highest and the lowest data points, when one includes the highest and lowest values that are present. The most prevalent limiting factor linked to the application of crude range and range is based on these tools being so reliant on the extreme ends of a distribution set, which typically complicates most analytic processes with the presence of variance. When one refers to the standard deviation, this term describes the precise level of variation or dispersion from the mean that can be detected. When the standard deviation is a low number, this is reflective of data points that are closely situated to the mean, while on the other hand standard deviations that are high numbers are derived from data points that span a wide spectrum.


Bickel, D.R. (2003). Robust and efficient estimation of the mode of continuous data: The mode as a viable measure of central tendency. Journal of statistical computation and simulation, 73(12), 899-912.

Manikandan, S. (2011). Measures of…… [read more]

Criminal Justice and Criminology Interpreting Simple Data Research Proposal

… ¶ … Criminal Justice and Criminology

Interpreting simple data

The data collection exercise involved posting a picture of a bear on Facebook. The caption of the picture asked viewers to provide their thoughts on the picture. This caption was kept… [read more]

ANOVA Study Analysis of Variance Research Paper

… Alternative Hypothesis: There is no significant difference between the treatments that the students are subjected to.


It is important to note here that in this case, there are two assumptions that will be made;

1. That there is a normal population distribution.

2. The variance associated with each variable is the same.

Types of errors likely in ANOVA case

Accuracy in statistics is the degree of closeness of a measurement of a quantity to the quantities true or actual value. Precision also termed as the reproducibility or repeatability is the degree to which repeated measurements under conditions that are not changed. A measurement in a system can be accurate but not precise and also accurate but not precise; it can also be neither or both. There are two categories of errors therefore experienced frequently in such a calculation. Type I and type II errors. Type I error is also known as error of the first kind and it occurs when the null hypothesis is true but is rejected. Type II error is also known as the error of the second kind and occurs when the null hypothesis is false but is not rejected erroneously. An example of the relationship between accuracy and precision if for instance when one reads out time right to the second even if one knows very well that the watch they are reading from is one minute slow, this means that the reading is precise but it is not accurate. An example of the relationship between type 1 and type 2 errors in our case would be that the type of treatment that the students are subjected does affect their marks. Type I error occurs when a conclusion is made that the treatment does not affect the performance in GMAT when it actually does. While a type II error is made when a conclusion is made that the treatment does affect the performance while it actually does not (Shera, 2006). In short, Type I is when the null hypothesis is rejected yet it should not and Type two is when the null hypothesis is accepted yet it should not be accepted.


Shera, J (2006). Statistical Errors (Type I, Type II,…… [read more]

Forensics One of the Most Important Statistical Discussion Chapter

… ¶ … Forensics

One of the most important statistical concept that deals with psychological research is the population being studied. Although it may sound preliminary, deciding who to be studied is the most primary, and thus important, subjects in all of research. Specifically, for psychological research, population selection dictates the basis of many of the subjects involved. In something as subtle and imprecise as psychology, the population the research is selected from is that much more important.

Another statistical idea that is of great importance is the relationship between validity and reliability. Once again, subtle and nuanced definitions of these words help create arguments that become documented and eventually known as facts. Understanding the difference and important of each of these terms can also explain misunderstandings that occur in seemingly well thought out experiments and research. Either way, both concepts can lead to learning and improvement, each in their own way.

Some other statistical ideas are very interesting. One such idea is the ability to predict the future with statistical inference. Although nothing is guaranteed in life, through mathematical relationships, statistics can help create images of the future in predictive and systematic ways. This discovery is truly overlooked in many instances, and in others, too heavily relied on. Finding a balanced and reasoned approach to the incorporation of statistical inference to science remains an interesting challenge.

Dunifon (2005) raised another interesting point in dealing with statistical concepts. He wrote that "experiment is the only way to truly determine whether a treatment causes an outcome." I agree with this very interesting concept,…… [read more]

Russia's Contributions to Science Essay

… Russia's Contribution To Science

Russian contribution to the field of science is famous due to many reasons including the invention of Radio by a. Popov, development of the periodic table by D. Mendeleev, the creation of principals in relation to… [read more]

Random Variable for Each Statement as Being Term Paper

… ¶ … Random Variable for Each Statement as Being Discreet or Continuous by

(a) the number of freshman in the required course, English 101

A) Discreet B) Continuous

(b) the number of phone calls between Florida and New York on Thanksgiving day.

A) Discreet B) Continuous

(c) the height of a radomly selected student.

A) Discreet B) Continuous

(d) the number of spills that occur in a local hospital.

A) Discreet B) Continuous

(e) the braking time of a car.

A) Discreet B) Continuous

Provide an appropriate response.

List the four requirements for a binomial distribution.

(i) Observations are independent

(ii) Outcome of observation is either a success or failure

(iii) Probability of outcome is the same

(iii) Fixed number of observations

Identify each of the variables in the binomial probability formula.

