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4434[Shiny App] Probability Distribution Viewer
https://www.merlot.org/merlot/viewMaterial.htm?id=1142697
Probability distributions, p-values, and percentiles are fundamental topics taught to introductory statistics students. Often, students are presented these topics with static images in textbooks, but frequently do not have access to a dynamic and interactive tool they can use for exploration. For example, in the case of p-values, introductory students are frequently shown how to use tables to obtain a range of possible p-values associated with their test statistic along with a guiding image, but this is often difficult for students to understand. The goal of this application is to provide the student with an intuitive, simple, and comprehensive visualization of the three aforementioned topics. At the moment, many but not all continuous distributions (Beta, Cauchy, Chi-Squared, Exponential, F, Gamma, Logistic, Log Normal, Normal, Student’s t, Uniform, and Weibull) are available in the application. Support for discrete may be added in the future.When the app first renders, the user is shown by default the standard normal distribution. The student may vary the both the distribution and parameters corresponding to the distribution of their choice from the options at the top under "Distribution." This enables the student to see how the shape of their selected distribution changes as these values change. For students that are interested in visualizing probabilities and percentiles, the probability viewer app easily allows the student to select between two types of inputs. The student can also select between different shaded tail visualizations for the inputted percentiles or probabilities. Whether the student chooses to input a percentile or probability, the app will automatically calculate the value that the student did not input, corresponding to the one was. After inputting all required values, a graph appears with the appropriate distribution, the percentile and probability pairs, and the appropriate shading.Wed, 23 Mar 2016 11:43:47 -0700[Shiny App] Maximum Likelihood Estimation for the Binomial Distribution
https://www.merlot.org/merlot/viewMaterial.htm?id=1142684
This app provides an introduction to the concept of maximum likelihood estimation by working through the example of the binomial distribution. The first tab shows the probability mass function (pmf) of the binomial distribution. The user specifies the parameters to see various pmfs, and is guided to understand that this function takes the number of successes (x) as input and provides a probability as output.The pmf is then contrasted with the likelihood function in the second tab. Here the user specifies the fixed "parameters" (x and n) for the likelihood function, and the likelihood curve is graphed. Here the user sees that the input to the function is p rather than x, and the text explains that inputs and parameters have effectively been switched. The user is guided to input various values of x and discover that the likelihood function is always maximized atp=x/n. The third tab displays the likelihood and log likelihood side-by-side so the user understands they achieve the maximum in the same location.Wed, 23 Mar 2016 11:41:43 -0700[Shiny App] Performance of Wilcoxon-Mann-Whitney Test vs. t-test
https://www.merlot.org/merlot/viewMaterial.htm?id=1142633
The goal of this app is to compare the performance of a nonparametric to a parametric test for the difference in two population means. Specifically, performance is measured in the app either by Type I error rate or power, and the two respective tests for comparison are the Wilcoxon-Mann-Whitney (WMW) test and the two-sample t-test. Recall that for the test conditions to be satisfied, the two-sample t-test requires either the two population distributions to be normal or large enough sample sizes while the WMW test requires the two population distributions to have the same shape. Users have the option to produce different scenarios and conclude the better test either through a lower Type I error rate (if the two population means are the same) or a higher power (if they are not).When users first launch the app, they are presented with the goal of the study. Then, a game demonstrates to users the difficulty of identifying the population distributions of sample data. Following the first two introductory tabs, users can proceed to comparing performance. They have the option to choose a tab corresponding to their choice of the population distributions. Within each tab, either a single comparison or comparisons over a range can be conducted. The settings available for users to adjust are sample sizes, population means, significance level, number of simulations, and range of comparison values. In addition, visualizations are implemented to communicate results to users. For a single comparison, the outputs are distributions of the test statistics and gauges. For comparisons over a range, the output illustrates the performance of the two tests in each comparison.Wed, 23 Mar 2016 11:23:49 -0700[Shiny App] Benford's Law: Data Examples (Census and Stock Exchange)
https://www.merlot.org/merlot/viewMaterial.htm?id=1142629
The first-digit distribution of many US Census variables is known to closely follow Benford's Law. We will consider several census variables available fromCounty Totals Dataset: Population, Population Change and Estimated Components of Population Change. The app will apply a goodness of fit test of the observed frequencies of first-digits for the selected variable. The variables under consideration are: Annual Resident Total Population Estimate (2010 to 2013), Annual Births (2010 to 2013), Annual Deaths (2010 to 2013). We also consider several census variables available from US Census State & County QuickFacts. The app will apply a goodness of fit test of the observed frequencies of first-digits for the selected variable. The variables under consideration are: Housing Units (2013), Households (2008-12), Veterans (2008-12), Nonemployer Establishments (2012), Private Nonfarm Establishments (2012), Private Nonfarm Employment (2012), Retail Sales (2007). Finally the app will download information from the Wall Street Journal website from themost recent end of day market data. The data will be based on various market variables for all companies listed in one of four stock markets. The app will apply a goodness of fit test of the observed frequencies of first-digits for the selected variable in the specified stock market.Wed, 23 Mar 2016 11:19:19 -0700[Shiny App] Benford's Law: Sequences (Additive, Power, Prime Number)
