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Introduction to Statistics
Purpose of Course showclose
In this course, you will look at the properties behind the basic concepts of probability and statistics and focus on applications of statistical knowledge. You will learn about how statistics and probability work together. The subject of statistics involves the study of methods for collecting, summarizing, and interpreting data. Statistics formalizes the process of making decisions, and this course is designed to help you use statistical literacy to make better decisions. Note that this course has applications for the natural sciences, economics, computer science, finance, psychology, sociology, criminology, and many other fields.
We read data in articles and reports every day. After finishing this course, you should be comfortable evaluating the author’s use of this data. You will be able to extract information from the articles and display that information effectively. You will also be able to understand the basics of how to draw statistical conclusions.
This course will begin with descriptive statistics and the foundation of statistics. You will then learn about probability and random distributions, the latter of which enable us to work with several aspects of random events and their applications. Finally, you will examine a number of ways to investigate the relationships between various characteristics of data. By the end of this course, you should have a grasp on what statistics represent, how to use them to organize and display data, and how to test data to make effective conclusions.
Learning Outcomes showclose
- Define descriptive statistics and statistical inference.
- Distinguish between a population and a sample.
- Explain the purpose of measures of location, variability, and skewness.
- Calculate probabilities.
- Explain the difference between how probabilities are computed for discrete and continuous random variables.
- Recognize and understand discrete probability distribution functions, in general.
- Identify confidence intervals for means and proportions.
- Explain how the central limit theorem applies in inference.
- Calculate and interpret confidence intervals for one population average and one population proportion.
- Differentiate between Type I and Type II errors.
- Conduct and interpret hypothesis tests.
- Compute regression equations for data.
- Use regression equations to make predictions.
- Conduct and interpret Analysis of Variance (ANOVA).
Course Requirements showclose
√ Have access to a computer.
√ Have continuous broadband Internet access.
√ Have the ability/permission to install plug-ins or software (Adobe Reader, Flash, etc.).
√ Have the ability to download and save files and documents to a computer.
√ Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).
√ Have read the Saylor Student Handbook.
Unit Outline show close
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Unit 1: Data and Descriptive Statistics
In today’s world we access and use large volumes of data every day. The first step of data analysis is to accurately summarize this data, both graphically and numerically, so that we can understand what the data is saying. To be able to use and interpret data correctly is essential to making informed decisions. In this unit you will learn about descriptive statistics, which is used to summarize and display data. After completing this unit, you will know what you can do to present data that you have collected.
Unit 1 Time Advisory show close
For example, suppose you are interested in buying a new mobile phone with a particular type of a camera. Suppose you are not sure about the price of any of the phones with this feature, so you log on to a website that provides you with a sample data set of prices, given your requirements. Now, looking at all the prices in the sample can sometimes be confusing. A better way to compare might be to look at the median price and the variation of prices. The median and variation are two of several ways that you can describe data. You can also graph the data so that it is easier to see what the price distribution looks like. In this unit, you will study numerical and graphical ways to describe and display data. You will understand the principles of calculating common descriptive statistics for measuring center, variability, and skewness in data. You will learn to not only calculate but to interpret these measurements and graphs.
Descriptive statistics describe what the data show. Numerical descriptive measures computed from data are called statistics. Numerical descriptive measures of the population, however, are called parameters. Remember: Descriptive statistics are, as their name suggests, “descriptive.” They do not generalize beyond the data considered. Inferential statistics can be used to generalize the findings from sample data to a broader population.
Unit 1 Learning Outcomes show close
- 1.1 Introduction to Statistics
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1.1.1 Statistics, Probability, and Key Terms
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 1: Sampling and Data
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 1: Sampling and Data: “Section 1: Sampling and Data” (PDF), “Section 2: Statistics” (PDF), “Section 3: Probability” (PDF), and “Section 4: Key Terms” (PDF)
Instructions: Read sections 1 - 4. Sections 1 and 2 provide a brief introduction to the field of statistics and the concepts of statistics. Section 3 introduces the concepts of probability, including some real-world examples. Finally, Section 4 presents a number of key terms related to statistical sampling and data.
These readings should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 1: Sampling and Data
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1.1.2 Data and Sampling
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 1: Sampling and Data
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 1: Sampling and Data: “Section 5: Data” (PDF) and “Section 6: Sampling” (PDF)
Instructions: Read sections 5 and 6. Section 5 discusses the initial concepts of qualitative data, quantitative continuous data, and quantitative discrete data as used in statistics. Section 6 introduces the concept of statistical sampling.
These readings should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 1: Sampling and Data
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1.1.3 Variation and Frequency
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 1: Sampling and Data
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 1: Sampling and Data: “Section 7: Variation” (PDF), “Section 8: Answers and Rounding Off” (PDF), and “Section 9: Frequency” (PDF)
Instructions: Read sections 7 - 8. Section 7 will briefly discuss statistical variability within data and samples, Section 8 is a brief review of how you should round your answers, and Section 9 will introduce the concepts of frequency, relative frequency, and cumulative relative frequency.
These readings should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 1: Sampling and Data”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 1: Sampling and Data” (YouTube)
Instructions: Watch lecture 1. This lecture will review the topics from Chapter 1.
This lecture should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Assessment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 1 Practice: Sampling and Data”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 1: Practice: Sampling and Data” (PDF)
Instructions: Solve problems 1 - 6 under the section titled “Key Terms” and problems 7 - 12 under the section titled “Discussion Questions.” Next, click on the hyperlink titled “Homework” and solve problems 1, 3, 5, 7, 9, 11, 16, 19, 21, and 25-27. To check your solutions, go to the end of the document.