P (x) = __ n!__ . px . qn-x

(n -- x)! x!

n = number of trials x = number of successes p = probability of success q = probability of failure

Also, explain what the fraction __ n!____ computes.

(n -- x)!x!

Number of ways to select 'x' items from 'n' given items

4. Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial.

n = 12, x = 5, p = 0.25, q = 0.75

P (x = 5) = __ 12!__ . (0.25)5 . (0.75)(12-5)

(12 -- 5)! 5!

= __ 12!__ . (0.25)5 . (0.75)7

7! 5!

= 12 x 11 x 10 x 9 x 8 x 7!_ . (0.000977). (0.133484)

7! x 5 x 4 x 3 x 2 x 1

= 0.103241


1. The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at the freezing point of water. Test on a large sample of these instruments reveal that at the freezing point of water, some thermometers give reading below 0° (denoted by negative numbers ). Assume that the mean reading is 0°C and that standard deviation of the reading is 1.00°C. Also assume that the readings are normally distributed. If one thermometer is randomly selected the, find the probability that at the freezing point of water, the reading is less than 1.57°C.



1.5 + 0.07 = 1.57


This is already standardized,

2. If Z. is the standard variable, find the probability, that Z.…… [read more]

Solved Problems Term Paper

… Statistical Estimates

(the Number That Appeared The Most In The Data)

median: (7. 30 + 7.60) / 2 =


range: 8.90-6.60 = 2.30 (Highest number = 8.90, lowest number = 6.60)

Computing standard deviation sample standard deviation =

sample mean =

sample standard deviation =

Using the empirical rule

68% of values lie within mean ± standard deviation

95% of lie within mean ± 2 standard deviation

% lie within mean ± 3 standard deviation

Given mean = 120cm, standard deviation = 12cm

Let x represent percentage of values,

Then, mean ± standard deviation = 120 mm Hg ±12 mm Hg

=108 mm Hg

132 mm Hg

mean ± 2 standard deviation = 120 mm Hg ±24 mm Hg

= 96 mm Hg

144 mm Hg

Thus, the approximate percentage of women between 96 mm Hg and 144mm Hg is 95%

Implementing Chebychev's theorem

Z-score (critical value)

Given that -2.37 is less than -2.00 or 2.37 is greater than 2.00, we may consider this value as unusual

Chapter 4

1. Probability estimate

2. Probability odds

3. Probability odds

The outcomes could be 1 or 2 or 3 or 4, and these are mutually exclusive events. Thus,

P (rolling a number less than 5) = P (1 or 2 or 3 or 4)

= P (1) + P (2) + P (3) + P (4)

= (1/6) + (1/6) + (1/6) + (1/6)

= 4/6 = 2/3 [Answer…… [read more]

Speeches and Presentations Essay

… Speech Organization

When a presentation is made there are certain verbal and visual supports that can be used to aid it. This supports are quite important in that they help in clarity as they make ideas that are complicated clear. They also help in developing interest in the audience as they make the main points more vivid. Finally this supports make the presentation more convincing because they provide evidence that enhance the claims made. This paper will therefore look into these supports and the impact they had on a presentation that I attended. Verbal supports include definitions, examples, stories, statistics, comparisons, quotations, citing sources and so on. On the other hand the visual supports include objects, diagrams, list and tables, photographs and so on.

What captured my interest in the presentation?

The verbal support that captured my interest in the presentation is the use of "Examples." I really liked the way the use of examples has been used in the presentation was. This helped me as the audience in understanding well what the presentations were all about. The brief illustrations of the points allowed me to get exactly what the presenter was talking about and it became very effective since several of the examples were given out.

What confused in the presentation

The use of statistics really confused me. This is because there were many numbers that the speaker used to present their ideas. The choice of the statistics that was applied and hence the statistics presented were very overwhelming to me. Another verbal support that confused me was the use of comparisons. This is because I did not seem to clearly get the figurative analogies that were being applied. They were very confusing to me since I could not seem to get the validity of the comparisons that were being made.

What bored me in the presentation

What made me bored in the presentation was in the way the stories were narrated. For instance, the stories given was non-fiction, hence did not create materials that helps the…… [read more]

Norway Brand Statistical Summary and Hypotheses Decisions Data Analysis Chapter

… Norway Brand

Statistical Summary and Hypotheses Decisions

The statistical method used to compare the experimental group to the control group was straightforward and fairly standard. First, with an established confidence interval of 95% and a significance or alpha of .05,… [read more]

Offline During the Final Exam Week A-Level Coursework

… ¶ … offline during the final exam week. Once you have complete the exam, input your exam into the final exam shell in the exam folder on the course web-page. Good luck

Census statistics show that college graduates make more… [read more]

Teach Geometry Dear Parent Essay

… Children develop their math vocabulary and learn to use appropriate terms. They have an opportunity to connect new understanding to prior knowledge. Math is not simply rote learning of facts and equations. The Common Core State Standards (CCSS), already adopted by forty-five states, were designed to facilitate higher-order thinking and problem solving skills ("Common core standards adoption by state," 2012). These abilities will better prepare students for the real world. Students will communicate with their teachers and with their peers to figure out different ways to solve problems. There is focus on problem solving as a process, so students will be able to understand where they went wrong and so they will be able to solve similar problems in the future.

