Probability and Statistics IV: Confidence Intervals and Hypothesis Tests from edX

This course covers two important methodologies in statistics – confidence intervals and hypothesis testing.

Confidence intervals are encountered in everyday life, and allow us to make probabilistic statements such as: “Based on the sample of observations we conducted, we are 95% sure that the unknown mean lies between A and B,” and “We are 95% sure that Candidate Smith’s popularity is 52% +/- 3%.” We begin the course by discussing what a confidence interval is and how it is used. We then formulate and interpret confidence intervals for a variety of probability distributions and their parameters.

Hypothesis testing allows us to pose hypotheses and test their validity in a statistically rigorous way. For instance, “Does a new drug result in a higher cure rate than the old drug?” or “Is the mean tensile strength of item A greater than that of item B?” The second half the course begins by motivating hypothesis tests and how they are used. We then discuss the types of errors that can occur with hypothesis testing, and how to design tests to mitigate those errors. Finally, we formulate and interpret hypothesis tests for a variety of probability distributions and their parameters.

What's inside

Learning objectives

Identify what a confidence interval is and how it is used
Formulate and interpret confidence intervals for a variety of probability distributions and their parameters
Determine what a hypothesis test is and how it is used

Identify the types of errors that can occur with hypothesis testing, and how to design tests to mitigate those errors
Formulate and interpret hypothesis tests for a variety of probability distributions and their parameters
Upon completion of this course, learners will be able to:

Identify what a confidence interval is and how it is used
Formulate and interpret confidence intervals for a variety of probability distributions and their parameters
Determine what a hypothesis test is and how it is used
Identify the types of errors that can occur with hypothesis testing, and how to design tests to mitigate those errors
Formulate and interpret hypothesis tests for a variety of probability distributions and their parameters
Upon completion of this course, learners will be able to:

Syllabus

“FCPS” refers to the free text, A First Course in Probability and Statistics: free access is provided via a PDF file or as a book

Module 1: Confidence Intervals• Lesson 1: Introduction to Confidence Intervals (FCPS §6.1)• Lesson 2: Normal Mean (variance known) (FCPS §6.2)• Lesson 3: Difference of Two Normal Means (variances known) (FCPS §6.3)• Lesson 4: Normal Mean (variance unknown) (FCPS §6.4)• Lesson 5: Difference of Two Normal Means (unknown equal variances) (FCPS §6.5.1)• Lesson 6: Difference of Two Normal Means (variances unknown) (FCPS §6.5.2)• Lesson 7: Difference of Paired Normal Means (variances unknown) (FCPS §6.5.3) • Lesson 8: Normal Variance (FCPS §6.6)• Lesson 9: Ratio of Variances of Two Normals (FCPS §6.7)• Lesson 10: Bernoulli Proportion (FCPS §6.8)Module 2: Hypothesis Testing• Lesson 1: Introduction to Hypothesis Testing (FCPS §7.1)• Lesson 2: The Errors of Our Ways (FCPS §7.1)• Lesson 3: Normal Mean Test with Known Variance (FCPS §7.2.1) • Lesson 4: Normal Mean Test with Known Variance: Design (FCPS §7.2.2)• Lesson 5: Two-Sample Normal Means Test with Known Variances (FCPS §7.2.3)• Lesson 6: Normal Mean Test with Unknown Variance (FCPS §7.3.1) • Lesson 7: Two-Sample Normal Means Tests with Unknown Variances (FCPS §7.3.2)• Lesson 8: Two-Sample Normal Means Test with Paired Observations (FCPS §7.3.2)• Lesson 9: Normal Variance Test (FCPS §7.4.1)• Lesson 10: Two-Sample Normal Variances Test (FCPS §7.4.2)• Lesson 11: Bernoulli Proportion Test (FCPS §7.4.3)• Lesson 12: Two-Sample Bernoulli Proportions Test (FCPS §7.4.4)• Lesson 13: Goodness-of-Fit Tests: Introduction (FCPS §7.5.1)• Lesson 14: Goodness-of-Fit Tests: Examples (FCPS §7.5.2)• Lesson 15 [OPTIONAL]: Goodness-of-Fit Tests: Honors Example (FCPS §7.5.3)