This assessment should take approximately 2 hours to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 1: Sampling and Data
- 1.2 Descriptive Statistics: Displaying Data
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1.2.1 Stem and Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 1: Descriptive Statistics” (PDF), “Section 2: Displaying Data” (PDF), “Section 3: Stem and Leaf Graphs (Stemplots), Line Graphs and Bar Graphs” (PDF)
Instructions: Read sections 1 - 3. Section 1 provides an overview of the chapter, and in Section 2 you will see the ways graphs and charts can be used to provide visual representations of data. Section 3 introduces the use of stem-and-leaf graphs, line graphs, and bar graphs for describing a set of data visually.
These readings should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Reading Pictographs,” “Reading Bar Graphs,” “Reading Line Graphs,” “Reading Pie Graphs (Circle Graphs),” “Stem-and-leaf Plots,” and “Misleading Line Graphs”
Link: Khan Academy’s Statistics: “Reading Pictographs” (YouTube), “Reading Bar Graphs” (YouTube), “Reading Line Graphs” (YouTube), “Reading Pie Graphs (Circle Graphs)” (YouTube), “Stem-and-leaf Plots” (YouTube), and “Misleading Line Graphs” (YouTube)
Instructions: View all six lectures. First, view the first five lectures; these explain how to read different types of graphs and show examples of each type of graph. Then, watch the last video titled “Misleading Line Graphs”, which will give an example of a misleading line graph and how it can distort patterns and trends.
Viewing these lectures and taking notes should take approximately 30 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics
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1.2.2 Histograms
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 4: Histograms”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 4: Histograms” (PDF)
Instructions: Read section 4. It introduces a method of displaying data called a histogram.
This reading should take you approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 4: Histograms”
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1.2.3 Box Plots
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 5: Box Plots”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 5: Box Plots” (PDF)
Instructions: Click the link above and read this section. Section 5 discusses box plots, which give a good graphical image of the concentration a given data set.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Box-and-Whisker Plots” and “Reading Box-and-Whisker Plots”
Link: Khan Academy’s Statistics: “Box-and-Whisker Plots” (YouTube) and “Reading Box-and-Whisker Plots” (YouTube)
Instructions: Click on the above link and view the lecture titled “Box-and-Whisker Plots”. In this video, you will learn about box plots, which give a good graph of the concentration of the data. Then, view the lecture titled “Reading Box-and-Whisker Plots” to learn how to read and interpret a box plot.
Viewing these lectures should take approximately 15 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 5: Box Plots”
- 1.3 Descriptive Statistics: Measures
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1.3.1 Measures of Location
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 6: Measures of the Location of the Data”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 6: Measures of the Location of the Data” (PDF)
Instructions: Read section 6. It explains percentiles and quartiles.
This reading, including the examples and problems, should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 6: Measures of the Location of the Data”
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1.3.2 Measures of Center
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 7: Measures of the Center of the Data”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 7: Measures of the Center of the Data” (PDF)
Instructions: Read section 7. It discusses ways to measure descriptive statistical information using the center of the data.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “The Average” and “Sample vs. Population Mean”
Link: Khan Academy’s Statistics: “The Average” (YouTube) and “Sample vs. Population Mean” (YouTube)
Instructions: Click the link above and view lecture titled “The Average." This lecture provides an introduction to descriptive statistics and central tendency as well as ways to measure the average of a set using median, mean, and mode. Then, view the lecture titled “Sample vs. Population Mean.” This lecture discusses the difference between the mean of a sample and the mean of a population.
Viewing these lectures and taking notes should take approximately 20 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Mean, Median, and Mode” and “Range and Mid-range”
Link: Khan Academy’s Statistics: “Mean, Median, and Mode” (YouTube) and “Range and Mid-range” (YouTube)
Instructions: View the linked lecture “Mean, Median, and Mode.” In this lecture, you will learn about some of the most widely used measures central tendency. Then, view the lecture titled “Range and Mid-range” to learn how to calculate range and mid-range for a given data set.
Viewing these lectures and taking notes should take approximately 15 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 7: Measures of the Center of the Data”
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1.3.3 Skewness and the Mean, Median, and Mode
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 8: Skewness and the Mean, Median, and Mode”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 8: Skewness and the Mean, Median, and Mode” (PDF)
Instructions: Read the section linked above. Section 8 discusses skewness and symmetry and how it relates to the mean, median, and mode.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 8: Skewness and the Mean, Median, and Mode”
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1.3.4 Measures of Spread
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 9: Measures of the Spread of the Data”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics “Section 9: Measures of the Spread of the Data” (PDF)
Instructions: Read the section linked above. Section 9 explains standard deviation as a measure of variation in data.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 2: Descriptive Statistics”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 2: Descriptive Statistics” (YouTube)
Instructions: View the linked lecture “Video Lecture 2: Descriptive Statistics” to learn how to measure and display data. You will learn how to display data numerically and graphically as well as different measures of data.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Variance of a Population,” “Sample Variance,” “Standard Deviation,” and “Alternate Variance Formulas”
Link: Khan Academy’s Statistics: “Variance of a Population” (YouTube), “Sample Variance” (YouTube), “Standard Deviation” (YouTube), and “Alternate Variance Formulas” (YouTube)
Instructions: View all four lectures. First, view “Variance of a Population.” Second, watch “Sample Variance,” which explains how to estimate the variance of a population using the variance of a sample. Then, watch the video titled “Standard Deviation,” which will provide a review for what you have learned so far as well as an introduction to the standard deviation. Finally, view “Alternate Variance Formulas” to learn alternates for the variance of a population.
