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Boundless Lecture Slides

Boundless Lecture Slides. Available on the Boundless Teaching Platform. Free to share, print, make copies and changes. Get yours at www.boundless.com. Using Boundless Presentations. Boundless Teaching Platform

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Boundless Lecture Slides

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  1. Boundless Lecture Slides Available on the Boundless Teaching Platform Free to share, print, make copies and changes. Get yours at www.boundless.com

  2. Using Boundless Presentations Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Get started now at: • The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. http://boundless.com/teaching-platform • Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. If you have any questions or problems please email: educators@boundless.com Free to share, print, make copies and changes. Get yours at www.boundless.com

  3. About Boundless • Boundless is an innovative technology company making education more affordable and accessible for students everywhere. The company creates the world’s best open educational content in 20+ subjects that align to more than 1,000 popular college textbooks. Boundless integrates learning technology into all its premium books to help students study more efficiently at a fraction of the cost of traditional textbooks. The company also empowers educators to engage their students more effectively through customizable books and intuitive teaching tools as part of the Boundless Teaching Platform. More than 2 million learners access Boundless free and premium content each month across the company’s wide distribution platforms, including its website, iOS apps, Kindle books, and iBooks. To get started learning or teaching with Boundless, visit boundless.com. Free to share, print, make copies and changes. Get yours at www.boundless.com

  4. The Normal Curve Continuous Random Variables Normal Approximation Measurement Error Expected Value and Standard Error ] Normal Approximation for Probability Histograms Continuous Random Variables Free to share, print, make copies and changes. Get yours at www.boundless.com

  5. Continuous Random Variables > The Normal Curve The Normal Curve • Continuous Probability Distributions • The Uniform Distribution • The Exponential Distribution • The Normal Distribution • Graphing the Normal Distribution • The Standard Normal Curve • Finding the Area Under the Normal Curve Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/continuous-random-variables-10/the-normal-curve-39/

  6. Continuous Random Variables > Normal Approximation Normal Approximation • The Normal Approximation to the Binomial Distribution • The Scope of the Normal Approximation • Calculating a Normal Approximation • Change of Scale Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/continuous-random-variables-10/normal-approximation-40/

  7. Continuous Random Variables > Measurement Error Measurement Error • Bias • Chance Error • Outliers Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/continuous-random-variables-10/measurement-error-41/

  8. Continuous Random Variables > Expected Value and Standard Error Expected Value and Standard Error • Expected Value • Standard Error Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/continuous-random-variables-10/expected-value-and-standard-error-42/

  9. Continuous Random Variables > Normal Approximation for Probability Histograms Normal Approximation for Probability Histograms • Probability Histograms • Probability Histograms and the Normal Curve • Conclusion Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/continuous-random-variables-10/normal-approximation-for-probability-histograms-43/

  10. Appendix Free to share, print, make copies and changes. Get yours at www.boundless.com

  11. Continuous Random Variables Key terms • Accuracythe degree of closeness of measurements of a quantity to that quantity's actual (true) value • bell curveIn mathematics, the bell-shaped curve that is typical of the normal distribution. • best fit lineA line on a graph showing the general direction that a group of points seem to be heading. • binomial distributionthe discrete probability distribution of the number of successes in a sequence of independent yes/no experiments, each of which yields success with probability • Box–Muller transformationA pseudo-random number sampling method for generating pairs of independent, standard, normally distributed (zero expectation, unit variance) random numbers, given a source of uniformly distributed random numbers. • central limit theoremThe theorem that states: If the sum of independent identically distributed random variables has a finite variance, then it will be (approximately) normally distributed. • central limit theoremThe theorem that states: If the sum of independent identically distributed random variables has a finite variance, then it will be (approximately) normally distributed. • central limit theoremThe theorem that states: If the sum of independent identically distributed random variables has a finite variance, then it will be (approximately) normally distributed. • correlationOne of the several measures of the linear statistical relationship between two random variables, indicating both the strength and direction of the relationship. • cumulantAny of a set of parameters of a one-dimensional probability distribution of a certain form. • cumulative distribution functionThe probability that a real-valued random variable with a given probability distribution will be found at a value less than or equal to . • datumA measurement of something on a scale understood by both the recorder (a person or device) and the reader (another person or device). Free to share, print, make copies and changes. Get yours at www.boundless.com