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Studies confidence intervals and hypothesis testing, fundamental statistical methods

Employs clear and comprehensive course modules to guide learning

Taught by David Goldsman, an experienced and respected instructor in the field

Requiring a basic understanding of probability and statistics, this course may not be suitable for complete beginners

Covers a wide range of topics within confidence intervals and hypothesis testing, providing a solid foundation in these areas

Leverages the free text, "A First Course in Probability and Statistics," for additional support and reference

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Probability and Statistics IV: Confidence Intervals and Hypothesis Tests with these activities:

Review fundamental probability and statistical concepts

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Reviewing these foundational concepts will help establish a solid basis for the course and enhance comprehension of more advanced material.

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Review basic concepts of probability, including events, outcomes, and probability measures
Go over different probability distributions, such as binomial, normal, and Poisson distributions
Revisit fundamental statistical concepts, including mean, median, mode, and standard deviation

Read 'Mathematical Statistics with Applications' by Wackerly and Mendenhall

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This book provides a comprehensive treatment of the theoretical foundations and practical applications of statistics, enhancing your understanding of the concepts covered in the course.

View Mathematical Statistics with Applications on Amazon

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Read the assigned chapters corresponding to the course material
Take notes and highlight important concepts
Complete the practice exercises at the end of each chapter

Practice confidence interval calculations

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Engaging in practice drills will strengthen your understanding of the procedures involved in calculating and interpreting confidence intervals, enhancing your ability to apply these concepts in real-world situations.

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Solve practice problems on determining sample size for a given confidence level and margin of error
Practice calculating confidence intervals for different population parameters, such as mean and proportion
Interpret the results of confidence intervals, including the level of confidence and precision

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Participate in peer study groups

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Engaging in peer study sessions will provide opportunities to clarify concepts, discuss different perspectives, and reinforce your understanding through collaborative learning.

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Connect with classmates through discussion forums or social media groups
Organize regular study sessions to focus on specific topics or problem-solving
Take turns presenting concepts, leading discussions, and providing feedback to others

Create a comprehensive study guide

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Compiling a comprehensive study guide will help organize and reinforce your understanding of the course material, serving as a valuable resource for exam preparation and future reference.

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Gather notes, assignments, quizzes, and other course materials
Organize the materials into logical sections and subsections
Summarize key concepts, formulas, and examples in a concise and easy-to-understand manner
Include practice questions and exercises for self-assessment

Create a visual representation of hypothesis testing

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Creating a visual representation will aid in visualizing and comprehending the key elements and steps involved in hypothesis testing, bolstering your understanding of this crucial statistical technique.

Browse courses on Hypothesis Testing

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Design a flowchart or infographic that outlines the process of hypothesis testing
Incorporate examples to illustrate the different types of hypotheses and their corresponding statistical tests
Explain the concepts of statistical significance and Type I and Type II errors
Present your visual representation to classmates or a mentor for feedback

Explore online tutorials on advanced statistical methods

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Seeking out and engaging with online tutorials will expose you to additional statistical techniques and concepts, expanding your knowledge and broadening your horizons in the field of statistics.

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Identify reputable websites or platforms that offer tutorials on advanced statistical methods
Select topics that align with your interests or areas you want to improve
Follow the tutorials, taking notes and completing any practice exercises provided

Career center

Learners who complete Probability and Statistics IV: Confidence Intervals and Hypothesis Tests will develop knowledge and skills that may be useful to these careers:

Data Scientist

A Data Scientist combines programming skills with knowledge of mathematics and statistics to extract meaningful insights from data. This course in Probability and Statistics IV can be a valuable aid in developing foundational knowledge for a career as a Data Scientist. By learning how to formulate and interpret confidence intervals and hypothesis tests, you can enhance your ability to analyze data, draw conclusions, and make informed decisions as a Data Scientist.