Viewing these lectures and taking notes should take approximately 1 hour to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Assessment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Practice 1: Center of the Data” and “Practice 2: Spread of the Data”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Practice 1: Center of the Data” (PDF) and “Practice 2: Spread of the Data” (PDF)
Instructions: Solve all the problems in the two sections. Next, click on the hyperlink titled “Homework” (PDF) and solve problems 1, 3, 5, 7, 9, 12, 14, 15, 17, 21, and 24-30. To check your solutions, go to the end of the document.
This assessment should take approximately 3 hours to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics: “Section 9: Measures of the Spread of the Data”
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Unit 2: Probability Topics
After you have learned to describe and display data, how can you use the sample data to draw conclusions about the populations? To answer this question, you need probability, a subject we will explore over the course of this unit.
Unit 2 Time Advisory show close
If you open a newspaper, you are likely to read headlines like: “Should risky medical procedures be allowed?”, “30% chance of a hurricane this weekend”, “Analysts expect the gas prices to increase this summer”, and “Two brothers meet by accident after being separate for more than 20 years.” All of these topics deal with everyday life and all of them have to do with probability.
The world seems to be full of apparently unpredictable events. Probability theory is a tool that was created to deal with such events more effectively. For example, before getting a surgery, patients want to know the probability that the surgery might fail; before taking medication, we want to know the probability that there will be side effects; before leaving the house, we want to know the probability that it will rain today. Probability deals with the likelihood of an event occurring. It is a measure that takes on values between 0 and 1, inclusive, with 0 representing impossible events and 1 representing certainty. The ability to calculate probability allows us to make better decisions.
Probabilities affect our everyday lives. In this unit you will learn what probability and its properties are, how probability behaves, and how to calculate and use it. You will study the fundamentals of probability and will work through examples that cover different types of probability problems. These basic probability concepts will provide a foundation for understanding more statistical concepts. You probably already (intuitively) use concepts from probability, but this unit will help you formally and precisely predict the likelihood of an event occurring given certain constraints.
Whether you are evaluating how likely it is to get more than 50% of the questions correct on a quiz if you guess randomly; predicting the likelihood that the next storm will arrive by the end of the week; or exploring the relationship between the number of hours students spend at the gym and their performance on an exam, an understanding of the fundamentals of probability is crucial. Make sure you spend time on this unit. Your goal will be to become comfortable with the basic machinery of probability theory and its applications.
Unit 2 Learning Outcomes show close
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2.1 Probability Overview
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 3: Probability Topics: “Section 1: Probability Topics” and “Section 2: Terminology”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 3: Probability Topics: “Section 1: Probability Topics” (PDF) and “Section 2: Terminology” (PDF)
Instructions: Read the sections linked above. Section 1 introduces the concept of probability, and Section 2 defines key terms related to probability. Note that these readings cover sub-subunits 2.1.1 and 2.1.2.
These readings should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 3: Probability Topics: “Section 1: Probability Topics” and “Section 2: Terminology”
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2.1.1 Probability Topics
Note: This topic is covered by the reading assigned in subunit 2.1. In particular, focus on the Section 1 reading of Illowsky and Dean’s Collaborative Statistics.
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2.1.2 Probability Terminology
Note: This topic is covered by the reading assigned in subunit 2.1. In particular, focus on the Section 2 reading of Illowsky and Dean’s Collaborative Statistics.
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2.2 Independent and Mutually Exclusive Events
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 3: Probability Topics: “Section 3: Independent and Mutually Exclusive Events”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Section 3: Independent and Mutually Exclusive Events” (PDF)
Instructions: Read the section linked above. Section 3 explains the concept of independent events, where the probability of event A does not have any effect on the probability of event B, and mutually exclusive events, where events A and B cannot occur at the same time.
This reading should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 3: Probability Topics: “Section 3: Independent and Mutually Exclusive Events”
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2.3 Two Basic Rules of Probability
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 3: Probability Topics: “Section 4: Two Basic Rules of Probability”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Section 4: Two Basic Rules of Probability” (PDF)
Instructions: Read the section linked above. Section 4 introduces the multiplication and addition rules used when calculating probabilities. Note that this reading covers the topics outlined in sub-subunits 2.3.1 and 2.3.2.
This reading, including examples and problems, should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 3: Probability Topics: “Section 4: Two Basic Rules of Probability”
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2.3.1 The Multiplication Rule
Note: This topic is covered by the reading assigned in subunit 2.3. Focus on the explanation of the multiplication rule as well as the examples and sample problems provided.
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2.3.2 The Addition Rule
Note: This topic is covered by the reading assigned in subunit 2.3. Focus on the explanation of the multiplication rule as well as the examples and sample problems provided.
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2.4 Contingency Tables, Venn Diagrams, and Tree Diagrams
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 3: Probability Topics: “Section 5: Contingency Tables,” “Section 6: Venn Diagrams,” and “Section 7: Tree Diagrams”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 3: Probability Topics: “Section 5: Contingency Tables” (PDF), “Section 6: Venn Diagrams” (PDF), and “Section 7: Tree Diagrams” (PDF)
Instructions: Read the sections linked above. Section 5 introduces the contingency table as a way of determining conditional probabilities. Section 6 discusses Venn diagrams, and Section 7 explains the utility of tree diagrams as a method for making some probability problems easier to solve.
