Welcome to GAMES! A Gentle Approach to Math, Excel, and Stats: Course 2

Who is this course for?

This course is for a diverse set of learners in business, economics and other social sciences including (a) those who may lack a strong enough math and statistics background from high school, (b) learners seeking to refresh their math and statistics knowledge in preparation for upper years in university and college, (c) learners wishing to re-skill or up-skill by switching majors or adding a minor and lacking the necessary math and statistics tools.

What does this course cover?

The course has three primary segments.

Segment 1 covers some advanced topics related to the material in course 1: integration, logarithms and exponential functions and their relation with modelling growth, multivariable calculus and unconstrained optimization with many variables. Optimization and multivariable functions are common in many areas of the social sciences; this part of the course will help prepare learners in understanding these concepts and analyzing them with mathematical rigor.

Segment 2 introduces learners to the basic idea of probability, the relation between events, and the important concepts of conditional, marginal and total probability, and Bayes’ rule. The idea of probability distributions across random variables is then developed, along with their expectation and variance. The segment also introduces learners to some of the most common probability distributions, such as the Binomial and the Normal distribution. Examples from the social sciences and the use of Excel / Google sheets enhance the applicability of the concepts for learners.

Segment 3 focuses on some of the most essential statistical tools for learners, an important area given the ever-increasing usage of data in all disciplines. One is introduced to the most common tools for visually representing data, and then exposed to the various descriptive measures for data such as the average, the standard deviation, covariance and correlation. Learners are also introduced to linear regression and the basics of statistical inference such as estimation, confidence intervals for estimates, and hypothesis testing. Real-life data examples and the use of Excel / Google sheets for statistical computation makes the segment of immediate practical use.

There is a natural progression in terms of the segments. Some parts of segment 2 on probability draw from segment 1 (especially the concepts of integration), while parts of segment 3 on statistics build on the basics of probability distributions to develop understanding of the important concepts in statistical inference.

Learning objectives

By following the lectures and attempting the embedded and practice questions, the learner should be able to:

• Develop an understanding of the basics of integration, multivariable calculus, and unconstrained optimization with many variables.
• Comprehend the core concepts in probability, including dependent and independent events, conditional and marginal probability, and Bayes’ rule.
• Develop an understanding of expectation, variance and covariance of random variables, and be introduced to some of the most common probability distributions such as the Binomial, the Poisson and the Normal distributions.
• Represent statistical data visually as well as in terms of descriptive statistics, using spreadsheet software such as Excel / Google sheets.
• Develop an understanding of linear regression and the basics of statistical inference, such as estimation, generation of confidence intervals, testing of hypotheses and significance of regression coefficients.

Course resources

Learners have access to the following course resources:

• Close-captioned lecture videos
• Embedded questions (with solutions) in the lecture videos to encourage retention and provide immediate feedback
• Full lecture transcripts
• Blank and annotated lecture slides
• Practice problems, including some statistical problems with real-world data (data-sets provided in both .xls and .ods formats)
• Solutions to the practice questions, including Excel computations (provided in both .xls and .ods formats)

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) https://creativecommons.org/licenses/by-nc-nd/4.0/ Unless otherwise noted, all content in this video was created by Catherine Pfaff, Sumon Majumdar, and Robert J. McKeown. You are free to copy and share this material in any format, but you must give appropriate credit to the authors. This project is made possible with funding by the Government of Ontario and through eCampusOntario’s support of the Virtual Learning Strategy. To learn more about the Virtual Learning Strategy visit: https://vls.ecampusontario.ca/.

Watch Our Introduction to GAMES Video

Acknowledgements

There are many who helped us directly and indirectly with developing the two courses. The three of us have been teaching segments of the material covered here in various courses that we have taught over the years. We would like thank the many, many students in our courses who have indirectly helped us develop, revise and fine-tune the material and sharpen our teaching.

We would like to thank Alan Ableson, Andrew McEachern, William Nelson, Andrew Skelton and Victoria Sytsma for their subject-matter expertise, Randy Ellis for allowing Catherine use his Lightboard studio, and Robert Winkler, Yelin Su and the Teaching Commons at York University for reviewing and user-testing the accessibility of the courses.

