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Stanford Online

Statistical learning with r.

SOHS-YSTATSLEARNING

Stanford School of Humanities and Sciences

This is an introductory-level course in supervised learning, with a focus on regression and classification methods. The syllabus includes: linear and polynomial regression, logistic regression and linear discriminant analysis; cross-validation and the bootstrap, model selection and regularization methods (ridge and lasso); nonlinear models, splines and generalized additive models; tree-based methods, random forests and boosting; support-vector machines. Some unsupervised learning methods are discussed: principal components and clustering (k-means and hierarchical).

This is not a math-heavy class, so we try and describe the methods without heavy reliance on formulas and complex mathematics. We focus on what we consider to be the important elements of modern data analysis. Computing is done in R. There are lectures devoted to R, giving tutorials from the ground up, and progressing with more detailed sessions that implement the techniques in each chapter. We also offer a separate version of the course called Statistical Learning with Python - the chapter lectures are the same, but the lab lectures and computing are done using Python. 

The lectures cover all the material in An Introduction to Statistical Learning, with Applications in R by James, Witten, Hastie and Tibshirani, with Applications in R (second addition) by James, Witten, Hastie and Tibshirani (Springer, 2021). The pdf for this book is available for free on the book website .

Prerequisites

Introductory level understanding of core concepts in statistics, linear algebra, and computing.

Core Competencies

Overview of statistical learning

• Linear regression
• Classification
• Resampling methods
• Linear model selection and regularization
• Moving beyond linearity
• Tree-based methods
• Support vector machines
• Deep learning
• Survival modeling
• Unsupervised learning
• Multiple testing
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Teach yourself statistics

Statistics and Probability

This website provides training and tools to help you solve statistics problems quickly, easily, and accurately - without having to ask anyone for help.

Online Tutorials

Learn at your own pace. Free online tutorials cover statistics, probability, regression, analysis of variance, survey sampling, and matrix algebra - all explained in plain English.

• Advanced Placement (AP) Statistics . Full coverage of the AP Statistics curriculum.
• Probability . Fundamentals of probability. Clear explanations with pages of solved problems.
• Linear Regression . Regression analysis with one or more independent variables.
• ANOVA . Analysis of variance made easy. How to collect, analyze, and interpret data.
• Survey Sampling . How to conduct a statistical survey and analyze survey data.
• Matrix Algebra . Easy-to-understand introduction to matrix algebra.

Practice and review questions reinforce key points. Online calculators take the drudgery out of computation. Perfect for self-study.

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• AP Tutorial : Study our free, AP statistics tutorial to improve your skills in all test areas.
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• Lectures: Mon/Wed 3-4:20pm in Thornt110
• Sections: Fri 1:30-2:30pm in 200-030
• Osbert Bastani (office hours: Tue 10am-12pm, Thu 10am-12pm in Gates 438)
• Peng Xu (office hours: Mon 10am-12pm, Wed 10am-12pm in Huang Basement)
• Homeworks (30%) : there will be three homeworks (plus a warmup which does not count towards your grade), centered around proving properties of statistical procedures. You will also implement and run some of the algorithms as a reality check. Each homework must be turned in at the beginning of class (hard copy), as well as electronically by 11pm . Only the hard copy version will be graded; the electronic copy is just for our records. You are encouraged to use LaTeX to typeset your homeworks; we've provided a template for your convenience.
• Exam (40%) : open-book, open-notes. Problems will be like the homeworks, but simpler. You can use laptops as long as you turn off the wireless. Date: Feb. 25 6-9pm.
• Paper reviews (30%) : you will write two 2-4 page reviews of papers. The goal is to learn to read technically demanding papers critically, and hopefully in the process, generate novel research ideas. Your review should not only summarize the main result of the paper, but critique it, instantiate it on examples, discuss its overall significance, and suggest possible future directions. See this Google doc for detailed guidelines and a list of papers. Each review should be submitted electronically by 11pm . Instead of doing the last paper review, with approval from the course staff on the project topic, you can do a final project, perhaps extending one of the earlier reviews to produce novel results. The project can be done in pairs.

To submit electronically, open up a terminal, (i) copy your submission file(s) (e.g., hw0.pdf ) to cardinal.stanford.edu : scp <your submission file(s)> <your SUNetID>@cardinal.stanford.edu: and (ii) run the submit script: ssh <your SUNetID>cardinal.stanford.edu python /usr/class/cs229t/WWW/submit.py <hw0|hw1|hw2|hw3|p1|p2|p3> . You can submit multiple times; each submission will just replace the previous one.

ECE 543: Statistical Learning Theory (Spring 2021)

About   |   schedule   |   coursework, announcements.

• Homework 4 is posted, due by the end of the day on Tuesday, April 27.
• Recordings of all lectures up to April 15 are now available.
• Homework 3 is posted, due by the end of the day on Tuesday, April 6.
• Recordings of all lectures up to March 23 are now available.
• There was a typo in Problem 1 of Homework 2. Revised version is posted.
• In-class notes and video recordings of Lectures 7-12 are now available.
• Homework 2 is posted, due by the end of the day on Tuesday, March 16.
• There will be TA office hours this week on Tuesday, February 16, 9:00-10:00 am.
• Since Wednesday, February 17, is a no-instruction day, there will be no TA office hours that week. Homework 1 is now due by the end of Wednesday, February 24.
• In-class notes and video recording of Lectures 4-6 are now available.
• Information about homework submissions and final project is available on the coursework page .
• Homework 1 is posted, due by the end of the day on Thursday, February 18.
• In-class notes and video recording of Lectures 2 and 3 are now available.
• In-class notes and video recording of Lecture 1 are now available.
• Zoom link and passcode for the course are posted on Piazza .

About this course

• Lecture notes by Bruce Hajek and Maxim Raginsky
• Shai Shalev-Shwartz and Shai Ben-David, Understanding Machine Learning , Cambridge University Press, 2014

Regular weekly schedule

Synopsis (摘要)

This course is open to graduates and senior undergraduates in applied mathematics, statistics, and engineering who are interested in learning from data. It covers hot topics in statistical learning, also known as machine learning, featured with various in-class projects in computer vision, pattern recognition, computational advertisement, bioinformatics, and social networks, etc. An emphasis this year is on deep learning with convolutional neural networks. Prerequisite: linear algebra, basic probability and multivariate statistics, convex optimization; familiarity with R, Matlab, and/or Python, Torch for deep learning, etc.

Reference (参考教材)

An Introduction to Statistical Learning, with applications in R. By James, Witten, Hastie, and Tibshirani

ISLR-python, By Jordi Warmenhoven .