These readings should take approximately 2 hours to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 3: Probability Topics”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 3: Probability Topics” (YouTube)
Instructions: View the linked lecture “Video Lecture 3: Probability Topics,” in which you will learn about elementary probability theory.
This lecture should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Assessment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 3: Probability Topics: “Practice 1: Contingency Tables” and “Practice 2: Calculating Probabilities”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 3: Probability Topics: “Practice 1: Contingency Tables” (PDF) and “Practice 2: Calculating Probabilities” (PDF)
Instructions: View the links “Practice 1: Contingency Tables” and “Practice 2: Calculating Probabilities.” Solve all the problems in these two sections. Next, click on the link titled “Homework” and solve problems 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, and 28-32. To check your solutions, go to the end of the document.
This assessment should take approximately 4 hours to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 3: Probability Topics: “Section 5: Contingency Tables,” “Section 6: Venn Diagrams,” and “Section 7: Tree Diagrams”
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Unit 3: Random Variables and Distributions
In the last unit, you learned how to calculate probabilities in the framework of sample spaces, outcomes, and events. In this unit, you will build on those ideas and learn about random variables. A random variable describes the outcomes of a statistical experiment. A statistical distribution describes the numbers of times each possible outcome occurs in a sample. The values of a random variable can vary with each repetition of an experiment. Intuitively, a random variable is an observable that takes on values with certain probabilities.
Unit 3 Time Advisory show close
A random variable can be classified as being either discrete or continuous, depending on the values it assumes. Suppose you count the number of people who go to a coffee shop between 4:00 pm and 5:00 pm and the amount of money that they spend in that hour. In this case, the number of people is an example of a discrete random variable and the amount of money they spend is an example of a continuous random variable.
In this unit, you will study probability problems involving random distributions. You will also learn about both discrete and continuous random variables and their applications. Finally, you will study an important example of a continuous distribution, the normal distribution, which is a bell-shaped distribution used widely across many disciplines.
A note from the textbook: “The values of discrete and continuous random variables can be ambiguous. For example, if X is equal to the number of miles (to the nearest mile) you drive to work, then X is a discrete random variable. You count the miles. If X is the distance you drive to work, then you measure values of X, and X is a continuous random variable. How the random variable is defined is very important.”[1]
[1] Illowsky, Barbara and Susan Dean. Collaborative Statistics. Connexions. Accessed March 22, 2010, http://cnx.org/content/col10522/1.38/.
Unit 3 Learning Outcomes show close
- 3.1 Discrete Random Variables and Discrete Probability Distributions
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3.1.1 Probability Distribution Functions
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 4: Discrete Random Variables: “Section 1: Discrete Random Variables” and “Section 2: Probability Distribution Function for a Discrete Random Variable”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 4: Discrete Random Variables: “Section 1: Discrete Random Variables” (PDF) and “Section 2: Probability Distribution Function for a Discrete Random Variable” (PDF)
Instructions: Read through each section. Be sure to complete the activities or problems included in each reading. Section 1 provides an introduction to discrete random variables, and Section 2 examines the discrete probability distribution function and its characteristics.
These readings should take approximately 45 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Introduction to Random Variables” and “Probability Density Functions”
Link: Khan Academy’s Statistics: “Introduction to Random Variables” (YouTube) and “Probability Density Functions” (YouTube)
Instructions: View the linked lecture “Introduction to Random Variables.” It will provide an introduction to random variables and probability distribution functions. Then, view linked lecture “Probability Density Functions” to learn about probability density functions for continuous random variables.
Viewing these lectures and taking notes should take approximately 30 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 4: Discrete Random Variables: “Section 1: Discrete Random Variables” and “Section 2: Probability Distribution Function for a Discrete Random Variable”
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3.1.2 Expected Value and Standard Deviation
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 4: Discrete Random Variables: “Section 3: Mean or Expected Value and Standard Deviation”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Section 3: Mean or Expected Value and Standard Deviation” (PDF)
Instructions: Read the section linked above. Section 3 explores the Law of Large Numbers.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Expected Value: E(X)”
Link: Khan Academy’s Statistics: “Expected Value: E(X)” (YouTube)
Instructions: View the lecture linked above. In this lecture you will learn how to calculate expected value.
This lecture should take approximately 15 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 4: Discrete Random Variables: “Section 3: Mean or Expected Value and Standard Deviation”
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3.1.3 Common Discrete Probability Distributions
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 4: Discrete Random Variables
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 4: Discrete Random Variables: “Section 4: Common Discrete Probability Distribution Functions” (PDF), “Section 5: Binomial” (PDF), “Section 6: Geometric (optional)” (PDF), “Section 7: Hypergeometric (optional)” (PDF), and “Section 8: Poisson” (PDF)
Instructions: Read the sections 4 - 8. Section 5 introduces characteristics of a binomial experiment and the binomial probability distribution function. Section 6 describes the geometric experiment and the geometric probability distribution. Section 7 describes the properties of a hypergeometric experiment and hypergeometric probability distribution. Section 8 describes the characteristics of a Poisson experiment and the Poisson probability distribution.
