Samples and ProbabilityType of Unit: ConceptualPrior KnowledgeStudents should be able to:Understand the concept of a ratio.Write ratios as percents.Describe data using measures of center.Display and interpret data in dot plots, histograms, and box plots.Lesson FlowStudents begin to think about probability by considering the relative likelihood of familiar events on the continuum between impossible and certain. Students begin to formalize this understanding of probability. They are introduced to the concept of probability as a measure of likelihood, and how to calculate probability of equally likely events using a ratio. The terms (impossible, certain, etc.) are given numerical values. Next, students compare expected results to actual results by calculating the probability of an event and conducting an experiment. Students explore the probability of outcomes that are not equally likely. They collect data to estimate the experimental probabilities. They use ratio and proportion to predict results for a large number of trials. Students learn about compound events. They use tree diagrams, tables, and systematic lists as tools to find the sample space. They determine the theoretical probability of first independent, and then dependent events. In Lesson 10 students identify a question to investigate for a unit project and submit a proposal. They then complete a Self Check. In Lesson 11, students review the results of the Self Check, solve a related problem, and take a Quiz.Students are introduced to the concept of sampling as a method of determining characteristics of a population. They consider how a sample can be random or biased, and think about methods for randomly sampling a population to ensure that it is representative. In Lesson 13, students collect and analyze data for their unit project. Students begin to apply their knowledge of statistics learned in sixth grade. They determine the typical class score from a sample of the population, and reason about the representativeness of the sample. Then, students begin to develop intuition about appropriate sample size by conducting an experiment. They compare different sample sizes, and decide whether increasing the sample size improves the results. In Lesson 16 and Lesson 17, students compare two data sets using any tools they wish. Students will be reminded of Mean Average Deviation (MAD), which will be a useful tool in this situation. Students complete another Self Check, review the results of their Self Check, and solve additional problems. The unit ends with three days for students to work on Gallery problems, possibly using one of the days to complete their project or get help on their project if needed, two days for students to present their unit projects to the class, and one day for the End of Unit Assessment.

Students will apply their knowledge of statistics learned in sixth grade. They will determine the typical class score from a sample of the population, and reason about the representativeness of the sample.Students analyze test score data from a fictitious seventh grade class and make generalizations about district-wide results. They then compare the data to a second seventh grade class and reason about whether these are random samples. Students will review measures of center and spread as they find evidence to draw conclusions about the data.Key ConceptsSample size will be considered as it affects the conclusions of an analysis of a population.Students will review tools that they used in sixth grade to analyze data, such as measures of center and spread, and different types of graphs.Goals and Learning ObjectivesExplore sample size.Look at the effects of using a nonrandom sample.Review tools used to analyze data.

Students begin to develop intuition about appropriate sample size by conducting an experiment. They compare different sample sizes and whether increasing the sample size improves the results.Key ConceptsSampling is a way to discover unknown characteristics about a population. The size of the sample is important in determining the accuracy of the results. Ratio and proportion are used to compare the sample to the population.Goals and Learning ObjectivesStudents will use sampling to determine the number of different color marbles in a jar.Students will explore sample size compared to population size.

Students are introduced to the concept of sampling as a method of determining characteristics of a population. They consider how a sample can be random or biased, and think of methods for randomly sampling a population to ensure that it is representative.The idea of sampling is connected to probability; a relatively small set of data (a random sample/number of trials) can be used to generalize about a population (or determine probability). A larger sample (more trials) will give more confidence in the conclusions, but how large of a sample is needed?Students also discuss what random means and how to generate a random sample. Random samples are compared to biased samples and give insight into how statistics can be misleading (intentionally or otherwise).Key ConceptsRandom samples are related to probability. In probability, the number of trials is a sample used to generalize about the probability of an event. The results in probability are random if we are looking at equally likely outcomes. If a data sample is not random, the conclusions about the population will not reflect it.Terminology introduced in this lesson:population: the entire set of objects that can be considered when asking a statistical questionsample: a subset of a population; can be random, where each object in the population is equally likely to be in the sample, or biased, where not every object in the population is equally likely to be in the sampleGoals and Learning ObjectivesIntroduce sampling as a method to generalize about a population.Discuss the concept of a random sample versus a biased sample.Determine methods to generate random samples.Understand that biased samples are sometimes used to mislead.SWD: Some students with disabilities will benefit from a preview of the goals in this lesson. Students can highlight the critical features and/or concepts and will help them to pay close attention to salient information.

Students critique and improve their work on the Self Check, then work on additional problems. Students revise the Self Check problem from the previous lesson and discuss their strategies.Key ConceptsStudents apply what they have learned to date to solve the problems in this lesson.Goals and Learning ObjectivesApply knowledge of sampling and data analysis to solve problems.Determine a random, representative sample that is nonbiased and of adequate sample size.Generalize about a population based on sampling.Compare data sets.

This course begins with the premise that the 1960s mark a great dividing point in the history of 20th century Western musical culture, and explores the ways in which various social and artistic concerns of composers, performers, and listeners have evolved since that decade. It focuses on works by classical composers from around the world. Topics include the impact of rock, as it developed during the 1960s - 70s; the concurrent emergence of post serial, neotonal, minimalist, and new age styles; the globalization of Western musical traditions; the impact of new technologies; and the significance of music video, video games, and other versions of multimedia. The course interweaves discussion of these topics with close study of seminal musical works, evenly distributed across the four decades since 1960; works by MIT composers are included.