We had many students at Queen’s and York University who reviewed the videos and the embedded questions, helped us generate examples, edited and formatted the videos and contributed in myriad ways, all of which have immensely enhanced the quality of the courses. We would like to thank Rebecca Carter and Vladimir Fenenko for their research and examples in our teaching material. Sean Steele and Alexis Blair Hamilton were essential in creating the interactive videos and transcripts with a tight deadline. Sebastian Vaillancourt, Haozhi (Howie) Hong, and Thurston Han edited many of our videos. Azeezah Jafry, Constance Anson, Diana Liu Xu, Rajbalinder Singh Ghatoura, Tao Tang, and Thashvin Ramnauth provided actionable feedback on our interactive videos.

Finally, we would like to thank Aya Javid at York University and Jenna Dijkema at Queen’s University for handling administrative matters related to the creation of the courses.

This project is made possible with funding by the Government of Ontario and through eCampusOntario’s support of the Virtual Learning Strategy. To learn more about the Virtual Learning Strategy visit: https://vls.ecampusontario.ca/.

Modules

Module 1: Integration 1

This module introduces students to integration, a somewhat advanced concept in calculus. It builds intuition for the basic idea through the use of Riemann sums and calculating the area under a curve using left and right end points. Taking the limit of the Riemann sums then establishes the concept of a definite integral. Having developed the concept for this fundamental idea, the module ends by talking about the practical issues of integrability and net area.

By the end of the module, the student should be able to:

• Develop an understanding of the concept of integration.
• Relate how integration is related to the area under a curve.
• Understand integrability and net area

Lecture Materials - This page includes the lecture notes from the videos and lecture transcripts.

Lectures

• 1.1 Integration Purpose Preamble
• 1.2 Overview of Integration and Its Relationship to Differentiation
• 1.3 Riemann Sum & Integration Warm-Up Example
• 1.4 Goal of Riemann Sums
• 1.5 Symbolically Dividing an Interval into n Subintervals
• 1.6 Approximating Area Under Curve Using Left Endpoints
• 1.7 Approximating Area Under Curve Using Left Endpoints & Funny Useful Notation
• 1.8 Approximating Area Under Curve Using Right Endpoints
• 1.9 Limit of Riemann Sums is Area Under Curve
• 1.10 General Application of Approximating Area Under Curve
• 1.11 Specific Application of Approximating Area Under Curve
• 1.12 Riemann Sum -> Definite Integrals
• 1.13 Integrability
• 1.14 Integrals and "Net Area"
• 1.15 Net Change Theorem

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 2: Integration 2

This module continues with integration, now introducing it as an antiderivative i.e. the opposite of differentiation. The basis for this is the Fundamental Theorem of Calculus. Developing this two-way connection between integration and differentiation leads one to the idea of an indefinite integral, and the relationship between definite and indefinite integrals. The second half of the module focusses on properties of integrals and some of the common techniques for computing integrals.

By the end of the module, the student should be able to:

• Develop an understanding of the concept of integration as an antiderivative
• Understand indefinite integrals
• Understand properties of integrals and techniques for integration

Lecture Materials - This page includes the lecture notes from the videos and lecture transcripts..

Lectures

• 2.1 Antiderivatives Introduced
• 2.2 Antiderivatives for a Function are Related
• 2.3 Antiderivative Rules and Examples
• 2.4 Fundamental Theorems of Calculus
• 2.5 The Definite Integral is Linear
• 2.6 Further Definite Integral Properties
• 2.7 Indefinite Integrals Introduced
• 2.8 Relationship Between Definite and Indefinite Integrals
• 2.9 (First) Properties of the Indefinite Integral
• 2.10 Solving for the Constant in an Indefinite Integral
• 2.11 Substitution for Indefinite Integrals Introduced
• 2.12 Identifying Substitution Situations
• 2.13 Substitution for Indefinite Integrals Example 1
• 2.14 Substitution for Indefinite Integrals Example 2
• 2.15 Substitution Rule for Definite Integrals
• 2.16 Comparison Properties for Integrals

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 3: Logarithms and Exponential Functions and Growth of Functions

This module introduces students to the idea of logarithms and exponential functions. Having established the basic idea, it then delves into an understanding of the connection of these functions with growth. The module then studies growth rates of exponential functions and compares the consequences of exponential versus linear growth. The module also looks at some practical examples of exponential growth and ends by introducing the useful idea of the logarithmic scale.