These readings, including examples, should take approximately 3 hours to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Binomial Distribution 1,” “Binomial Distribution 2,” “Binomial Distribution 3,” and “Binomial Distribution 4”
Link: Khan Academy’s Statistics: “Binomial Distribution 1” (YouTube), “Binomial Distribution 2” (YouTube), “Binomial Distribution 3” (YouTube), and “Binomial Distribution 4” (YouTube)
Instructions: View lectures 1 - 4. In the first three lectures, you will learn about the binomial distribution. In the fourth lecture, you will learn to use excel to visualize the basketball binomial distribution presented in the third video.
Viewing and taking notes on these lectures should take approximately 1 hour to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Assessment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 4: Discrete Random Variables
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 4: Discrete Random Variables: “Practice 1: Discrete Distribution” (PDF), “Practice 2: Binomial Distribution” (PDF), “Practice 3: Poisson Distribution” (PDF), “Practice 4: Geometric Distribution” (PDF), and “Practice 5: Hypergeometric Distribution” (PDF)
Instructions: Click on the above links and solve all the problems. Next, click on the link titled “Homework” and solve problems 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 29, and 31. To check your solutions, go to the end of the document. Solve all of the problems before checking the solutions.
This assessment should take approximately 4 hours to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 4: Discrete Distributions”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 4: Discrete Distributions” (YouTube)
Instructions: View the Lecture 4 learn about discrete random variables.
This lecture should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Expected Value of Binomial Distribution”
Link: Khan Academy’s Statistics: “Expected Value of Binomial Distribution” (YouTube)
Instructions: View the lecture above. You will learn about the expected value of a binomial distributed random variable.
This lecture should take approximately 20 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Lecture: Lecture: Khan Academy’s Statistics: “Poisson Process 1” and “Poisson Process 2”
Link: Khan Academy’s Statistics: “Poisson Process 1” (YouTube) and “Poisson Process 2” (YouTube)
Instructions: View both lectures. In these lectures, you will learn about the Poisson processes and the Poisson distribution and its derivation.
This lecture should take approximately 30 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 4: Discrete Random Variables
- 3.2 Continuous Random Variables
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3.2.1 Continuous Probability Functions
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 5: Continuous Random Variables
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 5: Continuous Random Variables: “Section 1: Continuous Random Variables” (PDF) and “Section 2: Continuous Probability Functions” (PDF)
Instructions: Read the sections 1 and 2. These sections introduce the continuous probability function.
These readings should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 5: Continuous Random Variables
-
3.2.2 Uniform Distribution
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 5: Continuous Random Variables: “Section 3: The Uniform Distribution”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Section 3: The Uniform Distribution” (PDF)
Instructions: Read the section linked above. Section 3 describes the properties of the uniform distribution.
This reading, including examples and problems, should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 5: Continuous Random Variables: “Section 3: The Uniform Distribution”
-
3.2.3 Exponential Distribution
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 5: Continuous Random Variables: “Section 4: The Exponential Distribution”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 5: Continuous Random Variables: “Section 4: The Exponential Distribution” (PDF)
Instructions: Read the section linked above. Section 4 introduces the properties of the exponential distribution.
This reading should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 5: Continuous Random Variables”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 5: Continuous Random Variables” (YouTube)
Instructions: View Lecture 5 to learn about continuous random variables and to study two specific continuous random variables.
This lecture should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Assessment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 5: Continuous Random Variables
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 5: Continuous Random Variables: “Practice 1: Uniform Distribution” (PDF) and “Practice 2: Exponential Distribution” (PDF)
Instructions: Solve all the problems in the two sections linked above. Next, click on the link titled “Homework” and solve problems 3, 5, 7, 9, 11, 13, and 15 - 20. To check your solutions, go to the end of the document. Solve all of the problems before checking the solutions.
This assessment should take approximately 3 hours and 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 5: Continuous Random Variables: “Section 4: The Exponential Distribution”
- 3.3 Normal Distribution
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3.3.1 The Standard Normal Distribution
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution: “Section 1: The Normal Distribution” (PDF) and “Section 2: The Standard Normal Distribution” (PDF)
Instructions: Read the sections linked above. These sections will introduce the normal distribution.
These readings should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Law of Large Numbers”
Link: Khan Academy’s Statistics: “Law of Large Numbers” (YouTube)
Instructions: View the lecture provided above. You will learn about the law of large numbers.
Viewing this lecture and taking notes should take approximately 15 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Normal Distribution Excel Exercise,” “Introduction to Normal Distribution,” and “Normal Distribution Problems: Qualitative Sense of Normal Distributions”
Link: Khan Academy’s Statistics: “Normal Distribution Excel Exercise” (YouTube), “Introduction to Normal Distribution” (YouTube), and “Normal Distribution Problems: Qualitative Sense of Normal Distributions” (YouTube)
Instructions: View the lectures linked above. In the lecture “Normal Distribution Excel Exercise” (26 minutes), you will see a presentation on a spreadsheet, which will show that the normal distribution approximates the binomial distribution for a large number of trials. View the other two lectures on normal distribution.
Dedicate approximately 1 hour and 30 minutes for viewing and taking notes.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution
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3.3.2 Z-scores
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution: “Section 3: Z-scores”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution: “Section 3: Z-scores” (PDF)
Instructions: Read the section linked above. Section 3 introduces the Z-score, which tells you how many standard deviations the value X lays above or below the mean.