  12. Continuous Random Variables • discrete random variableobtained by counting values for which there are no in-between values, such as the integers 0, 1, 2, …. • empirical ruleThat a normal distribution has 68% of its observations within one standard deviation of the mean, 95% within two, and 99.7% within three. • empirical ruleThat a normal distribution has 68% of its observations within one standard deviation of the mean, 95% within two, and 99.7% within three. • entropyA measure which quantifies the expected value of the information contained in a message. • Erlang distributionThe distribution of the sum of several independent exponentially distributed variables. • independentnot dependent; not contingent or depending on something else; free • integralthe limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed • interquartile rangeThe difference between the first and third quartiles; a robust measure of sample dispersion. • law of large numbersThe statistical tendency toward a fixed ratio in the results when an experiment is repeated a large number of times. • Lebesgue measureThe unique complete translation-invariant measure for the -algebra which contains all -cells—in and which assigns a measure to each -cell equal to that -cell's volume (as defined in Euclidean geometry: i.e., the volume of the -cell equals the product of the lengths of its sides). • normal approximationThe process of using the normal curve to estimate the shape of the distribution of a data set. • normal approximationThe process of using the normal curve to estimate the shape of the distribution of a data set. Free to share, print, make copies and changes. Get yours at www.boundless.com

  13. Continuous Random Variables • normal probability plota graphical technique used to assess whether or not a data set is approximately normally distributed • normal probability plota graphical technique used to assess whether or not a data set is approximately normally distributed • normalizationThe process of removing statistical error in repeated measured data. • outliera value in a statistical sample which does not fit a pattern that describes most other data points; specifically, a value that lies 1.5 IQR beyond the upper or lower quartile • p-valueThe probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. • Poisson processA stochastic process in which events occur continuously and independently of one another. • Precisionthe ability of a measurement to be reproduced consistently • random erroran error which is a combination of results both higher and lower than the desired measurement; precision error • random erroran error which is a combination of results both higher and lower than the desired measurement; precision error • random variablea quantity whose value is random and to which a probability distribution is assigned, such as the possible outcome of a roll of a die • real numberAn element of the set of real numbers; the set of real numbers include the rational numbers and the irrational numbers, but not all complex numbers. • regressionAn analytic method to measure the association of one or more independent variables with a dependent variable. Free to share, print, make copies and changes. Get yours at www.boundless.com

  14. Continuous Random Variables • regression lineA smooth curve fitted to the set of paired data in regression analysis; for linear regression the curve is a straight line. • standard normal distributionThe normal distribution with a mean of zero and a standard deviation of one. • standard scoreThe number of standard deviations an observation or datum is above the mean. • systematic erroran error which consistently yields results either higher or lower than the correct measurement; accuracy error • systematic erroran error which consistently yields results either higher or lower than the correct measurement; accuracy error • weighted averagean arithmetic mean of values biased according to agreed weightings • z-scoreThe standardized value of observation from a distribution that has mean and standard deviation . • z-scoreThe standardized value of observation from a distribution that has mean and standard deviation . • z-scoreThe standardized value of observation from a distribution that has mean and standard deviation . Free to share, print, make copies and changes. Get yours at www.boundless.com

  15. Continuous Random Variables Normal Probability Density The normal distribution is described by this probability density function. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Normal distribution."CC BY-SAhttps://en.wikipedia.org/wiki/Normal_distributionView on Boundless.com

  16. Continuous Random Variables Low Accuracy, High Precision This target shows an example of low accuracy (points are not close to center target) but high precision (points are close together). In this case, there is more systematic error than random error. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."High precision Low accuracy."CC BY-SAhttp://en.wikipedia.org/wiki/File:High_precision_Low_accuracy.svgView on Boundless.com

  17. Continuous Random Variables Probability Histogram This probability histogram shows the probabilities that 0, 1, 2, 3, or 4 heads will show up on four tosses of a fair coin. Free to share, print, make copies and changes. Get yours at www.boundless.com Hofstra University."Histogram Generator."CC BYhttp://people.hofstra.edu/stefan_waner/stats/histogram.htmlView on Boundless.com

  18. Continuous Random Variables Normal Probability Plot The data points do not deviate far from the straight line, so we can assume the distribution is approximately normal. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Normprob."CC BY-SAhttp://en.wikipedia.org/wiki/File:Normprob.pngView on Boundless.com

  19. Continuous Random Variables Memoryless Exponential Distributions If a random variable T is exponentially distributed, its conditional probability obeys this formula. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Exponential distribution."GNU FDLhttp://en.wikipedia.org/wiki/Exponential_distributionView on Boundless.com