ISLR-Python: Labs and Applied, by Matt Caudill .

The Elements of Statistical Learning. 2nd Ed. By Hastie, Tibshirani, and Friedman

statlearning-notebooks , by Sujit Pal, Python implementations of the R labs for the StatLearning: Statistical Learning online course from Stanford taught by Profs Trevor Hastie and Rob Tibshirani.

Instructors:

Time and venue:.

TuTh 4:30-5:50pm Rm4504 (Lift 25/26), Academic Bldg Piazza discussion forum: sign-up link

Homework and Projects:

Weekly homeworks, monthly mini-projects, and a final major project. No final exam. For 3-project plan, homework and projects will be counted in grading by 20-20-20-40 in percentage.

Grading scheme: [ description ]

Teaching Assistant (助教):

Mr. ZHU, Weizhi, Email: statml.hw (add "AT gmail DOT com" afterwards)

Tutorial Material

Schedule (时间表)

• [ slides in pdf ]
• Homework 1 [pdf] . Deadline: 09/28/2015, Monday. Mark on the head of your homework: Name - Student ID .
• Homework 2 [pdf] . Deadline: 10/12/2015, Monday. Mark on the head of your homework: Name - Student ID .
• Project 1 [pdf] . Deadline: 10/12/2015, Monday. Team work with no more than FIVE (5) collaborators.
• Jiechao XIONG, A Dynamic Approach to Variable Selection
• Homework 3 [pdf] . Deadline: 10/19/2015, Monday. Mark on the head of your homework: Name - Student ID .
• Homework 4 [pdf] . Deadline: 10/26/2015, Monday. Mark on the head of your homework: Name - Student ID .
• Homework 5 [pdf] . Deadline: 11/02/2015, Monday. Mark on the head of your homework: Name - Student ID .
• Xuening ZHU, Network Vector Regression
• Homework 6 [pdf] . Deadline: 11/09/2015, Monday. Mark on the head of your homework: Name - Student ID .

Datasets (to-be-updated)

• [Animal Sleep Data] Animal species sleeping hours vs. other features
• [Anzhen Heart Data] Heart Operation Effect Prediction , provided by Dr. Jinwen Wang, Anzhen Hospital
• [Beer Data] 877 beers dataset , provided by Mr. Richard Sun, Shanghai
• [Crime Data] Crime rates in 59 US cities during 1970-1992
• [Real-Time-Bidding Algorithm Competition Data] Contest Website
• [红楼梦人物事件矩阵] a 376-by-475 matrix (374-by-475 updated by WAN, Mengting) for character-event appearance in A Dream of Red Mansion (Xueqin Cao) [374 Characters.txt (for R/read.table)] [HongLouMeng374.csv] [HongLouMeng376.xls] [.mat] [readme.m]

• [Keywords Pricing] Keywords and profit index in paid search advertising, by Hansheng Wang (Guanghua, PKU). [readme.txt] [data in csv]
• [Radon Data] Radon measurements of 12,687 houses in US
• [Wells Data] Switch unsafe wells for arsenic pollution in Bangladesh
• to-be-done...

Stat 928, Spring 2011

Statistical learning theory.

Statistical learning theory studies the statistical aspects of machine learning and automated reasoning, through the use of (sampled) data. In particular, the focus is on characterizing the generalization ability of learning algorithms in terms of how well they perform on ``new'' data when trained on some given data set. The focus of the course is on: providing the the fundamental tools used in this analysis; understanding the performance of widely used learning algorithms (with a focus on regression and classification); understanding the ``art'' of designing good algorithms, both in terms of statistical and computational properties. Potential topics include: concentration of measure; empirical process theory; online learning; stochastic optimization; margin based algorithms; feature selection; regularization; PCA.

Prerequisites:

Requirements:, instructor:.

Time and location:

Schedule and notes:

• lecture notes pdf
• lecture 9 notes pdf
• lecture 10 notes pdf
• lecture 11 notes pdf
• lecture 12 notes pdf
• Review: Norms and Dual Norms
• lecture 14 notes pdf
• further reading: S. M. Kakade, S. Shalev-Shwartz, A. Tewari. Regularization Techniques for Learning with Matrices. pdf
• lecture 16 notes pdf
• lecture 17 notes pdf
• lecture 18 notes pdf
• lecture 19 notes pdf
• lecture 20 notes pdf
• lecture 21 notes pdf
• lecture 22 notes pdf
• lecture 23 notes pdf
• lecture 24 notes pdf
• lecture 25 notes pdf

1.1 Definitions of Statistics, Probability, and Key Terms

For each of the following eight exercises, identify: a. the population, b. the sample, c. the parameter, d. the statistic, e. the variable, and f. the data. Give examples where appropriate.

A fitness center is interested in the mean amount of time a client exercises in the center each week.

Ski resorts are interested in the mean age that children take their first ski and snowboard lessons. They need this information to plan their ski classes optimally.

A cardiologist is interested in the mean recovery period of her patients who have had heart attacks.

Insurance companies are interested in the mean health costs each year of their clients, so that they can determine the costs of health insurance.

A politician is interested in the proportion of voters in his district who think he is doing a good job.

A marriage counselor is interested in the proportion of clients she counsels who stay married.

Political pollsters may be interested in the proportion of people who will vote for a particular cause.

A marketing company is interested in the proportion of people who will buy a particular product.

Use the following information to answer the next three exercises: A Lake Tahoe Community College instructor is interested in the mean number of days Lake Tahoe Community College math students are absent from class during a quarter.

What is the population she is interested in?

• all Lake Tahoe Community College students
• all Lake Tahoe Community College English students
• all Lake Tahoe Community College students in her classes
• all Lake Tahoe Community College math students

Consider the following:

X X = number of days a Lake Tahoe Community College math student is absent

In this case, X is an example of a:

• population.

The instructor’s sample produces a mean number of days absent of 3.5 days. This value is an example of a:

1.2 Data, Sampling, and Variation in Data and Sampling

For the following exercises, identify the type of data that would be used to describe a response (quantitative discrete, quantitative continuous, or qualitative), and give an example of the data.

number of tickets sold to a concert

percent of body fat

favorite baseball team

time in line to buy groceries

number of students enrolled at Evergreen Valley College

most-watched television show

brand of toothpaste

distance to the closest movie theatre

age of executives in Fortune 500 companies

number of competing computer spreadsheet software packages

Use the following information to answer the next two exercises: A study was done to determine the age, number of times per week, and the duration (amount of time) of resident use of a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every 8th house in the neighborhood around the park was interviewed.