This reading should take less than 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Normal Distribution Problems: Z-score,” “Normal Distribution Problems: Empirical Rule,” and “Standard Normal Distribution and the Empirical Rule”
Link: Khan Academy’s Statistics: “Normal Distribution Problems: Z-score” (YouTube), “Normal Distribution Problems: Empirical Rule” (YouTube), and “Standard Normal Distribution and the Empirical Rule” (YouTube)
Instructions: View the lectures linked above. In these lecture, you will learn about Z-scores and how to use the empirical rule to estimate probabilities for normal distributions.
These lectures should take approximately 30 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution: “Section 3: Z-scores”
-
3.3.3 Areas to the Left and Right of X
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution: “Section 4: Areas to the Left and Right of x”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution: “Section 4: Areas to the Left and Right of x” (PDF)
Instructions: Read the section linked above. Section 4 shows you how to interpret probability as the area under a curve.
This reading should take less than 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution: “Section 4: Areas to the Left and Right of x”
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3.3.4 Calculations of Probabilities
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution: “Section 5: Calculations of Probabilities”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution: “Section 5: Calculations of Probabilities” (PDF)
Instructions: Read the section linked above. Section 5 provides examples in which you will use the normal distribution to calculate probabilities.
This reading, including examples and problems, should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 6: The Normal Distribution”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 6: The Normal Distribution” (YouTube)
Instructions: View Lecture 6 to learn about the normal distribution.
This lecture should take less than 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here http://cnx.org/content/m17567/latest/See a broken link? Please let us know!
- Assessment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution: “Practice: The Normal Distribution”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution: “Practice: The Normal Distribution” (PDF)
Instructions: Click on the above link “Practice: The Normal Distribution.” Solve all the problems in this section. Next, click on the link titled “Homework” and solve problems 1, 3, 5, 7, 9, 11, and 12 - 19. To check your solutions, go to the end of the document. solve all of the problems before checking the solutions.
This assessment should take approximately 2 hours to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 6: The Normal Distribution: “Section 5: Calculations of Probabilities”
-
Unit 4: Central Limit Theorem and Confidence Intervals
In this unit, you will learn how to use the central limit theorem and confidence intervals, the latter of which enable us to estimate unknown population parameters. The central limit theorem provides us a way to make inference from samples of non-normal populations. This theorem states that given any population (regardless of whether or not it is a normal distribution), as the sample size increases, the sampling distribution of the means approaches a normal distribution. It is a powerful theorem because it allows us to assume that given a large enough sample, the sampling distribution will be normally distributed. The central limit theorem is one of the most important ideas in statistics, so be sure to spend time on it.
Unit 4 Time Advisory show close
You will also learn about confidence intervals, which provide a way to estimate a population parameter. Instead of giving just a one-number estimate of a variable, a confidence interval gives a range of likely values for it. This is useful because sample results will vary from sample to sample, so a range of values is better than a one-number estimate. After completing this unit, you will know how to construct confidence intervals and calculate their margin of error. You will learn to how to come up with a range of values for a parameter and the level of confidence for the intervals.
For example, suppose you want to know the amount of soda that an average high school student in New York drinks per day. The average volume of soda for the entire population of New York high school students who drink soda is the parameter you are trying to estimate. Suppose you take a random sample and find out the average amount is 0.5 liters. Then, you also want to know how much you expect the average to vary from one sample to the next, with a certain level of confidence. The number that you use to represent this precision, i.e. to measure how close you expect your results to be to the truth, is called the margin of error.
Unit 4 Learning Outcomes show close
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4.1 The Central Limit Theorem
- Lecture: Khan Academy’s Statistics: “Sampling Distribution of the Sample Mean” and “Sampling Distribution of the Sample Mean 2”
Link: Khan Academy’s Statistics: “Sampling Distribution of the Sample Mean” (YouTube) and “Sampling Distribution of the Sample Mean 2” (YouTube)
Instructions: View both lectures, which discuss the sampling distribution of the sample mean.
Viewing and taking notes on these lectures should take approximately 30 minutes.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Sampling Distribution of the Sample Mean” and “Sampling Distribution of the Sample Mean 2”
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4.1.1 The Central Limit Theorem for Sample Means (Averages)
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 7: The Central Limit Theorem: “Section 1: The Central Limit Theorem” and “Section 2: The Central Limit Theorem for Sample Means (Averages)”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 7: The Central Limit Theorem: “Section 1: The Central Limit Theorem” (PDF) and “Section 2: The Central Limit Theorem for Sample Means (Averages)” (PDF)
Instructions: Read the sections linked above. These sections will provide an introduction to the Central Limit Theorem, one of the most important concepts in this course.
These readings should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Central Limit Theorem”
Link: Khan Academy’s Statistics: “Central Limit Theorem” (YouTube)
Instructions: View the lecture above. It will provide an introduction to the central limit theorem and the sampling distribution of the mean.
It should take less than 15 minutes to view and take notes on the lecture.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Sampling Distribution of the Sample Mean” and “Sampling Distribution of the Sample Mean 2”
Link: Khan Academy’s Statistics: “Sampling Distribution of the Sample Mean” (YouTube) and “Sampling Distribution of the Sample Mean 2” (YouTube)
Instructions: View both lectures, which discuss the sampling distribution of the sample mean.
Viewing and taking notes on these lectures should take approximately 30 minutes.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Standard Error of the Mean” and “Sampling Distribution Example Problem”
Link: Khan Academy’s Statistics: “Standard Error of the Mean” (YouTube) and “Sampling Distribution Example Problem” (YouTube)
Instructions: View both lectures. These lectures will discuss the standard error of the mean, i.e. the standard deviation of the sampling distribution of the sample mean, and work out an example.