As far as studying geometry instead of "the basics," geometry is the basis for much of what we do in mathematics. The foundations young students will get in geometry will support their work later on in higher-level mathematics. For example, fractional amounts and percentages are most often represented using geometric shapes. Shapes drawn on a coordinate grid are analyzed in terms of algebraic relationships. Geometry can be thought of as "a conceptual glue" (Schwartz, 2008, p. 72) that connects many different areas within mathematics. With respect to real-world applications, analysis of two- and three-dimensional shapes and the study of geometric relationships are used in fields such as landscaping and architectural design. The ability to specify locations and describe spatial relationships is necessary in transportation, navigation, and construction. Transformations and symmetry are useful in packaging and product design, as well as artistic expression. Geometry made possible the programming of computer graphics and the intuitive interface with computers (Schwartz, p. 72).

Your child's teacher


Chard, D.J., Baker, S.K., Clarke, B., Jungjohann, K., Davis, K., and Smolkowski, K. (2008).

Preventing early mathematics difficulties: The feasibility of a rigorous kindergarten mathematics curriculum. Learning Disability Quarterly 31(1), pp. 11-20.

Common core standards adoption by state. (2012). ASCD. Retrieved from adoption-map.aspx

Cooke, B.D., and Buccholz, D. (2005). Mathematical communication in the classroom: A teacher makes a difference. Early Childhood Education Journal 32(6), pp. 365-369).

Schwartz, J.E. (2008). Elementary…… [read more]

Database Developer (Based on Job Research Paper

… I instructed the optimizer to use a specific access path by using a hint.

d. Access Plan Execution

-- Executes the selected access plan.

I used the EXPLAIN PLAN command for looking at the execution plan of the SQL.

2. Provide examples of errors that you had during your professional experiences.

I had been frustrated with questions that included the following: why the query was running slow; why one query was going slower than another; I wondered whether my index was getting used, and if not, why not... The execution plan told me how the query would be executed, but I had trouble following the steps and had to be habituated to it. Syntactic analysis took a while.

As novitiate, I had difficulty in the beginning reading the code (in the results of the query) as well as understanding the different graphical, text and XML execution plans. The Graphical Plans were somewhat easier to read than the Text Plans, although the detailed data (of graph) was somewhat harder. The format that may have been the most obscure for me was the SHOWPLAN_TEXT

I had also been advised to do certain things for querying and working with the plan cache. I had to run a certain SQL script in order to see how long a plan takes to compile. In the beginning, I had to ask someone in order to understand the objects within the cache (in order to see how the optimizer and storage engine created my plan).

There were also differences between the esti-mated and actual execution plans. This probably occurred due to statistics being stale when over time data was modified and the statistics gradually became mismatched to actual data. I was told that I received bad execution plans because the statistical data was not up-to-date.

Sometimes, when I wanted a parallel query, I saw a completely different plan -- simply because the optimizer felt it could not support my request at that time. At least once or twice the estimated plan didn't work at all since it…… [read more]

Statistical Tests Can Provide More Information Data Analysis Chapter

… ¶ … statistical tests can provide more information than a single one, allowing for more meaningful assessments of a situation. This interaction of two statistical tests (as described below) demonstrates that in this scenario younger women are by far the most likely to be the best employees for this call center.

The Pearson r provides an answer to the question of whether or not two variables are related to each other. More than simply establishing whether a relationship exists or not, Pearson r determines how strong this relationship is and whether it is a direct or inverse relationship. In a direct relationship, if one variable goes up than so does another (or others).

For example, in general as an individual's height goes up, so does his/or her weight. In an inverse relationship, as one variable goes up another one goes down. An example of this would be: The fewer workers are assigned to construct a building the longer it will take to construct the building. Both of these relationships cited here make intuitive sense to us. We may never have considered them to be a part of the world of statistics,…… [read more]

Calculus and Definitions Assessment

… In simple terms the Riemann sum is used to define the definite integral of a function. We begin by considering a simple case, whereby the definition of the Riemann integral of a continuous function f over a rectangle R. In this case rather than having a one-variable case, we can overcome the tendency of connecting integration too strongly with anti-differentiation. (Buck 2003)

Formal definition

It is the definition of a function by use of graphs to define the limits; it uses Greek letters epsilon (?) and Delta (?). Epsilon always represents any distance on the limiting side and delta represents the distance on the x- axis. The limits of a given function clearly explain how that given function behaves when it nears the x value.