By the end of the module, the student should be able to:

• Understand the concept of an exponential function
• Develop an understanding of exponential growth and decay and their consequences in real-world examples
• Distinguish between exponential and linear growth

Lecture Materials - This page includes the lecture notes from the videos and lecture transcripts..

Lectures

• 3.1 What is an Exponential Function?
• 3.2 Log a(x) is the Inverse of a^x
• 3.3 Our Favorite Log Function: The Natural Logarithm
• 3.4 Careful with e^kx
• 3.5 Careful with ln(u) and 1/u
• 3.6 Comparing Function Growth Rate Introduced
• 3.7 L'Hopital's Rule Introduced
• 3.8 Examples of Using l'Hopital's Rule to Compare Function Growth
• 3.9 Exponential Growth and Decay Differential Equation
• 3.10 Population Growth Example
• 3.11 Comparing Exponential Growth Example
• 3.12 Linear vs Exponential Growth Example
• 3.13 The Logarithmic Scale
• 3.14 Inverse Square Law Example

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 4: Multivariable Calculus

This module introduces students to multivariable calculus. This is a major area of calculus and this module covers only some of the basic ideas that will be useful for students in the social sciences. The module begins by introducing functions of many variables with some graphs of 2-variable functions. Having established the basic idea, we then develop the important concept of partial derivatives. The second half of the module focuses on local and global extrema for multivariable functions and on the useful methodology for finding extrema.

By the end of the module, the student should be able to:

• Understand the concept of a function of many variables
• Understand and compute partial derivatives
• Develop an understanding of local and global extrema for multivariable functions and the methodology for finding extrema

Lecture Materials - This page includes the lecture notes from the videos and lecture transcripts.

Lectures

• 4.1 Multivariable Functions Introduced
• 4.2 Graphs of 2-Variable Functions
• 4.3 A Little Practice Graphing 2-Variable Functions
• 4.4 2- and 3-Dimensional Vector Basics
• 4.5 Dot Product of 2-Dimensional Vectors
• 4.6 Partial Derivatives Introduced
• 4.7 Partial Derivative Picture
• 4.8 Approximating 2-Variable Function with Tangent Plane
• 4.9 Gradient
• 4.10 Directional Derivatives
• 4.11 2nd Order Partial Derivatives by Example
• 4.12 f(x.y) Local Extrema
• 4.13 f(x,y) Global Extrema
• 4.14 Finding Global Extrema
• 4.15 Methodology for Finding Local Min/Max of f(x,y)
• 4.16 Finding Critical Points for f(x,y) Example
• 4.17 Classifying Local Minima, Local Maxima, Saddle Points Example

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 5: Basics of Probability

Uncertainty pervades human decision-making, the focus of much of the social sciences. In this module, we introduce the basic idea of probability, which provides a formal language for analyzing uncertainty. Starting from the main axioms, we develop how probabilities of different events can be calculated. The important idea of independence of events and its implication for calculating probabilities of compound events is also covered in the module. A diverse set of examples introduce the student to calculating probabilities in a variety of contexts.

By the end of the module, the student should be able to:

• Calculate probabilities of some simple and compound events
• Develop an understanding of the independence of events and its implications
• Develop an understanding of the interpretation of probability

Lecture Materials - This page includes basic and annotated slides that are used in lectures and lecture transcripts.

Lectures

• 5.1 Introduction to Probability
• 5.2 Terminology and Venn diagrams
• 5.3 Basic Axioms of Probability and the Complementary Rule
• 5.4 The Addition Rule
• 5.5 Extending the Addition Rule
• 5.6 An Application of the Extended Addition Rule and Interpretation of Probability
• 5.7 Relationship Between Events: Independence
• 5.8 Implications and Applications of Independence
• 5.9 Implications and Applications of Independence contd.
• 5.10 Another Application of Independence
• 5.11 Checking for Independence

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 6: Conditional Probability and Bayes' Rule

This module focuses on developing an understanding of the relationship between events. Using probability trees, we show how probabilities of combination of events can be calculated. The module then introduces the related ideas of Conditional Probability, Marginal and Total Probability and demonstrates them in the context of two-way frequency tables. The module ends by introducing students to the idea of reverse conditioning and the important concept of Bayes’ rule. Examples from politics and stock markets are used to demonstrate some of the concepts.