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Statistician

A Statistician collects, analyzes, interprets, and presents data. They use their knowledge of statistical methods to develop and test hypotheses, which can be used to make informed decisions in a variety of fields. This course in Probability and Statistics IV can provide you with a strong foundation in the principles of statistics, preparing you for a successful career as a Statistician.

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Quantitative Analyst

A Quantitative Analyst (Quant) uses mathematical and statistical models to analyze financial data and make investment decisions. This course in Probability and Statistics IV can help you build the quantitative skills needed to succeed as a Quant. By learning how to formulate and interpret confidence intervals and hypothesis tests, you can enhance your ability to assess risk and make informed investment decisions.

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Market Researcher

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Business Analyst

A Business Analyst uses data and statistical techniques to identify and solve business problems. They work with stakeholders to understand their needs, gather and analyze data, and develop recommendations for improvement. This course in Probability and Statistics IV can help you build the analytical skills needed to succeed as a Business Analyst. By learning how to formulate and interpret confidence intervals and hypothesis tests, you can enhance your ability to analyze data and make informed recommendations.

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Actuary

An Actuary uses mathematical and statistical techniques to assess risk and uncertainty. They work in a variety of fields, including insurance, finance, and healthcare. This course on Probability and Statistics IV can provide you with a strong foundation in the principles of statistics, which are essential for success as an Actuary.

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Data Engineer

A Data Engineer designs and builds the infrastructure that stores and manages data. They work with data scientists and other stakeholders to ensure that data is accessible, reliable, and secure. This course on Probability and Statistics IV may be useful for a Data Engineer who wants to build a foundation in statistical concepts and methods.

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Epidemiologist

An Epidemiologist investigates the causes and patterns of disease and other health problems in populations. They use statistical methods to analyze data and develop strategies for prevention and control. This course on Probability and Statistics IV may be useful for an Epidemiologist who wants to build a foundation in statistical concepts and methods.

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Forensic Scientist

A Forensic Scientist uses scientific methods to analyze evidence and provide expert testimony in legal proceedings. They use statistical methods to interpret data and draw conclusions about the evidence. This course on Probability and Statistics IV may be useful for a Forensic Scientist who wants to build a foundation in statistical concepts and methods.

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Operations Research Analyst

An Operations Research Analyst uses mathematical and statistical techniques to solve complex problems in business and industry. They work with stakeholders to identify and define problems, collect and analyze data, and develop recommendations for improvement. This course on Probability and Statistics IV may be useful for an Operations Research Analyst who wants to build a foundation in statistical concepts and methods.

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Risk Manager

A Risk Manager identifies, assesses, and mitigates risks to an organization. They use statistical methods to analyze data and develop strategies for risk management. This course on Probability and Statistics IV may be useful for a Risk Manager who wants to build a foundation in statistical concepts and methods.

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Quality Control Manager

A Quality Control Manager ensures that products and services meet quality standards. They use statistical methods to analyze data and identify areas for improvement. This course on Probability and Statistics IV may be useful for a Quality Control Manager who wants to build a foundation in statistical concepts and methods.

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Technical Writer

A Technical Writer creates and maintains technical documentation, such as user manuals, product descriptions, and training materials. They use statistical methods to analyze data and develop clear and concise documentation. This course on Probability and Statistics IV may be useful for a Technical Writer who wants to build a foundation in statistical concepts and methods.

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Project Manager

A Project Manager plans, executes, and closes projects. They use statistical methods to track progress and identify risks. This course in Probability and Statistics IV may be useful for a Project Manager who wants to build a foundation in statistical concepts and methods.

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Software Engineer

A Software Engineer designs, develops, and tests software applications. They use statistical methods to analyze data and identify trends. This course in Probability and Statistics IV may be useful for a Software Engineer who wants to build a foundation in statistical concepts and methods.

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