  20. Continuous Random Variables The Normal Distribution This image shows the equation for the normal distribution. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Normal distribution."CC BY-SAhttps://en.wikipedia.org/wiki/Normal_distributionView on Boundless.com

  21. Continuous Random Variables High Accuracy, Low Precision This target shows an example of high accuracy (points are all close to center target) but low precision (points are not close together). In this case, there is more random error than systematic error. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."High accuracy Low precision."CC BY-SAhttp://en.wikipedia.org/wiki/File:High_accuracy_Low_precision.svgView on Boundless.com

  22. Continuous Random Variables Low Accuracy, High Precision This target shows an example of low accuracy (points are not close to center target) but high precision (points are close together). In this case, there is more systematic error than random error. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."High precision Low accuracy."CC BY-SAhttp://en.wikipedia.org/wiki/File:High_precision_Low_accuracy.svgView on Boundless.com

  23. Continuous Random Variables Approximately Normal - Probability Plot This is a sample of size 50 from a normal distribution, plotted as a normal probability plot. The plot looks fairly straight, indicating normality. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Normprob."CC BY-SAhttp://en.wikipedia.org/wiki/File:Normprob.pngView on Boundless.com

  24. Continuous Random Variables Catching a Bus The Uniform Distribution can be used to calculate probability problems such as the probability of waiting for a bus for a certain amount of time. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Arriva T6 nearside."Public domainhttps://en.wikipedia.org/wiki/File:Arriva_T6_nearside.JPGView on Boundless.com

  25. Continuous Random Variables High Accuracy, Low Precision This target shows an example of high accuracy (points are all close to center target) but low precision (points are not close together). In this case, there is more random error than systematic error. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."High accuracy Low precision."CC BY-SAhttp://en.wikipedia.org/wiki/File:High_accuracy_Low_precision.svgView on Boundless.com

  26. Continuous Random Variables -Score Table The -score table is used to calculate probabilities for the standard normal distribution. Free to share, print, make copies and changes. Get yours at www.boundless.com Mandy's Notes."Z-Score Chart."CC BYhttp://www.mandysnotes.com/Statistics/Z-Scores/Z-Score-ChartView on Boundless.com

  27. Continuous Random Variables Normal Area 1 This graph shows the area below 8.5. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax CNX."David Lane, Normal Approximation to the Binomial. June 12, 2013."CC BY 3.0http://cnx.org/content/m11162/latest/View on Boundless.com

  28. Continuous Random Variables The Bell Curve The graph of a normal distribution is known as a bell curve. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikispaces."killianhma0809 - Normal Distribution."CC BY-SAhttp://killianhma0809.wikispaces.com/Normal+DistributionView on Boundless.com

  29. Continuous Random Variables Height of a Bell Curve The height of the graph at any x value can be found through this equation. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikispaces."killianhma0809 - Normal Distribution."CC BY-SAhttp://killianhma0809.wikispaces.com/Normal+DistributionView on Boundless.com

  30. Continuous Random Variables Normal Area 2 This graph shows the area below 7.5. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax CNX."David Lane, Normal Approximation to the Binomial. June 12, 2013."CC BY 3.0http://cnx.org/content/m11162/latest/View on Boundless.com

  31. Continuous Random Variables Normal Approximation Approximation for the probability of 8 heads with the normal distribution. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax CNX."David Lane, Normal Approximation to the Binomial. June 12, 2013."CC BY 3.0http://cnx.org/content/m11162/latest/View on Boundless.com

  32. Continuous Random Variables Boxplot Versus Probability Density Function Boxplot and probability density function of a normal distribution . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia."Boxplot vs PDF."CC BY-SAhttp://commons.wikimedia.org/wiki/File:Boxplot_vs_PDF.svgView on Boundless.com

  33. Continuous Random Variables Mean of Exponentially Distributed Random Variable Random variable X and rate parameter of . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Exponential distribution."GNU FDLhttp://en.wikipedia.org/wiki/Exponential_distributionView on Boundless.com

  34. Continuous Random Variables Variance of an Exponentially Distributed Random Variable Random variable X and rate parameter of . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Exponential distribution."GNU FDLhttp://en.wikipedia.org/wiki/Exponential_distributionView on Boundless.com

  35. Continuous Random Variables Graph 1 Bell curve visualizing a normal distribution with a relatively small standard deviation. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikispaces."killianhma0809 - Normal Distribution."CC BY-SAhttp://killianhma0809.wikispaces.com/Normal+DistributionView on Boundless.com