By the end of the module, the student should be able to:

• Calculate probabilities of dependent and independent events
• Calculate Conditional, Marginal and Total Probability
• Develop an understanding of reverse conditioning and Bayes’ rule

Lecture Materials - This page includes basic and annotated slides that are used in lectures and lecture transcripts.

Lectures

• 6.1 Module 5 Recap and Introduction to Dependent Events
• 6.2 Probability Trees and General Multiplication Rule
• 6.3 An Application and the Law of Total Probability
• 6.4 The Law of Total Probability contd. and an Application
• 6.5 The Law of Total Probability at Work contd.: An Application
• 6.6 Probability Trees with Independent Events and Introduction to Finding Conditional Probabilities
• 6.7 Finding Conditional Probability contd. and Introduction to Marginal Probability
• 6.8 Two Way Frequency Tables: An Example
• 6.9 Finding Conditional Probability From a Two-Way Frequency Table
• 6.10 The Many Applications of Bayes' Rule
• 6.11 Bayes' Rule
• 6.12 Bayes' Rule at Work: An Example from the Stock Markets
• 6.13 The Many Applications of Bayes' Rule

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Course 2 Term Test

This test covers modules 1-6. Try to complete the test in less than 90 minutes without looking at your notes or the internet - just a piece of paper and a calculator. After completing the test on your own with no outside help, compare your answers to those in the solution file.

Course 2 Term Test

Module 7: Discrete Random Variables

This module begins by revisiting the important concept of Bayes’ rule with examples from medical testing and credit histories. We then introduce discrete random variables and develop an understanding of their probability distribution, in particular their p.m.f. and c.d.f. and the relationship between the two. The important concepts of expectation and variance of a discrete random variable are subsequently studied. The module ends by introducing students to three common discrete distributions: the Bernoulli, the Binomial and the Poisson distribution, and how their probabilities can be computed using Excel/Google sheets.

By the end of the module, the student should be able to:

• Develop an understanding of the p.m.f. and c.d.f. of a discrete random variable
• Develop an understanding of the expectation and variance of a discrete random variable
• Understand the Bernoulli, the Binomial and the Poisson distributions, and use Excel/Google sheets to calculate probabilities for these distributions

Lecture Materials - This page includes basic and annotated slides and Excel/ods files that are used in lectures and lecture transcripts.

Lectures

• 7.1 Module 6 Recap and Review of Bayes' Rule
• 7.2 Bayes' Rule Extended with an Application to Bank Lending and Credit Histories
• 7.3 Introduction to Random Variables and the Prob. Mass Function
• 7.4 Deriving P.M.F. for Discrete Random Variables: Two Examples
• 7.5 Introduction to the Cumulative Distribution Function (C.D.F.)
• 7.6 Properties of the C.D.F.
• 7.7 Deriving P.M.F. from C.D.F.
• 7.8 Expectation of a Random Variable
• 7.9 Properties of the Expected Value, with an Application
• 7.10 Variance of a Random Variable
• 7.11 Common Discrete Distributions 1 & 2: Bernoulli and Binomial
• 7.12 More on the Binomial Distribution
• 7.13 Common Discrete Distributions 3: Poisson
• 7.14 Excel: Calculating Expectation and Variance
• 7.15 Excel: Calculating Binomial and Poisson probabilities

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 8: Continuous Random Variables

This module begins by covering a residual topic from the previous module: Covariance and Correlation between two random variables. We then introduce continuous random variables and develop an understanding of their probability distribution, in particular their p.d.f. and c.d.f. The expectation and variance of a continuous random variable are subsequently studied. The final segment of the module introduces students to two common continuous distributions: the Uniform and the Normal distribution. It delves into the important properties of the Normal distribution in preparation for their use in the Statistics modules later.

By the end of the module, the student should be able to:

• Understand Covariance and Correlation between two random variables
• Develop an understanding of the p.d.f. and c.d.f. of a continuous random variable
• Develop an understanding of the expectation and variance of a continuous random variable
• Understand properties of the Uniform and the Normal Distribution

Lecture Materials - This page includes basic and annotated slides and Excel/ods files that are used in lectures and lecture transcripts.