“Number of times per week” is what type of data?

• qualitative
• quantitative discrete
• quantitative continuous

“Duration (amount of time)” is what type of data?

Airline companies are interested in the consistency of the number of babies on each flight, so that they have adequate safety equipment. Suppose an airline conducts a survey. Over Thanksgiving weekend, it surveys six flights from Boston to Salt Lake City to determine the number of babies on the flights. It determines the amount of safety equipment needed by the result of that study.

• Using complete sentences, list three things wrong with the way the survey was conducted.
• Using complete sentences, list three ways that you would improve the survey if it were to be repeated.

Suppose you want to determine the mean number of students per statistics class in your state. Describe a possible sampling method in three to five complete sentences. Make the description detailed.

Suppose you want to determine the mean number of cans of soda drunk each month by students in their twenties at your school. Describe a possible sampling method in three to five complete sentences. Make the description detailed.

List some practical difficulties involved in getting accurate results from a telephone survey.

List some practical difficulties involved in getting accurate results from a mailed survey.

With your classmates, brainstorm some ways you could overcome these problems if you needed to conduct a phone or mail survey.

The instructor takes her sample by gathering data on five randomly selected students from each Lake Tahoe Community College math class. The type of sampling she used is

• cluster sampling
• stratified sampling
• simple random sampling
• convenience sampling

A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every eighth house in the neighborhood around the park was interviewed. The sampling method was:

• simple random

Name the sampling method used in each of the following situations:

• A woman in the airport is handing out questionnaires to travelers asking them to evaluate the airport’s service. She does not ask travelers who are hurrying through the airport with their hands full of luggage, but instead asks all travelers who are sitting near gates and not taking naps while they wait.
• A teacher wants to know if her students are doing homework, so she randomly selects rows two and five and then calls on all students in row two and all students in row five to present the solutions to homework problems to the class.
• The marketing manager for an electronics chain store wants information about the ages of its customers. Over the next two weeks, at each store location, 100 randomly selected customers are given questionnaires to fill out asking for information about age, as well as about other variables of interest.
• The librarian at a public library wants to determine what proportion of the library users are children. The librarian has a tally sheet on which she marks whether books are checked out by an adult or a child. She records this data for every fourth patron who checks out books.
• A political party wants to know the reaction of voters to a debate between the candidates. The day after the debate, the party’s polling staff calls 1,200 randomly selected phone numbers. If a registered voter answers the phone or is available to come to the phone, that registered voter is asked whom he or she intends to vote for and whether the debate changed his or her opinion of the candidates.

A “random survey” was conducted of 3,274 people of the “microprocessor generation” (people born since 1971, the year the microprocessor was invented). It was reported that 48% of those individuals surveyed stated that if they had $2,000 to spend, they would use it for computer equipment. Also, 66% of those surveyed considered themselves relatively savvy computer users.

• Do you consider the sample size large enough for a study of this type? Why or why not?
• Based on your “gut feeling,” do you believe the percents accurately reflect the U.S. population for those individuals born since 1971? If not, do you think the percents of the population are actually higher or lower than the sample statistics? Why? Additional information: The survey, reported by Intel Corporation, was filled out by individuals who visited the Los Angeles Convention Center to see the Smithsonian Institute's road show called “America’s Smithsonian.”
• With this additional information, do you feel that all demographic and ethnic groups were equally represented at the event? Why or why not?
• With the additional information, comment on how accurately you think the sample statistics reflect the population parameters.

The Well-Being Index is a survey that follows trends of U.S. residents on a regular basis. There are six areas of health and wellness covered in the survey: Life Evaluation, Emotional Health, Physical Health, Healthy Behavior, Work Environment, and Basic Access. Some of the questions used to measure the Index are listed below.

Identify the type of data obtained from each question used in this survey: qualitative, quantitative discrete, or quantitative continuous.

• Do you have any health problems that prevent you from doing any of the things people your age can normally do?
• During the past 30 days, for about how many days did poor health keep you from doing your usual activities?
• In the last seven days, on how many days did you exercise for 30 minutes or more?
• Do you have health insurance coverage?

In advance of the 1936 Presidential Election, a magazine titled Literary Digest released the results of an opinion poll predicting that the republican candidate Alf Landon would win by a large margin. The magazine sent post cards to approximately 10,000,000 prospective voters. These prospective voters were selected from the subscription list of the magazine, from automobile registration lists, from phone lists, and from club membership lists. Approximately 2,300,000 people returned the postcards.

• Think about the state of the United States in 1936. Explain why a sample chosen from magazine subscription lists, automobile registration lists, phone books, and club membership lists was not representative of the population of the United States at that time.
• What effect does the low response rate have on the reliability of the sample?
• Are these problems examples of sampling error or nonsampling error?
• During the same year, George Gallup conducted his own poll of 30,000 prospective voters. These researchers used a method they called "quota sampling" to obtain survey answers from specific subsets of the population. Quota sampling is an example of which sampling method described in this module?

Crime-related and demographic statistics for 47 US states in 1960 were collected from government agencies, including the FBI's Uniform Crime Report . One analysis of this data found a strong connection between education and crime indicating that higher levels of education in a community correspond to higher crime rates.

Which of the potential problems with samples discussed in 1.2 Data, Sampling, and Variation in Data and Sampling could explain this connection?

YouPolls is a website that allows anyone to create and respond to polls. One question posted April 15 asks:

“Do you feel happy paying your taxes when members of the Obama administration are allowed to ignore their tax liabilities?” (lastbaldeagle. 2013. On Tax Day, House to Call for Firing Federal Workers Who Owe Back Taxes. Opinion poll posted online at: http://www.youpolls.com/details.aspx?id=12328 (accessed May 1, 2013).)

As of April 25, 11 people responded to this question. Each participant answered “NO!”

Which of the potential problems with samples discussed in this module could explain this connection?

A scholarly article about response rates begins with the following quote:

“Declining contact and cooperation rates in random digit dial (RDD) national telephone surveys raise serious concerns about the validity of estimates drawn from such research.”(Scott Keeter et al., “Gauging the Impact of Growing Nonresponse on Estimates from a National RDD Telephone Survey,” Public Opinion Quarterly 70 no. 5 (2006), http://poq.oxfordjournals.org/content/70/5/759.full (accessed May 1, 2013).)