These lectures should take approximately 30 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 7: The Central Limit Theorem: “Section 1: The Central Limit Theorem” and “Section 2: The Central Limit Theorem for Sample Means (Averages)”
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4.1.2 The Central Limit Theorem for Sums
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 7: The Central Limit Theorem: “Section 3: The Central Limit Theorem for Sums”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 7: The Central Limit Theorem: “Section 3: The Central Limit Theorem for Sums” (PDF)
Instructions: Read the section linked above. Section 3 will explain the central limit theorem for sums.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 7: The Central Limit Theorem: “Section 3: The Central Limit Theorem for Sums”
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4.1.3 Using the Central Limit Theorem
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 7: The Central Limit Theorem: “Section 4: Using the Central Limit Theorem”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 7: The Central Limit Theorem: “Section 4: Using the Central Limit Theorem” (PDF)
Instructions: Read the section linked above. Section 4 covers how and when to use the Central Limit Theorem.
This reading, including examples and problems, should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 7: The Central Limit Theorem”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 7: The Central Limit Theorem” (YouTube)
Instructions: View the lecture titled “Video Lecture 7: The Central Limit Theorem” to study the Central Limit Theorem.
This lecture should take less than 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Assessment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 7: The Central Limit Theorem: “Practice: The Central Limit Theorem”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 7: The Central Limit Theorem: “Practice: The Central Limit Theorem” (PDF)
Instructions: Solve all the problems in this section. Next, click on the link titled “Homework” and solve problems 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19 - 25. To check your solutions, go to the end of the document. Solve all of the problems before checking the solutions.
This assessment should take approximately two and a half hours to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 7: The Central Limit Theorem: “Section 4: Using the Central Limit Theorem”
- 4.2 Confidence Intervals
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4.2.1 Confidence Interval, Single Population Mean, Population Standard Deviation Known, Normal
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 8: Confidence Intervals: “Section 1: Confidence Intervals” and “Section 2: Confidence Interval, Single Population Mean, Population Standard Deviation Known, Normal”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 8: Confidence Intervals: “Section 1: Confidence Intervals” (PDF) and “Section 2: Confidence Interval, Single Population Mean, Population Standard Deviation Known, Normal” (PDF)
Instructions: Read sections 1 and 2. Section 1 introduces the idea of a confidence interval, which is an estimate for an unknown population parameter. Section 2 explains how to construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known.
These readings should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Confidence Interval 1”
Link: Khan Academy’s Statistics: “Confidence Interval 1” (YouTube)
Instructions: View the lecture linked above. You will learn to estimate the probability that the true population mean lies within a given range around a sample mean.
This lecture should take less than 20 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Margin of Error 1” and “Margin of Error 2”
Link: Khan Academy’s Statistics: “Margin of Error 1” (YouTube) and “Margin of Error 2” (YouTube)
Instructions: View both lectures linked above. You will learn to find the 95% confidence interval for a problem.
This lecture should take approximately 30 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Confidence Interval Example”
Link: Khan Academy’s Statistics: “Confidence Interval Example” (YouTube)
Instructions: View the lecture linked above. In this lecture, you will work out a confidence interval example.
This lecture should take approximately 20 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 8: Confidence Intervals: “Section 1: Confidence Intervals” and “Section 2: Confidence Interval, Single Population Mean, Population Standard Deviation Known, Normal”
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4.2.2 Confidence Interval, Single Population Mean, Standard Deviation Unknown, Student-T
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 8: Confidence Intervals: “Section 3: Confidence Interval, Single Population Mean, Standard Deviation Unknown, Student-T”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 8: Confidence Intervals: “Section 3: Confidence Interval, Single Population Mean, Standard Deviation Unknown, Student-T” (PDF)
Instructions: Read the section linked above. Section 3 explains how to construct a confidence interval for a single population mean μ, where the population standard deviation is unknown, using a Student-T distribution.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Small Sample Size Confidence Intervals” and “Small Sample Hypothesis Test”
Link: Khan Academy’s Statistics: “Small Sample Size Confidence Intervals” (YouTube) and “Small Sample Hypothesis Test” (YouTube)
Instructions: View both lectures linked above. These lectures will discuss constructing small size confidence intervals using t-distributions.
Viewing and taking notes on these lectures should take approximately 30 minutes.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 8: Confidence Intervals: “Section 3: Confidence Interval, Single Population Mean, Standard Deviation Unknown, Student-T”
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4.2.3 Confidence Interval for a Population Proportion
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 8: Confidence Intervals: “Section 4: Confidence Interval for a Population Proportion”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 8: Confidence Intervals: “Section 4: Confidence Interval for a Population Proportion” (PDF)
Instructions: Read the section 4. It explains how to construct a confidence interval for a population proportion.
This reading should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 8: Confidence Intervals”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 8: Confidence Intervals” (YouTube)
Instructions: View the lecture 8 to study confidence intervals, which are used to estimate a parameter.