Consider the following functions g (x) and f (x), this functions are as a result of definition of real numbers. The following relationship exist x ?

This relationship exist only when there is a positive constant C. such that for all sufficiently large values of x, f (x) is at most C. multiplied by g (x) in absolute value. That is, f (x) = O (g (x)) if and only if there exists a positive real number C. And a real number x0 such that

In general the growth rate are of much interest in that the variable x which goes to infinity is often left unstated, and one writes more simply that f (x) = O (g (x)).

Additional explanation indicates that it doesn't matter how close a function can be to a limit, it is always necessary to find the corresponding x value which is closer to the given value and using the new notations of epsilon (?) and delta (?), we make f (x) within ? Of L, the limit, and later determine x within ? Of C. (Bradley et al.,.2000)

Again, since this is tricky, let's resume our example from before: f (x) =x2at x=2. To start, let's say we want f (x) to be within .01 of the limit. We know by now that the limit should be 4, so we say: for ?= 0.1, there is some ? so that as long as, then

To show this, we can pick any delta (?) that is bigger than 0, so long as it works. For example, you might pick .000000001, because you are absolutely sure that if x is within .00000000000001 of 2, then f (x) will be within .01 of 4. This works for. But we can't just pick a specific value for, like .01, because we said in our definition "for every." This means that we need to be able to show an infinite number of s, one for each.

In summary indefinite integration exists when the limits of integration are not given, this means that the upper and the lower limits are not given. Definite integration occurs when the limits are given and therefore you need to calculate the area that is when x=c to x=d

Examples… [read more]

Mathematical Modeling Term Paper

… ¶ … Mathematical Modeling

Although even complex mathematical modeling is certainly not new, the process has been facilitated enormously in recent years by the introduction of computer-based modeling applications. Despite these innovations, there are still some significant limitations to mathematical modeling that must be taken into account when using these techniques. To gain some additional insights in this area, this paper provides a review of the relevant literature to identify the benefits and limitations of mathematical modeling, a discussion concerning the use of mathematical modeling in the author's profession and the extent to which such modeling is used as value-added to other kinds of empirical research, and the extent to which it is used in place of other kinds of empirical research. A summary of the research and important findings are presented in the conclusion.

Review and Analysis

Serious interest in mathematical modeling emerged during the mid-20th century when computer science was in its infancy but the need for ways to simulate real-world situations became pronounced. According to Maxwell (2004), "The federal government and many private enterprises have used mathematical modeling since the late 1950s as aids in developing policies, conducting research and development, and engineering complex systems" (p. 67). Today, computer-driven mathematical modeling applications have a number of real-world applications, including gambling and sports simulations as well as modeling human interactions for couples therapy and other "people prediction" applications (Albert, 2002). In this regard, Oliver and Myers report that, "Game theory provides a rich history of considering the strategies derived from various payoff structures, rules about repeating the game, and how players communicate" (p. 34). Mathematical modeling has proven efficacy in other settings as well, including the entire range of economic analyses (Oliver & Myers, n.d.) and even enormously complex weather prediction applications (Kirlik, 2006). Moreover, mathematical modeling has been used to good effect in helping researchers better understand how physiological processes operate at the molecular level. For example, Peter (2008) reports that, "Mathematical models allow researchers to investigate how complex regulatory processes are connected and how disruptions of these processes may contribute to the development of disease" (p. 49).

Furthermore, mathematical modeling can facilitate the systematic analyses of various "what-if"-type scenarios (Oliver & Myers, n.d.), formulate new hypotheses to serve as the basis for regimens of therapeutic interventions and even to evaluate the appropriateness of specific molecules for therapeutic purposes (Peter, 2008). According to Peter, "Numerous mathematical methods have been developed to address different categories of biological processes, such as metabolic processes or signaling and regulatory pathways. Today, modeling approaches are essential for biologists, enabling them to analyze complex physiological processes, as well as for the pharmaceutical industry, as a means for supporting drug discovery and development programs" (2008, p. 50). In fact, some authorities suggest that the limits of mathematical modeling are fundamentally human-based rather than technologically restricted. In this regard, Maxwell (2004) points out that, "Mathematical modeling and computer simulation are limited only by the ingenuity of the person or team conducting the analysis. They have… [read more]

Score Z Scores Z Research Proposal

… 33

P= Area of the curve beyond 1.33

=0.0918 or 9.18%

b) Less than 80 minutes

Using formula Z=X-? + ?




P= Area of the curve below -0.67



Between 45-75

Using the formula Z=X-? + ?

Z (45) =45-60/15


Z (75) = 75-60/15


P between -1 and 1

= 0.3413 + 0.3413


% =68.26%

Question 3

Bob takes an online IQ test and finds that his IQ according to the test is 134. Assuming that the mean IQ is 100, the standard deviation is 15, and the distribution of IQ scores is normal, what proportion of the population would score higher than Bob? Lower than Bob?