Lectures

• 8.1 Module 7 Recap
• 8.2 Covariance and Correlation
• 8.3 An Example of Computing Covariance, and Properties of Covariance
• 8.4 Correlation Coefficient: Definition, Example and Properties
• 8.5 Introduction to Continuous Random Variables and their C.D.F.
• 8.6 P.D.F. for continuous random variables
• 8.7 More on the P.D.F.
• 8.8 Connections between the C.D.F. and the P.D.F.
• 8.9 Expectation and Variance of a Continuous Random Variable
• 8.10 Common Continuous Distributions 1: Uniform Distribution
• 8.11 Common Continuous Distributions 2: Normal Distribution
• 8.12 Calculating Probabilities for the Normal distribution: With and Without Excel
• 8.13 Properties of the Normal Distribution
• 8.14 Percentiles, and Module Recap

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 9: Introduction to Statistics and Data Visualization

The analysis of data is an integral part of most areas in the social sciences. The widespread availability of data and related statistical software has only increased its importance. This module begins by introducing students to the various types of data and variables that are usually encountered in the social sciences. We then focus on some of the common tools that are used to visually represent different types of data: Bar/Column charts, Pie charts, Time trends, Scatter plots and Histograms. Using real data from a diverse set of areas, we demonstrate how Excel/Google sheets can be used to generate the particular type of chart or plot that is most appropriate for the given situation.

By the end of the module, the student should be able to:

• Understand the various types of data and variables that are common in the social sciences
• Develop an understanding of the appropriate visual representation tool to use for a given situation
• Generate Bar/Column charts, Pie charts, Time trends, Scatter plots and Histograms using Excel/Google sheets

Lecture Materials - This page includes basic and annotated slides and Excel/ods files that are used in lectures and lecture transcripts.

Lectures

• 9.1 The Role of Statistics in the Social Sciences
• 9.2 Sources of Data
• 9.3 Types of Variables
• 9.4 Visual Presentation of Categorical Variable Data: Column Chart/Bar Graph
• 9.5 Excel: Creating a Column Chart/Bar Graph 1
• 9.6 Excel: Creating a Column Chart/Bar Graph/Pie Chart 2
• 9.7 Excel: Creating a Column Chart/Bar Graph/Pie Chart 3
• 9.8 Excel: Grouped and Stacked Column Chart/Bar Graph
• 9.9 Visual Presentation of Quantitative Variable Data: Time Trend
• 9.10 Excel: Creating a Scatter Plot 1
• 9.11 Excel: Creating a Scatter Plot 2
• 9.12 Visual Presentation of a Quantitative Variable's Distribution: Histogram 1 (Excel)
• 9.13 Excel: Creating a Histogram 2 and Module Recap

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 10: Average and Variability

A data series can have dozens, hundreds, or even millions of separate observations. People in business and the social sciences need single statistics to describe large quantities of information. The most common statistics are measures of central tendency which summarize the most common or most likely values in the data. Measures of central tendency are also known as averages: the median, the mean, and the mode. Another common descriptive statistic relates to variability or dispersion - a measure of spread. Common measures of variability include the range, IQR, variance, and standard deviation. These descriptive statistics are useful tools for describing and summarizing data. Variance and standard deviation are also useful in more advanced statistical testing and forecasting.

By the end of this module, the student should be able to:

• Compute the median, the mean, and the mode, range, IQR, variance, and standard deviation by hand and using real-world and simulated data.
• Create a stem-and-leaf plot Excel or another spreadsheet program and use it to understand the distribution of values in a series.
• Create a frequency PDF and CDF Table by hand and using Excel or another spreadsheet program.

Lecture Materials - This page includes basic and annotated slides and Excel/ods files that are used in lectures and lecture transcripts.