This course is an investigation into the history and aesthetics of music and technology as deployed in experimental and popular musics from the 19th century to the present. Through original research, creative hands-on projects, readings, and lectures, the following topics will be explored. The history of radio, audio recording, and the recording studio, as well as the development of musique concrète and early electronic instruments. The creation and extension of musical interfaces by composers such as Harry Partch, John Cage, Conlon Nancarrow, and others. The exploration of electromagnetic technologies in pickups, and the development of dub, hip-hop, and turntablism. The history and application of the analog synthesizer, from the Moog modular to the Roland TR-808. The history of computer music, including music synthesis and representation languages. Contemporary practices in circuit bending, live electronics, and electro-acoustic music, as well as issues in copyright and intellectual property, will also be examined. No prerequisites.

In this lesson students explore the creative concepts and technological practices on which Hip Hop music was constructed, investigating what it means to "sample" from another style, who has used sampling and how. Then, students experience the technology first hand using the Soundbreaking Sampler TechTool. Students will follow patterns of Caribbean immigration and the musical practices that came to New York City as a result of those patterns, finally considering the ways in which Hip Hop reflects them. Moving forward to the late 1980s and early 90s, what some consider Hip Hop's "Golden Age," this lesson explores how sampling might demonstrate a powerful creative expression of influence or even a social or political statement. Finally, this lesson encourages students to consider the conceptual hurdle Hip Hop asked listeners to make by presenting new music made from old sounds.

In this lesson, students use examples from visual art as well as rap to enter into a "structured academic controversy" that explores the concept of originality.

The course serves as an introduction to the theory and practice behind many of today’s communications systems. 6.450 forms the first of a two-course sequence on digital communication. The second class, 6.451 Principles of Digital Communication II, is offered in the spring.
Topics covered include: digital communications at the block diagram level, data compression, Lempel-Ziv algorithm, scalar and vector quantization, sampling and aliasing, the Nyquist criterion, PAM and QAM modulation, signal constellations, finite-energy waveform spaces, detection, and modeling and system design for wireless communication.

The course serves as an introduction to the theory and practice behind many of today’s communications systems. 6.450 forms the first of a two-course sequence on digital communication. The second class, 6.451, is offered in the spring.
Topics covered include: digital communications at the block diagram level, data compression, Lempel-Ziv algorithm, scalar and vector quantization, sampling and aliasing, the Nyquist criterion, PAM and QAM modulation, signal constellations, finite-energy waveform spaces, detection, and modeling and system design for wireless communication.

This course is an introduction to discrete applied mathematics. Topics include probability, counting, linear programming, number-theoretic algorithms, sorting, data compression, and error-correcting codes. This is a Communication Intensive in the Major (CI-M) course, and thus includes a writing component.

An introduction to the principles of tomographic imaging and its applications. It includes a series of lectures with a parallel set of recitations that provide demonstrations of basic principles. Both ionizing and non-ionizing radiation are covered, including x-ray, PET, MRI, and ultrasound. Emphasis on the physics and engineering of image formation.

This course introduces theoretical and practical principles of design of oceanographic sensor systems. Topics include: transducer characteristics for acoustic, current, temperature, pressure, electric, magnetic, gravity, salinity, velocity, heat flow, and optical devices; limitations on these devices imposed by ocean environments; signal conditioning and recording; noise, sensitivity, and sampling limitations; and standards. Lectures by experts cover the principles of state-of-the-art systems being used in physical oceanography, geophysics, submersibles, acoustics. For lab work, day cruises in local waters allow students to prepare, deploy and analyze observations from standard oceanographic instruments.

CK-12 Advanced Probability and Statistics introduces students to basic topics in statistics and probability but finishes with the rigorous topics an advanced placement course requires. Includes visualizations of data, introduction to probability, discrete probability distribution, normal distribution, planning and conducting a study, sampling distributions, hypothesis testing, regression and correlation, Chi-Square, analysis of variance, and non-parametric statistics.

CK-12 Advanced Probability and Statistics Teacher's Edition provides tips and enrichment activities for teaching CK-12 Advanced Probability and Statistics Student Edition. The solution and assessment guides are available upon request.

An important consideration when sampling from a population is that of random error (also known as sampling error), which results from chance variation in the members of any sample taken from a larger population. Random error may affect the conclusions you draw from a study by affecting the precision of a descriptive study, or the power of an analytic study. However, although the magnitude of random error can be quantified to some degree, its direction cannot be predicted due to its random nature. Random errors can be accounted for to some degree through the application of inferential statistics when presenting and interpreting results.

This course examines how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Topics covered include: randomized computation; data structures (hash tables, skip lists); graph algorithms (minimum spanning trees, shortest paths, minimum cuts); geometric algorithms (convex hulls, linear programming in fixed or arbitrary dimension); approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms.

This course develops skills in research design for policy analysis and planning. The emphasis is on the logic of the research process and its constituent elements. The course relies on a seminar format so students are expected to read all of the assigned materials and come to class prepared to discuss key themes, ideas, and controversies. Since the materials draw broadly on the social sciences, and since students have diverse interests and methodological preferences, ongoing themes in our discussions will be linking concepts to planning scholarship in general and considering how different epistemological orientations and methodological techniques map on to planning specializations.

Social psychologists are interested in the ways that other people affect thought, emotion, and behavior. To explore these concepts requires special research methods. Following a brief overview of traditional research designs, this module introduces how complex experimental designs, field experiments, naturalistic observation, experience sampling techniques, survey research, subtle and nonconscious techniques such as priming, and archival research and the use of big data may each be adapted to address social psychological questions. This module also discusses the importance of obtaining a representative sample along with some ethical considerations that social psychologists face.