Proportion higher than Bob would be the area of the curve beyond Bob's score, the proportion lower than Bob would be the area of the curve below Bob's score.

Using the formula Z=X-? + ?

Z =134-100/15



P Higher than Bob's Score = 0.0116

P Lower than Bob's Score =0.9984

References…… [read more]

Strategic Plan on Janix Healthcare Consultation Essay

… ¶ … people test hypotheses?

A hypothesis is a statement that predicts the outcome of an experiment or research. Every experiment must have a hypothesis statement which shows the aim of an experiment. It is usually an educated guess and indicates the expectations of a researcher. Carrying out a number of experiments, can either approve or disapprove a hypothesis Moschopoulos & Davidson, 1985.

A hypothesis is formed after literature study has been finished and the problem of the study stated. There are different types of hypothesis these include, Inductive hypothesis, based on specific observation, deductive hypothesis provides evidence that expands, supports or contradicts a theory. Non-directional hypothesis states the relationship or difference between variables, directional hypothesis defines the desired direction of the relation or difference between the variables. Null hypothesis states that there is either a significant difference between variables or no significant relationship between the variables Dembo & Peres, 1994()

Hypothesis should be specific while its concept should be clear. One should be able to test a hypothesis and should be related to the theory. A hypothesis should recognize certain variables, must be verifiable and must be in simple terms made easy to understand.

For example, in the article about the changing roles of teachers in an era of high stakes accountability, it is hypothesized that the teachers have changing roles as high stakes accountability becomes increasingly pervasive in their day-to-day work. In the second article on the supervisor perceptions of the quality of troops to teachers program, the hypothesis is that the T3 program increases the professional education level of the teachers and has a positive impact on the achievement of students.


A theory is a statement or a principle that has been well established so as to explain and describe the cause and effect of a certain research investigation. A theory summarizes a hypothesis .These hypotheses have been accepted to be true by a number of experiments. A theory is what hypotheses become when they are accepted to be true. A theory remains valid until evidence that disputes it arises. Some of the scientific theories that are true include Newton's theory of gravity. This theory has proved to be true because it has enabled man to send others to the moon and even launch satellites. Other theories include Maxwell's theory of electromagnetism, periodic table theory, Einstein's theory of relativity, Quantum theory and Darwin's theory of evolution. These theories have been tested and accepted by the scientific community. Components of a theory can be improved on upon further experiments in the future or changed, but the overall truth of the theory is not changed Loosen, 1997()

Sources of theories and hypothesis

The activities that happen in our daily life may serve as a source for developing a hypothesis. An example of…… [read more]

How Do We Combat Math Anxiety? Research Paper

… Math Anxiety

How to Combat Math Anxiety

Causes for Anxiety

How to Begin to Help

What Schools Can Do

What Parents Can Do

Albert Einstein once stated, "Do not worry about your difficulties in mathematics; I assure you mine are… [read more]

Bayes Probability Can Bayes Confirmation Essay

… [footnoteRef:7] He was studying the paradox which arises from the use of Bayes theorem when trying to explain phenomena in the field of psychology. He states that; [7: PE Meehl, 'Theory-testing in psychology and physics: A methodological paradox', Philosophy of… [read more]

Devise a Standard of Existence Rule Essay

… ¶ … Existence / Rule for Existence

Existence is a philosophical question that has eluded thinkers for centuries. From as early as ancient Greek, philosophers have sought to define existence as a concept to encompass not only the physical world, but those objects that exist in different non-physical plains. It is these issues that present the challenge in determining the true meaning of existence and non-existence.

An object exists when it has a form that is not in violation of any universal rules or truths. A form is any physical, metaphysical, or cognitive presentation. Universal rules are those derived in science and mathematics such as gravity, mass, geometry, and algebra. Truths are those statements that are absolute and cannot be refuted. When held against this definition of existence, a horse, the number four, and a unicorn exist whereas the square circle does not exist.

A horse is the most obvious of the items that exists. The reason is the horse fulfills the definition perfectly. First, the horse exists in a form, two forms actually. The first form is its archetypical form. This is the form in the mind that creates the definition of a horse. A object exists in an archetypical form when the mentioning of the object brings a specific image or definition to the mind. In this case, when the word "horse" is mentioned, a person immediately conjures up the image of a four-legged mammal with hooves, main and tail. Horses also exist in a physical for as well. Their physical form exists in the third dimension along-side humans. This means that horses can be touched, smelled, heard, watched and interacted with. It is these features that even greater solidify the horse's existence within the mind. Now to address the second and third parts of the definition. A horse is not in violation of any universal rules or truths. Its very definition, in fact, is solely derived from its physical form and the observations thereof. So, a horse meets the full criteria of an existing object and therefor does exist.