The Pew Research Center for People and the Press admits:

“The percentage of people we interview – out of all we try to interview – has been declining over the past decade or more.” (Frequently Asked Questions, Pew Research Center for the People & the Press, http://www.people-press.org/methodology/frequently-asked-questions/#dont-you-have-trouble-getting-people-to-answer-your-polls (accessed May 1, 2013).)

• What are some reasons for the decline in response rate over the past decade?
• Explain why researchers are concerned with the impact of the declining response rate on public opinion polls.

1.3 Frequency, Frequency Tables, and Levels of Measurement

Fifty part-time students were asked how many courses they were taking this term. The (incomplete) results are shown below:

• Fill in the blanks in Table 1.33 .
• What percent of students take exactly two courses?
• What percent of students take one or two courses?

Sixty adults with gum disease were asked the number of times per week they used to floss before their diagnosis. The (incomplete) results are shown in Table 1.34 .

• Fill in the blanks in Table 1.34 .
• What percent of adults flossed six times per week?
• What percent flossed at most three times per week?

Nineteen immigrants to the U.S were asked how many years, to the nearest year, they have lived in the U.S. The data are as follows: 2 ; 5 ; 7 ; 2 ; 2 ; 10 ; 20 ; 15 ; 0 ; 7 ; 0 ; 20 ; 5 ; 12 ; 15 ; 12 ; 4 ; 5 ; 10 .

Table 1.35 was produced.

• Fix the errors in Table 1.35 . Also, explain how someone might have arrived at the incorrect number(s).
• Explain what is wrong with this statement: “47 percent of the people surveyed have lived in the U.S. for 5 years.”
• Fix the statement in b to make it correct.
• What fraction of the people surveyed have lived in the U.S. five or seven years?
• What fraction of the people surveyed have lived in the U.S. at most 12 years?
• What fraction of the people surveyed have lived in the U.S. fewer than 12 years?
• What fraction of the people surveyed have lived in the U.S. from five to 20 years, inclusive?

How much time does it take to travel to work? Table 1.36 shows the mean commute time by state for workers at least 16 years old who are not working at home. Find the mean travel time, and round off the answer properly.

Forbes magazine published data on the best small firms in 2012. These were firms which had been publicly traded for at least a year, have a stock price of at least $5 per share, and have reported annual revenue between $5 million and $1 billion. Table 1.37 shows the ages of the chief executive officers for the first 60 ranked firms.

• What is the frequency for CEO ages between 54 and 65?
• What percentage of CEOs are 65 years or older?
• What is the relative frequency of ages under 50?
• What is the cumulative relative frequency for CEOs younger than 55?
• Which graph shows the relative frequency and which shows the cumulative relative frequency?

Use the following information to answer the next two exercises: Table 1.38 contains data on hurricanes that have made direct hits on the U.S. Between 1851 and 2004. A hurricane is given a strength category rating based on the minimum wind speed generated by the storm.

What is the relative frequency of direct hits that were category 4 hurricanes?

• Not enough information to calculate

What is the relative frequency of direct hits that were AT MOST a category 3 storm?

1.4 Experimental Design and Ethics

How does sleep deprivation affect your ability to drive? A recent study measured the effects on 19 professional drivers. Each driver participated in two experimental sessions: one after normal sleep and one after 27 hours of total sleep deprivation. The treatments were assigned in random order. In each session, performance was measured on a variety of tasks including a driving simulation.

Use key terms from this module to describe the design of this experiment.

An advertisement for Acme Investments displays the two graphs in Figure 1.14 to show the value of Acme’s product in comparison with the Other Guy’s product. Describe the potentially misleading visual effect of these comparison graphs. How can this be corrected?

The graph in Figure 1.15 shows the number of complaints for six different airlines as reported to the US Department of Transportation in February 2013. Alaska, Pinnacle, and Airtran Airlines have far fewer complaints reported than American, Delta, and United. Can we conclude that American, Delta, and United are the worst airline carriers since they have the most complaints?

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/introductory-statistics/pages/1-introduction

• Authors: Barbara Illowsky, Susan Dean
• Publisher/website: OpenStax
• Book title: Introductory Statistics
• Publication date: Sep 19, 2013
• Location: Houston, Texas
• Book URL: https://openstax.org/books/introductory-statistics/pages/1-introduction
• Section URL: https://openstax.org/books/introductory-statistics/pages/1-homework

© Jun 23, 2022 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

Statistical Learning

Course material for bst 263, general information.

• An Introduction to Statistical Learning , by Gareth James, Daniela Witten, Trevor Hastie, and Robert Tibshirani.
• The Elements of Statistical Learning , by Trevor Hastie, Robert Tibshirani, and Jerome Friedman.

Lecture notes

• 1. Introduction (Course overview, Choosing among methods)
• 2. Probability and linear algebra basics
• 3. Measuring performance (K-nearest neighbors, MSE, Bias-variance, Classification error rate, Bayes optimal)
• knn.r (R code for KNN regression, MSE, Bias-variance tradeoff)
• knn-classifier.r (R code for KNN classifier, Error rate, Bayes optimal classifier)
• 4. Lab on KNN and measuring performance
• 5. Linear regression (Probabilistic model, Basis functions, Estimation, Uncertainty quantification)
• 6. Lab on Linear regression
• 7. Classification (Loss functions, Confusion matrix, ROC curve, Logistic regression, LDA/QDA)
• classification.r (R code for classification topics)
• 8. Lab on Classification
• 9. Cross-validation (k-fold CV, Choosing model settings with CV, Choosing # of folds)
• cv.r (R code for cross-validation topics)
• 10. Lab on Cross-validation
• 11. Penalized regression (Subset selection, Model selection, Ridge, Lasso, Elastic net)
• 12. Lab on Penalized regression
• 13. Principal components analysis (Intuition, Covariance method, SVD method, Principal components regression)
• 14. Lab on PCA
• (In progress)

Homework assignments

• Homework 1 (Probability and linear algebra basics)
• Homework 2 (Measuring performance, Bias-variance)
• Homework 3 (Linear regression)
• Homework 4 (Classification)
• Homework 5 (Cross-validation)
• Homework 6 (Penalized regression)
• Midterm solutions

STAT 430: Basics of Statistical Learning

University of illinois at urbana-champaign, fall 2017, dalpiaz, schedule - homework - quizzes - projects, syllabus - compass - r4sl.