https://www.merlot.org/merlot/viewMaterial.htm?id=1142618
This app examines the first-digit distribution of various sequences: Additive, Power, and Prime NumberAdditive Sequence: Consider a sequence of numbers where we fix the initial two numbers and then the value of each subsequent number is the sum of the previous two. We will call this an additive sequence. When the initial two numbers are both 1 then this yields the famous Fibonacci sequence. If you only consider the first digit of each number in an additive sequence and examine its distribution, is it the case that it closely follows Benford's Law? This app generates an additive sequence, for a given length and initial sequence numbers, and applies a goodness of fit test of the observed frequencies of first digits to Benford's Law.Power Sequence: Consider a sequence of the form b1, b2, …, bn, where b is called the base. We will call this a power sequence. If you only consider the first digit of each number in a power sequence and examine its distribution, is it the case that it closely follows Benford's Law? This app generates the power sequence, for a given b and n, and applies a goodness of fit test of the observed frequencies of first digits to Benford's Law.Prime Number Sequence: Consider the sequence of prime numbers less than or equal to some power of 10. An article from 2009 shows that the distribution of the first digit of these prime numbers is well described by what's known as Generalized Benford's Law (GBL) . This app generates the prime numbers less than or equal to 103, 104, 105, or 106 and applies a goodness of fit test of the observed frequencies of first digits to GBL.Wed, 23 Mar 2016 11:16:40 -0700[Shiny App] t-test with diagnostics
https://www.merlot.org/merlot/viewMaterial.htm?id=1142611
This app focuses on conducting a t-test and checking the normality condition. Both the one-sample and two-sample t-tests are implemented in this app. Recall that for the t-test to be valid either sample size(s) need to be large enough or the population distribution(s) needs to be a Normal distribution. To begin the app, data configuration is required. Users have the choice to either use sample data or upload their own data when first launching the app. Customization is needed in respect to the uploaded data. After selecting their option, users can proceed to visualizing the data. A histogram is presented for one sample while comparative boxplots are presented for two samples. In addition, summary statistics are also available for display.The hypothesis test tab displays the null and alternative hypotheses. The settings available for users to adjust are the hypothesized value, the direction for the alternative hypothesis, and the significance level. For users who are not familiar with the concept of the hypothesis test, they can click on a link that shows information in a popover. Additional information on the one-sample and two-sample t-tests is also available. When users have run the t-test, the output includes items such as the shaded t-distribution, t-statistic, and the p-value. The point estimate(s) and confidence interval can also be outputted by users’ request. In the normality condition tab, the Shapiro-Wilk normality test is performed and a Q-Q plot is displayed. In all relevant outputs throughout the app, sample interpretations from popovers are included for users to understand the results of the hypothesis test.Wed, 23 Mar 2016 11:13:02 -0700[Shiny App] Multiple Regression Visualization
https://www.merlot.org/merlot/viewMaterial.htm?id=1142595
Scatterplots are often useful to visualize the relationship between two quantitative variables. However, with Multiple Regression, there are more than one predictor variables used to model one response variable. Thus, a simple scatterplot is no longer adequate to graphically represent all of the variables. In the case of two predictor variables, we can illustrate this in three dimensions, or also in two dimensions with appropriate color schemes. This applet shows us these illustrations for a variety of datasets.Wed, 23 Mar 2016 11:05:59 -0700Statistics Intro: Mean, median and mode
https://www.merlot.org/merlot/viewMaterial.htm?id=1078399
Mean, median and modeWed, 11 Nov 2015 08:42:38 -0800Scale Ella
https://www.merlot.org/merlot/viewMaterial.htm?id=794378
The evil Scaleo has escaped from prison and is transforming the length, width, and height of objects until they become useless... or dangerous.Who can put things right? Our super heroine, Scale Ella, uses the power of scale factor to foil the villain.Scale Ella guides students in using factors, multiples, prime factorization, and relatively prime numbers to solve problems; developing, analyzing, and explaining methods for solving problems involving proportions, such as scaling and finding equivalent ratios; and using factors, multiples, prime factorization, and relatively prime numbers to solve problems.Wed, 25 Sep 2013 14:21:47 -0700Ratey the Math Cat
https://www.merlot.org/merlot/viewMaterial.htm?id=794389
Ratey the Math Cat supports students in understanding and using ratios and proportions to represent quantitative relationships, building new mathematical knowledge through problem solving, and solving problems that arise in mathematics and in other contexts.Wed, 25 Sep 2013 14:19:03 -0700