The number 4 also exists, except it does not exist in the same way that a horse exists. Unlike a horse, the number four is not a living thing, in the sense that it breaths, eats, or grows. It does still, nonetheless exist. The number 4, like the horse, has two forms. The first form is the archetypical form. When the number 4 is mentioned, those trained in mathematical law immediately conjure up the image. While this time the image is not as physical as it was with the horse, the concept can still be conjured within the mind and is solidified when tied to another object such as the horses. When 4 horses is mentioned, it becomes even easier to envision the number 4 in use. The second form that the number four can take is physical. Once again, unlike the horse, the physical form is not alive, but it still exists. This form, commonly referred to… [read more]

Nursing Research: Discussion Questions Quantitative Essay

… The human rights of subjects must always be protected in research -- and that includes not publishing data that could result in harm to individuals, who are treated in a particular manner, based upon inaccurate data.

Q4. A simple hypothesis states the relationship between two variables. A complex hypothesis states the relationship between three or more variables. A nondirectional hypothesis states that a relationship exists between two variables. A directional hypothesis predicts the relationship between the two variables (Burns 2010: 172-173). An associative hypothesis describes phenomenon that occur together, while causal hypothesis describes one phenomenon that causes another (Burns 2010: 168). A null hypothesis states that there is no relationship between two variables (and usually the researcher wants to disprove the null hypothesis). The research hypothesis states that there is a relationship between the two variables, which the researcher is usually trying to prove.

Q5. Quantitative research attempts to accumulate numerical data about a specific phenomenon. A quantitative literature review attempts to accumulate data from a vast array of different quantitative studies, to either describe or find out specific tendencies in the types of hypotheses tested regarding the phenomenon. Of course, it is rare that all studies will reach the same conclusion, so the researchers will evaluate the quality of the studies (for example, if a study produces an anomalous result, the author of the review will likely try to determine why this is the case, such as if there was too small a sampling size). The literature review may reach a conclusion about the phenomenon, based upon statistically analyzing the data. A qualitative research study merely assesses the variety of informational studies on a particular phenomenon to paint a clearer picture of…… [read more]

Linear Regression Models (Meier, Chapter 18 Article Review

… Linear Regression models (Meier, Chapter 18 / 19)

These are used in order to determine whether a correlation (or relationship) exists between one element and another and, if so, in which direction (negative or positive).

The two variables are plotted on a graph. Independent variable on the x line (horizon); y- variable (dependent) on the vertical line. The pattern between them is called the 'slope'. The point where X and Y intersect online is called 'intercept'.

The theorem used tells us that the slope of the line will be equal to the change in x (IV) given changes of y (DV). The shape of the slope (their direction and gradient) describes the relationship between X and Y.

Linear regression, as are the previous models, is used apply results population sample to population as a whole. >Linear regression is also useful for predicting occurrences in that sphere. For instance, linear regression may be used to determine whether there is a correlation between vehicle collisions and rainy days. If so, one can predict that the stormier the weather the greater the quantity of collisions.

Goodness of Fit

We will want to know the amount of error i.e. how well the regression line fits the data. The distance a point is from the regression line is known as error. A certain calculation exists to find this out. Another goodness of fit measure is the standard error of the estimation where a calculation is used to find out the extent to which the results of the sampled population will correspond to the population as a whole. Thirdly, the coefficient of determination is used to measure the total variation in the independent variable (X). Complex calculations exist for this. (All of these calculations can be worked out by special computer programs too).

Linear regression has various assumptions:

1. For any value in X, the errors in predicting Y are normally distributed with a mean of zero.

2. Errors do not get larger as X becomes larger; rather the errors remain constant throughout slope regardless of the X value.

3. The errors of Y and X are independent of one another.

4. Both IV and DV (X and Y) must be interval variables (i.e. numerical data).

5. The relationships between X and Y are linear.

Ignoring these assumptions will result in faulty statistical conclusions.

Topic 2: Comparing 2 Groups

A researcher may run the same study on two different groups with one, for instance, acting as control and the other as experimental. He may then want to know whether differences are observed between the two groups.

1. Research and null hypothesis are drawn up stating that: (a) significant difference will be found, (b) significant difference will not be found between both groups.

e.g. Alternative Hyp. H1: Employees who have taken *program will have higher job scores

Null hyp (H0): There is no difference in scores between employees who have taken program and employees who have not.

2. Mean and standard deviation of each group is calculated… [read more]

Improved Your Knowledge, Skills, Abilities, and Yourself Essay

… ¶ … Improved Your Knowledge, Skills, Abilities, and Yourself in This Session Through This Course

The mathematical skill that was taught in the course is necessary for a career that may be related to business, investment and analysis. In other words the course shows the path to business mathematics by the use of exponential, and logarithmic functions that management use in the Managerial information systems -- MIS and this is also used with functions, set theories and other allied matters thought to analyze data and solve problems related to the market, and necessary decisions to be made can be based on these knowledge. To that extent I feel I have gained a lot. The overall experience is that the course has given me the ability to attend to problems that I once feared. Due to this course the approach to math, which I always approached with trepidation has changed and am now ready and willing to go further in exploring mathematics, both for its academic interest and also as a useful tool for me in my daily work.