• First day of class! Course overview and syllabus discussion.
• Materials : Syllabus Slides , Full Syllabus
• ISL Videos : Opening Remarks and Examples , Supervised and Unsupervised Learning
• Quick probability review. Recapping some R basics.
• Reading : R4SL Chapter 2 , R4SL Chapter 3
• Slides : Probability Recap , R Introduction
• Lab : R Basics , R Basics Solutions
• Introduction to rmarkdown .
• Slides : rmarkdown Introduction
• No class! Labor Day
• More rmarkdown details and practice. What is a model?
• Reading : R4SL Chapter 3
• Lab : rmarkdown , rmarkdown Solutions
• Begin recap of regression basics.
• Reading : ISL 3.1 - 3.4, R4SL Chapter 4
• ISL Slides : Linear Regression
• ISL Videos : Simple Linear Regression , Hypothesis Testing , Interpreting Regression Coefficients , Model Selection and Qualitative Predictors , Interactions and Nonlinearity
• Deadline : Homework 00 Due
• Review using lm() for regression models in R .
• Reading : R4SL Chapter 4
• Introduce the supervised learning, regression, task. Discuss the test-train split and models that generalize to unseen data.
• Reading : ISL 2.1 - 2.2
• ISL Slides : Statistical Learning
• ISL Videos : Statistical Learning and Regression , Assessing Model Accuracy and Bias-Variance Trade-off
• Lab : Test-Train Split , Test-Train Split Solutions
• Continue discussion of regression in the context of statistical learning.
• Slides : Linear Models for Statistical Learning, Regression
• Deadline : Homework 01 Due
• Introduce KNN. Compare non-parametric methods to parametric methods. Discuss tuning parameters versus model parameters.
• Reading : R4SL Chapter 7 (Currently very sparse notes.)
• Continue discussion of KNN. Compare KNN to linear models. Some live coding examples.
• Finish discussion of KNN
• Deadline : Homework 02 Due
• Bias-Variance Tradeoff
• Reading : R4SL Chapter 8
• Slides : Bias-Variance Tradeoff
• Begin classification.
• Reading : ISL 4.1, R4SL Chapter 9
• Slides : Classification Introduction
• More classification. Introduction to logistic regression
• Reading : ISL 4.2 - 4.3, R4SL Chapter 10
• ISL Slides : Classification
• ISL Videos : Introduction to Classification
• Deadline : Homework 03 Due
• Continued discussion of logistic regression.
• Reading : ISL 4.3, R4SL Chapter 10
• ISL Videos : Logistic Regression , Multiple Logistic Regression
• KNN for classification.
• Reading : R4SL Chapter 12
• Generative methods in R .
• Reading : ISL 4.4, R4SL Chapter 11
• Deadline : Homework 04 Due
• Continued discussion of generative methods. Details of univariate LDA.
• ISL Videos : Linear Discriminant Analysis and Bayes Theorem , Univariate Linear Discriminant Analysis
• Continued discussion of generative methods. Multivariate LDA, QDA, Naive Bayes.
• ISL Videos: Multivariate Linear Discriminant Analysis , Quadratic Discriminant Analysis and Naive Bayes
• Some final thoughts on generative methods. Some recap of classification methods. Some R details.
• Deadline : Homework 05 Due
• Begin discussing Statistical Learning in practice.
• Deadline : None. No homework during quiz week.
• Cross-validation and caret .
• Reading: ISL 5.1, R4SL Chapter 20 , R4SL Chapter 21
• ISL Slides: Resampling
• ISL Videos: Validation Set Approach , k-fold Cross-Validation , Cross-Validation: The Right and Wrong Ways
• More cross-validation and caret .
• Deadline : Homework 06 Due
• Some comments on variable selection.
• Reading: ISL 6.1, R4SL Chapter 22
• ISL Slides: Model Selection
• ISL Videos: Best Subset Selection , Forward Stepwise Selection , Backward Stepwise Selection , Estimating Test Error I , Estimating Test Error II
• Entering the modern age. Introducing regularization.
• Reading: ISL 6.2, R4SL Chapter 24 - Regularization
• ISL Videos: Shrinkage Methods and Ridge Regression , The Lasso , Tuning Parameter Selection
• More on ridge and lasso. Using ridge and lasso in R .
• Reading: ISL 6.2, R4SL Chapter 24 - Regularization , R4SL Chapter 25 - Elastic Net
• Deadline : Homework 07 Due
• Elastic net.
• R4SL: Elastic Net
• Overview: Introduction to trees.
• Reading: ISL 8.1
• ISL Slides: Trees
• Additonal Slides: Part II: Tree-based Methods
• ISL Videos: Decision Trees , Pruning a Decision Tree , Classification Trees and Comparison with Linear Models
• Discussed some finer details of R .
• Deadline : Homework 08 Due
• Deadline : Final Project Group Choice
• Continuation of tree discussion. Introduction to ensemble methods, mostly random forests.
• Reading: ISL 8.2
• R4SL: Ensemble Methods
• Additonal Slides: Part I: Pruning, Bagging, Boosting
• ISL Videos: Bootstrap Aggregation (Bagging) and Random Forests , Boosting and Variable Importance
• Continuation of tree discussion. Introduction to ensemble methods, mostly boosting.
• Extensions of random forests and boosting, in R . Some summary of supervised learning.
• Additional Slides: Supervised Learning Review
• Reading: Extremely Randomized Trees, Ranger, XGBoost [ rmarkdown ]
• Reading: Statistical Modeling: The Two Cultures
• Reading: Do we Need Hundreds of Classifiers to Solve Real World Classification Problems?
• No class. Fall break.
• Deadline : Homework 09 Due
• No class . Consider a group meeting.
• Unsupervised learning. Clustering.
• Reading: ISL 10.1 - 10.3
• R4SL: Unsupervised Learning
• ISL Slides: Unsupervised Learning
• Additional Slides: Unsupervised Learning, Part I, Clustering
• ISL Videos: Unsupervised Learning and Principal Components Analysis , Exploring Principal Components Analysis and Proportion of Variance Explained , K-means Clustering , Hierarchical Clustering
• Unsupervised learning. Clustering in R .
• Deadline : Project proposals. No homework is due.
• Unsupervised learning. PCA. Clustering again.
• Additional Slides: Unsupervised Learning, Part II, PCA
• No class. Office hours 8 - 10 at David’s office. Work on projects!
• Discussion of graduate student project results. Thoughts on keeping up to date with data science and machine learning.
• Reading: Some Machine Learning and Data Science Resources
• Deadline : Homework 10 Due
• No class. Finals!
• Deadline : Final Project Report
• Deadline : Final Project Peer Review

Homework 00

• Due: Friday, September 8
• Assignment: [ html ] [ pdf ] [ zip ]
• Solution: [ zip ]

Homework 01

• Due: Friday, September 15

Homework 02

• Due: Friday, September 22

Homework 03

Homework 04.