  36. Continuous Random Variables Areas Under the Normal Curve This table gives the cumulative probability up to the standardized normal value . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikidot."Using Normal Distributions - IB Math Stuff."CC BY-SAhttp://ibmathstuff.wikidot.com/usingnormaldistributionsView on Boundless.com

  37. Continuous Random Variables -table The -score table is used to calculate probabilities for the standard normal distribution. Free to share, print, make copies and changes. Get yours at www.boundless.com http://statistics.wdfiles.com/local--files/ch7/normDistTable.pdf.CC BY-SAhttp://statistics.wdfiles.com/local--files/ch7/normDistTable.pdfView on Boundless.com

  38. Continuous Random Variables Normal Approximation The normal approximation to the binomial distribution for 12 coin flips. The smooth curve is the normal distribution. Note how well it approximates the binomial probabilities represented by the heights of the blue lines. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax CNX."David Lane, History of Normal Distribution. June 12, 2013."CC BY 3.0http://cnx.org/content/m11164/latest/View on Boundless.com

  39. Continuous Random Variables Central Limit Theorem A distribution being "smoothed out" by summation, showing original density of distribution and three subsequent summations Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Central limit thm."CC BY-SAhttp://en.wikipedia.org/wiki/File:Central_limit_thm.pngView on Boundless.com

  40. Continuous Random Variables Normal Distribution and Scales Compares the various grading methods in a normal distribution. Includes: standard deviations, cumulative percentages, percentile equivalents, -scores, -scores, and standard nine. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Normal distribution and scales."Public domainhttp://en.wikipedia.org/wiki/File:Normal_distribution_and_scales.gifView on Boundless.com

  41. Continuous Random Variables SDM This is the formula for the true standard deviation of the sample mean. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Standard error."GNU FDLhttp://en.wikipedia.org/wiki/Standard_errorView on Boundless.com

  42. Continuous Random Variables Graph 2 Bell curve visualizing a normal distribution with a relatively large standard deviation. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikispaces."killianhma0809 - Normal Distribution."CC BY-SAhttp://killianhma0809.wikispaces.com/Normal+DistributionView on Boundless.com

  43. Continuous Random Variables Law of Large Numbers An illustration of the law of large numbers using a particular run of rolls of a single die. As the number of rolls in this run increases, the average of the values of all the results approaches 3.5. While different runs would show a different shape over a small number of throws (at the left), over a large number of rolls (to the right) they would be extremely similar. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Largenumbers."Public domainhttp://en.wikipedia.org/wiki/File:Largenumbers.svgView on Boundless.com

  44. Continuous Random Variables Expected Value The computation of the expected value in our example. Free to share, print, make copies and changes. Get yours at www.boundless.com P.Mean Website."Stats: Expected value and moments (July 29, 2005)."CC BYhttp://www.pmean.com/05/Moments.aspView on Boundless.com

  45. Continuous Random Variables Probability of Number of Girls The probabilities of the number of girls in a family of three children. Free to share, print, make copies and changes. Get yours at www.boundless.com P.Mean Website."Stats: Expected value and moments (July 29, 2005)."CC BYhttp://www.pmean.com/05/Moments.aspView on Boundless.com

  46. Continuous Random Variables Expected Value of Girl Bonus The computation of the expected value of the girl bonus in our example. Free to share, print, make copies and changes. Get yours at www.boundless.com P.Mean Website."Stats: Expected value and moments (July 29, 2005)."CC BYhttp://www.pmean.com/05/Moments.aspView on Boundless.com

  47. Continuous Random Variables Finite Population Correction The error should be multiplied by the FPC when the sampling fraction is large. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Standard error."GNU FDLhttp://en.wikipedia.org/wiki/Standard_errorView on Boundless.com

  48. Continuous Random Variables Correlation Correction This factor results in an unbiased estimate of the true standard error when correlation exists. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Standard error."GNU FDLhttp://en.wikipedia.org/wiki/Standard_errorView on Boundless.com

  49. Continuous Random Variables SEM SEM is usually estimated by the sample estimate of the population standard deviation divided by the square root of the sample size. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Standard error."GNU FDLhttp://en.wikipedia.org/wiki/Standard_errorView on Boundless.com

  50. Continuous Random Variables Approximately Normal - Histogram This is a sample of size 50 from a normal distribution, plotted out as a histogram. The histogram looks somewhat bell-shaped, indicating normality. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Normhist."CC BY-SAhttp://en.wikipedia.org/wiki/File:Normhist.pngView on Boundless.com

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