2. Evaluation of the work you did during the session for the class and explanations of ways you could have performed better

I have been introduced and provided guidance and training to seek the results of complicated functions that can provide with answers to every day questions that I may have to answer in the course of my trade or occupation and generally in life. These include the data sets. The lessons on functions were tough and I believe that I could have put in better effort there. The functions are still hazy but I found the use of the calculation of simple things like interest, and the profitability etc. As something very useful to me. I did concentrate more on them and perhaps they may have caused the problems in my understanding of the other topics. I believe that I could have done better with the study of functions, and I performed rather well but I could have done better.

There were small gaps in my understanding probably a result of my anxiety…… [read more]

Human Factors Affecting Safe Operation Data Analysis Chapter

… ¶ … Human Factors Affecting Safe

Operation Of The UAV

Study of Selected Human Factors affecting safe operation of the UAV

This chapter presents the findings of the thesis. The survey questionnaires are collected from the 35 respondents. The data… [read more]

NBA Stats Term Paper

… 1 (for points). Mean height is 78.7 and mean points is 16.843, so calculating the covariance of this data pair would look like this:

(72 -- 78.7)(13.1-16.843) = 25.0781

In order to develop the value for each pair, two new columns were created in the Excel sheet to calculate the difference of the data point and the mean with the following formulas:

=a2-78.7 (for height, pasted into the other 99 rows which automatically adjust the cell value for each row, as =a3-78.7, a4-78.7, etc.), and =b2-16.843 (for points, also pasted).

A third column with the following formula created the product of each row in these two columns:

=c2*d2 (pasted for all rows).

The average function was used to determine the mean of this last row, which is the covariance: 2.1679. Standard deviations are 3.60274867 (for height) and 3.74805687 (for points), meaning Pearson's coefficient is calculated as:

2.1679/(3.60274867 * 3.74805687) = 0.160545859.

Linear regression attempts to find the line of best fit for a data sample. The basic equation of any line for variables x and y is given as y = a + bx. For a linear regression slope (b) is calculated as:

((n*?xy)-(?x)(?y)) / ((n*?x2)-(?x) 2)

Adding another column to the spreadsheet enabled the quick calculation of height * points (xy) for each data pair, and a column for x2 was also added; the SUM function was used to calculate ?xy (132771.2), ?x (7870), ?y (1684.3), and ?x2 (620654). With n (population size) 100, the slope (b) of the equation becomes:

((100*132771.2)-(7870)(1684.3))/((100*620654)-(78702) = 0.168708171.

The intercept (a) is calculated as:

(?y-b (?x))/n which with the substituted values becomes

(1684.3-(0.168708171*7870))/100 = 3.56566694.

The full linear regression equation for this data set, then, is given as:

points = 3.56566694 + (0.168708171*height)

Chi-square analysis cannot be applied…… [read more]

Mathematical Knowledge for Teaching Article Review

… Mathematical Knowledge in Education

Differentiating Types of Mathematical Knowledge and Relevance to Education

Ball, D.L., Lubienski, S., and Mewborn, D. (2001). "Research on teaching mathematics:

The unsolved problem of teachers' mathematical knowledge." (In Handbook of Research on Teaching. New York: Macmillan).

Generally, mathematics proficiency among teachers corresponds to higher achievement in their students. While that conclusion has been supported by a substantial volume of empirical research, much less empirical research has been devoted to trying to understand how and why teacher achievement in mathematics benefits student outcomes, or what it is about mathematics, specifically, that generates this apparent relationship. Most importantly, there is a need to understand whether and to what extent teacher mathematics achievement in different aspects of mathematics matters with regard to the positive effect on learners.

According to the authors of this article, there is a fundamental difference between teaching mathematics and teaching through mathematics. In many ways, that distinction helps explain why, in general, mathematics proficiency among teachers tends to correspond to better learning outcomes. More particularly, understanding that distinction may help explain why the positive benefit of mathematics knowledge among teachers is much more evident in connection with their academic study of mathematical method than in connection with their academic study of advanced mathematics. Furthermore, it could explain why advanced mathematical achievement among teachers also corresponds to higher incidence of negative affects on some learners whereas that is not true in the case of teachers whose high achievement in mathematics relates more to their non-pedagogical content knowledge than to their pedagogical content knowledge.

In principle, the value of teaching mathematics is much broader than the value of the substantive material, particularly in contemporary society that provides instant and accessible electronic calculation to solve the types of mathematical problems that could typically arise in everyday adult life. Study after study suggests that teachers who are more knowledgeable about mathematics tend to promote learning better than teachers who are less proficient in mathematics.