• Due: Friday, October 6

Homework 05

• Due: Friday, October 13

Homework 06

• Due: Friday, October 27

Homework 07

• Due: Friday, November 3

Homework 08

• Due: Friday, November 10

Homework 09

• Due: Monday, November 20

Homework 10

• Due: Wednesday, December 13
• Date: Wednesday, October 18
• Review: [ In-Class Practice Probelms ]
• Date: Wednesday, December 6

Group Final Project

• Group Choice - Friday, November 10, 11:59 PM
• Analysis Proposal - Friday, December 1, 11:59 PM
• Report Template
• Peer Evaluation - Thursday, December 21, 10:00 PM

Graduate Student Project

• Autograder - Saturday, December 9, 11:59 PM
• Report - Saturday, December 9, 11:59 PM

Advanced Topics in Statistical Learning: Spring 2024

Other resources

Statistics and Probability Worksheets

Welcome to the statistics and probability page at Math-Drills.com where there is a 100% chance of learning something! This page includes Statistics worksheets including collecting and organizing data, measures of central tendency (mean, median, mode and range) and probability.

Students spend their lives collecting, organizing, and analyzing data, so why not teach them a few skills to help them on their way. Data management is probably best done on authentic tasks that will engage students in their own learning. They can collect their own data on topics that interest them. For example, have you ever wondered if everyone shares the same taste in music as you? Perhaps a survey, a couple of graphs and a few analysis sentences will give you an idea.

Statistics has applications in many different fields of study. Budding scientists, stock market brokers, marketing geniuses, and many other pursuits will involve managing data on a daily basis. Teaching students critical thinking skills related to analyzing data they are presented will enable them to make crucial and informed decisions throughout their lives.

Probability is a topic in math that crosses over to several other skills such as decimals, percents, multiplication, division, fractions, etc. Probability worksheets will help students to practice all of these skills with a chance of success!

Most Popular Statistics and Probability Worksheets this Week

Mean, Median, Mode and Range Worksheets

Calculating the mean, median, mode and range are staples of the upper elementary math curriculum. Here you will find worksheets for practicing the calculation of mean, median, mode and range. In case you're not familiar with these concepts, here is how to calculate each one. To calculate the mean, add all of the numbers in the set together and divide that sum by the number of numbers in the set. To calculate the median, first arrange the numbers in order, then locate the middle number. In sets where there are an even number of numbers, calculate the mean of the two middle numbers. To calculate the mode, look for numbers that repeat. If there is only one of each number, the set has no mode. If there are doubles of two different numbers and there are more numbers in the set, the set has two modes. If there are triples of three different numbers and there are more numbers in the set, the set has three modes, and so on. The range is calculated by subtracting the least number from the greatest number.

Note that all of the measures of central tendency are included on each page, but you don't need to assign them all if you aren't working on them all. If you're only working on mean, only assign students to calculate the mean.

In order to determine the median, it is necessary to have your numbers sorted. It is also helpful in determining the mode and range. To expedite the process, these first worksheets include the lists of numbers already sorted.

• Calculating Mean, Median, Mode and Range from Sorted Lists Sets of 5 Numbers from 1 to 10 Sets of 5 Numbers from 10 to 99 Sets of 5 Numbers from 100 to 999 Sets of 10 Numbers from 1 to 10 Sets of 10 Numbers from 10 to 99 Sets of 10 Numbers from 100 to 999 Sets of 20 Numbers from 10 to 99 Sets of 15 Numbers from 100 to 999

Normally, data does not come in a sorted list, so these worksheets are a little more realistic. To find some of the statistics, it will be easier for students to put the numbers in order first.

• Calculating Mean, Median, Mode and Range from Unsorted Lists Sets of 5 Numbers from 1 to 10 Sets of 5 Numbers from 10 to 99 Sets of 5 Numbers from 100 to 999 Sets of 10 Numbers from 1 to 10 Sets of 10 Numbers from 10 to 99 Sets of 10 Numbers from 100 to 999 Sets of 20 Numbers from 10 to 99 Sets of 15 Numbers from 100 to 999

Collecting and Organizing Data

Teaching students how to collect and organize data enables them to develop skills that will enable them to study topics in statistics with more confidence and deeper understanding.

• Constructing Line Plots from Small Data Sets Construct Line Plots with Smaller Numbers and Lines with Ticks Provided (Small Data Set) Construct Line Plots with Smaller Numbers and Lines Only Provided (Small Data Set) Construct Line Plots with Smaller Numbers (Small Data Set) Construct Line Plots with Larger Numbers and Lines with Ticks Provided (Small Data Set) Construct Line Plots with Larger Numbers and Lines Only Provided (Small Data Set) Construct Line Plots with Larger Numbers (Small Data Set)
• Constructing Line Plots from Larger Data Sets Construct Line Plots with Smaller Numbers and Lines with Ticks Provided Construct Line Plots with Smaller Numbers and Lines Only Provided Construct Line Plots with Smaller Numbers Construct Line Plots with Larger Numbers and Lines with Ticks Provided Construct Line Plots with Larger Numbers and Lines Only Provided Construct Line Plots with Larger Numbers

Interpreting and Analyzing Data

Answering questions about graphs and other data helps students build critical thinking skills. Standard questions include determining the minimum, maximum, range, count, median, mode, and mean.

• Answering Questions About Stem-and-Leaf Plots Stem-and-Leaf Plots with about 25 data points Stem-and-Leaf Plots with about 50 data points Stem-and-Leaf Plots with about 100 data points
• Answering Questions About Line Plots Line Plots with Smaller Data Sets and Smaller Numbers Line Plots with Smaller Data Sets and Larger Numbers Line Plots with Larger Data Sets and Smaller Numbers Line Plots with Larger Data Sets and Larger Numbers
• Answering Questions About Broken-Line Graphs Answer Questions About Broken-Line Graphs
• Answering Questions About Circle Graphs Circle Graph Questions (Color Version) Circle Graph Questions (Black and White Version) Circle Graphs No Questions (Color Version) Circle Graphs No Questions (Black and White Version)
• Answering Questions About Pictographs Answer Questions About Pictographs

Probability Worksheets

• Calculating Probabilities with Dice Sum of Two Dice Probabilities Sum of Two Dice Probabilities (with table)

Spinners can be used for probability experiments or for theoretical probability. Students should intuitively know that a number that is more common on a spinner will come up more often. Spinning 100 or more times and tallying the results should get them close to the theoretical probability. The more sections there are, the more spins will be needed.