Lectures

• 10.1 How to Measure Central Tendency
• 10.2 The Average: Mean
• 10.3 The Weighted Mean
• 10.4 The Average: Median
• 10.5 Comparing Mean and Medians
• 10.6 Comparing the Mean, Median, and Mode
• 10.7 Variability
• 10.8 The Range
• 10.9 Standard Deviation I
• 10.10 Standard Deviation II
• 10.11 Sample Standard Deviation and Variance
• 10.12 Frequency PDF and CDF
• 10.13 Excel: Stem-Leaf Plot and Useful Tools
• 10.14 Excel: Central Tendency and IQR
• 10.15 Excel: Measures of Dispersion and the Frequency Function

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 11: Correlation and Linear Regression

Business leaders and social scientists need to understand the relationship between two or more variables. For example, determining whether a larger sales force produces more sales or whether increasing income inequality leads to more crime. Correlation coefficients can capture the relationship between two variables in a single statistic. This module explains how to calculate a correlation coefficient and describes the conditions under which the correlation coefficient is accurate. The concept of correlation is built upon to introduce simple linear regression - a simple but useful technique for forecasting. Lastly, a measure of error called the residual is introduced and used to verify whether the conditions for an accurate simple linear regression have been satisfied.

By the end of this module, the student should be able to:

• Calculate the correlation coefficient and use it to identify the direction and strength of the relationship between two variables.
• Explain the conditions under which the correlation coefficient and simple linear regression are accurate measures of the true underlying relationship between two variables.
• Graphically analyze the residuals from linear regression to determine if the conditions for an accurate prediction are satisfied.
• Calculate correlation, covariance, and a simple linear regression by hand and using Excel or another spreadsheet program.

Lecture Materials - This page includes basic and annotated slides and Excel/ods files that are used in lectures and lecture transcripts.

Lectures

• 11.1 Covariance and Correlation
• 11.2 Correlation Examples
• 11.3 Relationship Between Two Variables: Correlation
• 11.4 Interpreting a Correlation Coefficient
• 11.5 Correlation of Determination
• 11.6 The Accuracy of the Correlation Coefficient
• 11.7 Satisfying Correlation Conditions
• 11.8 Properties of the Correlation Coefficient
• 11.9 Making Predictions with Simple Linear Regression
• 11.10 Linear Regression Example
• 11.11 Residuals: a Measure of Error
• 11.12 Analyzing Residuals
• 11.13 Excel: Correlation with Education and Income
• 11.14 Excel: Correlation Matrix with Crime and Ice Cream
• 11.15 Excel: Linear Regression with Education, Crime, and Ice Cream
• 11.16 Excel: Linear Regression with Outliers

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Module 12: Basics of Statistical Inference

This module introduces students to the idea of drawing inference about the entire population based on observations on a small part of the population. It is a vast area of statistics, and this long-ish module covers only some of the basic ideas of statistical inference. Using insights from the Central Limit Theorem, we develop methods of estimation of population parameters, their standard errors and confidence intervals, and the testing of hypotheses. The module ends by connecting these ideas with the statistical significance of regression coefficients, including some examples from academic research papers.

By the end of the module, the student should be able to:

• Compute an estimate, its standard error and confidence intervals
• Test hypothesis for a single population
• Develop an understanding of the statistical significance of regression coefficients

Lecture Materials - This page includes basic and annotated slides and Excel/ods files that are used in lectures and lecture transcripts.

Lectures

• 12.1 Introduction to the Main Questions in Statistical Inference
• 12.2 Main Issues in Sampling
• 12.3 What is One Looking For: Parameter and Statistic
• 12.4 The Law of Large Numbers and the Central Limit Theorem
• 12.5 Estimate and Standard Error, With an Example
• 12.6 Excel: Computing Estimates and Standard Errors
• 12.7 Confidence Intervals: The Main Idea and Real Life Examples
• 12.8 Confidence Interval: Examples and Properties
• 12.9 Interpreting a Confidence Interval and Different CIs
• 12.10 Hypothesis Testing: The Basic Idea
• 12.11 The Mechanics of Hypothesis Testing
• 12.12 Example: Hypothesis Testing With Proportions and P-values
• 12.13 Two-sided vs One-sided Tests
• 12.14 Connecting Hypothesis Testing with Regressions
• 12.15 Examples from Research Papers and Module Recap

Practice Problems - After watching the videos, attempt these problems and check your answers against the solutions.

Course 2 Final Exam

This exam covers modules 1-12 with particular emphasis on modules 8-12. Try to complete the test in less than 120 minutes without looking at your notes or the internet - just a piece of paper and a calculator. After completing the test on your own with no outside help, compare your answers to those in the solution file.

Course 2 Final Exam