However, there is evidence that suggest that this relationship is much more complex than simply a direct transfer of pedagogical mathematical knowledge. For example, one unexpected finding is that the benefit of greater mathematics proficiency exists in the first grade. Presumably, all teachers are equally proficient at first-grade addition and subtraction; moreover, the academic study of mathematics in greater depth (i.e. post-calculus) should not have any impact on the level of teacher understanding of first-grade mathematics concepts. Similarly, there is no intuitive reason that either the mathematical proficiency of teachers or their highest level of mathematical study should translate to better teaching of elementary mathematical concepts. In that background, the correspondence between teachers having studied mathematical method and the highest identifiable benefits to learning seem to explain the basis of the phenomenon.

Specifically, mathematics (especially at the elementary level), can be taught rigidly and by rote rule or by conceptual understanding. Apparently, teachers with more extensive experience in studying mathematical method are better equipped to deliver mathematics lessons in a manner conducive to inspiring… [read more]

Butts, R.E. ). Galileo. In W.H. Newton-Smith Annotated Bibliography

… Butts, R.E. (2001). Galileo. In W.H. Newton-Smith (Author), a companion to the philosophy of science (pp. 149-152). Malden, MA: Blackwell Pub.

This excerpt from a reference work is a biographical sketch of Galileo, 17th

century Italian scientist. It outlines five crucial achievements that he made, a few of which include his divergence from Aristotelian theories of science, his advocacy of the real-world applications of mathematics, and his use of experimentation. The author outlines the origins of Galileo's scientific research, particularly in cosmology and his work with the telescope. His work also centered around making geometry less abstract. He pointed out how geometric laws worked in concert with both the natural and mechanical worlds. The most compelling point the author makes, which has application even today to teachers and researchers, is about Galileo's rejection of the dominant philosophies of the era. Though he created controversy, opposition drove his science to advance, leading ultimately to success.

Frodeman, R., & Parker, J. (2009). Intellectual merit and broader impact: The National

Science Foundation's broader impacts criterion and the question of peer review.

Social Epistemology, 23(3-4), 337-345.

The article examines how scientific discovery dictates social values, and how the philosophy of science has evolved. Science has historically been funded with an eye to how it will benefit society. The specific focus of the piece is on the NSF

peer review process and the change of criteria used to allocate funding that occurred in 1997. This change created two criteria: intellectual merit and broader impacts. The broader impacts criteria include education/outreach and an effort to broaden diversity. But the question remains: How is benefit to society determined and measured? The article also raises the question of whether these two criteria categories should be merged, and if intellectual merit is actually a subset of broader impact. This is brought up to point of the potential pitfalls of peer review and to call for a closer examination of its procedures. This is relevant to math education in that research into education practice should be viewed with a mind toward its application in the lives of students and its greater impact on society.

Lesser, L.M. (2000). Reunion of broken parts: Experiencing diversity in algebra.

Mathematics Teacher, 93(1), 62-67.

The author employs a central metaphor of the meaning of algebra, that being the reunion of broken parts. He compares this to the way students interact with algebra, given that they can feel disconnected from it and it is the job of teachers to provide real world applications that will make connections for the students. He takes aim particularly at the need to…… [read more]

Derivatives and Definite Integrals Word Term Paper

… This is done with a "time derivative" which defines the rate of change over time. The instantaneous velocity of an object is calculated by the coordinate derivative relative to time. To know how quickly the velocity of a given object will change in the course of time, another value called acceleration is defined. Thus, acceleration is the time-derivative of an object's velocity.

How are derivatives used to solve maximum and minimum problems? The maxima and the minima, also known as the extremum, are the values of the largest and smallest limits that a function may take in a given point. They are expressed as a set of greatest and least values. Derivatives are used to determine these points. The derivative of a function is interpreted geometrically as the slope of the curve of the mathematical function y (t), whereas the function of t is plotted. The derivative is noted as positive when a function increases toward a maximum; the maximum being horizontal at zero. It is considered negative just beyond the maximum. The second derivative notates the rate of change. It is called negative since the process of the slope, as described, is always getting smaller. The second derivative is always negative and corresponds to a maximum.

How is the definite integral used to solve area problems? A definite integral is an integral with upper and lower limits. It is used to calculate the area of certain regions in the plane. In the calculation of the area of a region, the region finds a limit above by the graph of a function f (x), below by the x -axis, and on two sides by vertical lines that correspond to the equations x = a and x = b.


Nave, R. Derivatives and integrals. Hyper Physics, Retrieved from

Kouba, D.A. The Calculus Page, U.C. Davis Department of Mathematics., Retrieved January 25, 2011

Weisstein, E. Wolfram Mathworld., Retrieved January 25, 2011… [read more]

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