• Calculating Probabilities with Number Spinners Number Spinner Probability (4 Sections) Number Spinner Probability (5 Sections) Number Spinner Probability (6 Sections) Number Spinner Probability (7 Sections) Number Spinner Probability (8 Sections) Number Spinner Probability (9 Sections) Number Spinner Probability (10 Sections) Number Spinner Probability (11 Sections) Number Spinner Probability (12 Sections)

Non-numerical spinners can be used for experimental or theoretical probability. There are basic questions on every version with a couple extra questions on the A and B versions. Teachers and students can make up other questions to ask and conduct experiments or calculate the theoretical probability. Print copies for everyone or display on an interactive white board.

• Probability with Single-Event Spinners Animal Spinner Probability ( 4 Sections) Animal Spinner Probability ( 5 Sections) Animal Spinner Probability ( 10 Sections) Letter Spinner Probability ( 4 Sections) Letter Spinner Probability ( 5 Sections) Letter Spinner Probability ( 10 Sections) Color Spinner Probability ( 4 Sections) Color Spinner Probability ( 5 Sections) Color Spinner Probability ( 10 Sections)
• Probability with Multi-Event Spinners Animal/Letter Combined Spinner Probability ( 4 Sections) Animal/Letter Combined Spinner Probability ( 5 Sections) Animal/Letter Combined Spinner Probability ( 10 Sections) Animal/Letter/Color Combined Spinner Probability ( 4 Sections) Animal/Letter/Color Combined Spinner Probability ( 5 Sections) Animal/Letter/Color Combined Spinner Probability ( 10 Sections)

Copyright © 2005-2024 Math-Drills.com You may use the math worksheets on this website according to our Terms of Use to help students learn math.

• Effective Techniques for Tackling Statistics Homework

How to Approach and Solve Advanced Statistics Homework

Statistics homework often involves real-world applications that can seem daunting at first. However, with a structured approach and understanding of fundamental concepts, you can effectively tackle even the most challenging problems. This guide will walk you through a step-by-step approach to solving assignments similar to the Karfones Inc. problem, focusing on optimization and linear programming. The initial step is understanding the problem statement, which includes identifying the objective, constraints, and variables. Once you comprehend these elements, the next phase involves formulating the problem mathematically by creating an objective function and defining the constraints in mathematical terms. For example, in the Karfones Inc. problem, the goal is to maximize profit subject to constraints on sales time and minimum sales goals. Graphical representation helps visualize the feasible region and identify the optimal solution, while incorporating additional constraints may require re-evaluating this region. For more complex scenarios, linear programming techniques like the Simplex method and software tools such as Excel Solver, R , or Python can be utilized. Ultimately, interpreting the solution in real-world context and considering practical implications ensures the results are actionable. Regular practice, seeking help when needed, and staying organized are key to mastering these assignments.

Understanding the Problem Statement

The first step in solving any statistics assignment is to thoroughly understand the problem statement. This section will break down the essential components you need to identify and consider.

Identify the primary goal of the problem. What are you trying to achieve? In the Karfones Inc. problem, the objective is to maximize profit. Understanding the objective helps you focus on what needs to be optimized or solved.

Constraints

Next, identify the constraints or limitations. These are the conditions that must be met for the solution to be valid. In the Karfones Inc. example, constraints include the available sales time and minimum sales goals for each model. These constraints shape the feasible region within which the solution must lie.

Determine the unknowns that need to be solved. Variables represent the elements you need to find to achieve the objective. In our example, the variables are the number of model X and model Y telephones sold. Defining the variables clearly is crucial for setting up the mathematical model.

Formulating the Problem Mathematically

Once you have a clear understanding of the problem, the next step is to translate it into a mathematical model. This involves creating an objective function and defining the constraints in mathematical terms.

Objective Function

The objective function represents what you are trying to optimize. For Karfones Inc., the objective function is the total profit, which can be expressed as:

Profit=40X+50Y

where (X) and (Y) are the units of model X and model Y telephones sold, respectively. This function needs to be maximized subject to the given constraints.

Constraints are the conditions that limit the solution. For the Karfones Inc. problem, the constraints include:

3X+5Y≤600(total sales time)

X≥25(minimum sales of model X)

Y≥25(minimum sales of model Y)

These inequalities must be satisfied for any solution to be valid. Writing down these constraints helps in identifying the feasible region.

Example Problem Setup

To illustrate, let's set up the problem for Karfones Inc.:

• Objective: Maximize Profit = 40X + 50Y
• Constraints:
• 3X + 5Y ≤ 600

This setup forms the basis for solving the problem using graphical or algebraic methods.

Graphical Representation

For problems involving two variables, a graphical method can be used to find the feasible region and the optimal solution. This section will guide you through plotting the constraints and identifying the feasible region.

Plotting the Constraints

Start by drawing the lines representing each constraint on a graph. Each inequality constraint is converted into an equation to plot the line. For example, for the constraint 3X + 5Y ≤ 600, you plot the line 3X + 5Y = 600.

Identifying the Feasible Region

The feasible region is the area where all the constraints overlap. This region represents all possible solutions that satisfy the constraints. It's typically a polygon bounded by the constraint lines.

Determining the Optimal Solution

Evaluate the objective function at each vertex (corner point) of the feasible region to find the maximum or minimum value. For linear programming problems, the optimal solution lies at one of these vertices.

Considering Additional Constraints

Sometimes, additional constraints are introduced, which require adjustments to the mathematical model and feasible region. Let's discuss how to handle new constraints effectively.

Incorporating New Constraints

If a new constraint is introduced, such as selling at least as many model Y telephones as model X, you need to update your mathematical model. For example, the new constraint can be written as:

Updating the Feasible Region

Incorporate this new constraint into your graph and identify the new feasible region. This might reduce the size of the feasible region or shift it entirely.

Re-evaluating the Solution

With the new constraint in place, re-evaluate the vertices of the updated feasible region to find the new optimal solution. The process is similar to the initial evaluation but with the adjusted constraints.

Solving Using Linear Programming Techniques

For more complex problems or those involving more than two variables, linear programming techniques such as the Simplex method are used. This section will introduce these methods and the tools available.

Simplex Method

The Simplex method is a popular algorithm for solving linear programming problems. It iterates through possible solutions to find the optimal one efficiently.

Software Tools

Several software tools and online solvers can assist with linear programming problems:

• Excel Solver: A powerful tool within Microsoft Excel that can handle linear programming problems by setting up the objective function and constraints.
• R Programming: Packages like lpSolve and optim are useful for solving linear programming problems in R. They provide functions to define and solve optimization problems.
• Python: Libraries such as PuLP and SciPy offer robust solutions for optimization problems. These libraries provide functionalities to define constraints, objective functions, and solve the linear programming model.

Practical Application

To solve a problem using these tools, you typically need to:

• Define the objective function.
• Specify the constraints.
• Use the solver to find the optimal solution.

Interpreting the Solution

Once you have the optimal solution, it's important to interpret it in the context of the problem. This section will guide you through checking constraints, analyzing results, and understanding real-world implications.

Checking the Constraints

Ensure that the solution meets all the given constraints. Verify that the values of variables satisfy each inequality or equation. This step is crucial to confirm the validity of the solution.

Analyzing the Results

Understand what the solution means for the real-world scenario. For instance, in the Karfones Inc. problem, determine how many units of each model should be sold to maximize profit.

Real-World Implications

Consider the practical implications of the solution. Assess whether the solution is feasible and aligns with the company's goals. In some cases, the optimal mathematical solution might need adjustments to fit real-world constraints better.

Practical Tips for Success

To excel in solving statistics assignments, follow these practical tips:

Practice Regularly

The more you practice, the more comfortable you will become with different types of problems. Regular practice helps reinforce concepts and improves problem-solving skills.

Seek Help When Needed

Don’t hesitate to use resources like assignment help websites or consult with your professors if you get stuck. Seeking assistance can provide new insights and approaches to the problem.

Stay Organized

Keep your work neat and methodical. Breaking down complex problems into smaller, manageable parts can make them easier to solve. Organization helps in keeping track of various steps and ensures a clear solution path.

Use Technology

Utilize available software and online tools to simplify the solving process. Technology can handle complex calculations and provide visualizations, making it easier to understand and solve problems.

Continuous Learning

Stay updated with new techniques and methods in statistics and linear programming. Continuous learning helps in adopting the best practices and improving problem-solving efficiency.

Collaboration

Work with peers on complex problems. Collaborative efforts can lead to better solutions and a deeper understanding of the concepts involved. Group discussions often bring out different perspectives and solutions.

Time Management

Allocate sufficient time for each step of the problem-solving process. Proper time management ensures that you can thoroughly analyze and solve the problem without rushing through any part.

Critical Thinking

Apply critical thinking to analyze and approach problems from different angles. Question assumptions and consider various scenarios to find the most robust solution.

Real-World Applications

Relate problems to real-world scenarios to understand their practical relevance. Real-world applications provide context and make it easier to grasp complex concepts.

Review and Reflect

After solving a problem, review your approach and solution. Reflect on what worked well and what could be improved. This reflection helps in learning from each assignment and improving for future problems.

By following these steps, you can approach similar statistics assignments with confidence. Understanding the problem, formulating it mathematically, using graphical methods, and applying linear programming techniques will help you find optimal solutions efficiently. Remember, practice and utilizing available resources are key to mastering these types of assignments. For personalized help, consider using StatisticsHomeworkHelper.com to get expert assistance tailored to your needs. This comprehensive approach will equip you with the skills and knowledge needed to tackle even the most complex statistics assignments successfully. With dedication and the right strategies, you can excel in your statistics coursework and achieve your academic goals.

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Discover Frequently Asked Math Questions and Their Answers

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• What is a quadrilateral that is not a parallelogram and not a trapezoid?
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• How do you solve #\frac { 5} { 8} + \frac { 3} { 2} ( 4- \frac { 1} { 4} ) - \frac { 1} { 8}#?
• What are some acronyms for PEMDAS?
• How do you find all the asymptotes for function #y=(3x^2+2x-1)/(x^2-4 )#?
• How do you determine whether the graph of #y^2+3x=0# is symmetric with respect to the x axis, y axis or neither?
• How do you determine whether the graph of #y^2=(4x^2)/9-4# is symmetric with respect to the x axis, y axis, the line y=x or y=-x, or none of these?
• How do you find the end behavior of #-x^3+3x^2+x-3#?
• How do you find the asymptotes for #(2x^2 - x - 38) / (x^2 - 4)#?
• How do you find the asymptotes for #f(x) = (x^2) / (x^2 + 1)#?
• How do you find the vertical, horizontal and slant asymptotes of: #(3x-2) / (x+1)#?
• How do you find the Vertical, Horizontal, and Oblique Asymptote given #s(t)=(8t)/sin(t)#?
• How do you find vertical, horizontal and oblique asymptotes for #(x^3+1)/(x^2+3x)#?
• How do you find vertical, horizontal and oblique asymptotes for #y = (4x^3 + x^2 + x + 5 )/( x^2 + 3x)#?
• What is a pooled variance?
• What is the mean, mode median and range of 11, 12, 13, 12, 14, 11, 12?
• What is the z-score of sample X, if #n = 81, mu= 43, St. Dev. =90, and E[X] =57#?
• The camera club has five members, and the mathematics club has eight. There is only one member common to both clubs. In how many ways could a committee of four people be formed with at least one member from each club?
• How many permutations are there of the letter in the word baseball?
• How do you evaluate 6p4?
• What is the probability of #X= 6# successes, using the binomial formula?
• A lottery has a $100 000 first prize, a $25 000 second prize, and five $500 third prizes. A total of 50000 tickets are sold. What is the probability of winning a prize in this lottery?
• When a event is reported, the probability that is a negative event is 30%. What is the probability that 3 out of 5 reported events are negative?
• What is the median of 5, 19, 2, 28, 25?
• How do you solve this trigonometric equation?
• What is the frequency of #f(theta)= sin 3 t - cos2 t #?
• How do you evaluate #Sin(pi/2) + 6 cos(pi/3) #?
• How do you find the values of all six trigonometric functions of a right triangle ABC where C is the right angle, given a=9, b=40, c=41?
• How do you express (5*pi)/4 into degree?
• Solve for θ on the interval [90°,180°]:2tanθ +19 = 0?
• Prove that #((cos(33^@))^2-(cos(57^@))^2)/((sin(10.5^@))^2-(sin(34.5^@))^2)= -sqrt2# ?
• A triangle has sides A, B, and C. The angle between sides A and B is #(pi)/3# and the angle between sides B and C is #pi/6#. If side B has a length of 26, what is the area of the triangle?
• How do you write the following in trigonometric form and perform the operation given #(sqrt3+i)(1+i)#?
• A triangle has sides A, B, and C. The angle between sides A and B is #(2pi)/3#. If side C has a length of #32 # and the angle between sides B and C is #pi/12#, what is the length of side A?

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