This lecture should take approximately 20 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Assessment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 8: Confidence Intervals
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 8: Confidence Intervals: “Practice 1: Confidence Intervals for Averages, Known Population Standard Deviation” (PDF), “Practice 2: Confidence Intervals for Averages, Unknown Population Standard Deviation” (PDF), and “Practice 3: Confidence Intervals for Proportions” (PDF)
Instructions: Click on the link titled “Practice 1: Confidence Intervals for Averages, Known Population Standard Deviation” and solve problems 1 - 13. Next, click on the links titled “Practice 2: Confidence Intervals for Averages, Unknown Population Standard Deviation” to solve problems 1 - 11 and “Practice 3: Confidence Intervals for Proportions” to solve problems 1 - 13. Finally, click on the link titled “Homework” and solve problems 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, and 21. To check your solutions, go to the end of the document. Solve all of the problems before checking the solutions.
This assessment should take approximately 3 hours and 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 8: Confidence Intervals: “Section 4: Confidence Interval for a Population Proportion”
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Unit 5: Hypothesis Testing
One of the major goals in statistics is to use the information you collect from a sample to get a better idea of the entire population in which you are interested. In this unit, you will learn about hypothesis testing, which will enable you to achieve that goal.
Unit 5 Time Advisory show close
A hypothesis test involves collecting and evaluating data from a sample to make a decision as to whether or not that data supports a claim made about a population. This unit will also teach you how to conduct hypothesis tests and to identify and differentiate between the errors associated with them.
Many times, you need answers to questions in order to make efficient decisions. For example, a restaurant owner might claim that his restaurant’s food costs 30% less than other restaurants in the area, or a phone company might claim that its phones last at least one year more than phones from other companies. The process of hypothesis testing is a way making decisions about claims like these. In this unit, you will learn to establish your assumptions through null and alternative hypotheses. Then, you will learn to compare sample characteristics to assumptions to see whether there is enough data to accept or reject the null hypothesis. The null hypothesis is, at first, assumed to be true and the one you hope to nullify, while the alternative hypothesis is a research hypothesis that you claim to be true. This means that you need to conduct the correct tests to be able to accept or reject the null hypothesis. The unit concludes with an introduction to Chi-distributions and their applications.
Unit 5 Learning Outcomes show close
- 5.1 Hypothesis Testing: Single Mean and Single Proportion
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5.1.1 Null and Alternate Hypotheses
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 9: Hypothesis Testing: Single Mean and Single Proportion
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 9: Hypothesis Testing: Single Mean and Single Proportion: “Section 1: Hypothesis Testing: Single Mean and Single Proportion” (PDF) and “Section 2: Null and Alternate Hypotheses” (PDF)
Instructions: Read the sections 1 and 2. Section 1 briefly introduces the process of hypothesis testing. Section 2 discusses the null hypothesis and the alternate hypothesis.
These readings should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 9: Hypothesis Testing: Single Mean and Single Proportion
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5.1.2 Type I and Type II Errors
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 9: Hypothesis Testing: Single Mean and Single Proportion: “Section 3: Outcomes and the Type I and Type II Errors”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 9: Hypothesis Testing: Single Mean and Single Proportion: “Section 3: Outcomes and the Type I and Type II Errors” (PDF)
Instructions: Read the section linked above. Section 3 explains the types of errors that can occur while performing a hypothesis test.
This reading should take less than 30 minutes to complete.
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- Lecture: Khan Academy’s Statistics: “Type I Errors”
Link: Khan Academy’s Statistics: “Type I Errors” (YouTube)
Instructions: Watch the brief video lecture, which discusses Type I Errors.
Viewing this lecture and taking notes should take you less than 15 minutes.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 9: Hypothesis Testing: Single Mean and Single Proportion: “Section 3: Outcomes and the Type I and Type II Errors”
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5.1.3 Distribution for Hypothesis Testing and More
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 9: Hypothesis Testing: Single Mean and Single Proportion
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 9: Hypothesis Testing: Single Mean and Single Proportion: “Section 4: Distribution Needed for Hypothesis Testing” (PDF) “Section 5: Assumption” (PDF), “Section 6: Rare Events” (PDF), “Section 7: Using the Sample to Support One of the Hypotheses” (PDF), “Section 8: Decision and Conclusion” (PDF), and “Section 9: Additional Information” (PDF)
Instructions: Read sections 4 - 9. The first three sections discuss assumptions about particular distributions associated with hypothesis testing. Section 7 explains how to use the sample to calculate the actual probability of getting the test result, called the p-value. Section 8 discusses how to make a decision about whether to reject or not reject the null hypothesis.
These readings should take approximately 1 hour and 30 minutes to complete.
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- Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 9: Hypothesis Testing with a Single Mean”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 9: Hypothesis Testing with a Single Mean” (YouTube)
Instructions: View the lecture 9 to learn how to perform hypothesis tests for a single mean.
This lecture should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Assessment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 9: Hypothesis Testing: Single Mean and Single Proportion
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 9: Hypothesis Testing: Single Mean and Single Proportion: “Practice 1: Single Mean, Known Population Standard Deviation” (PDF), “Practice 2: Single Mean, Unknown Population Standard Deviation” (PDF), and “Practice 3: Single Proportion” (PDF)
Instructions: Click on the above links and solve all the problems in each practice lesson. Next, click on the link titled “Homework” and solve problems 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, and 27. To check your solutions, go to the end of the document. Complete all assigned exercises before checking the solutions.
This assessment should take approximately 3 hours to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Large Sample Proportion Hypothesis Testing”
Link: Khan Academy’s Statistics: “Large Sample Proportion Hypothesis Testing” (YouTube)
Instructions: View video lecture, which discusses large sample proportion hypothesis testing.
This lecture should take less than 20 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Difference of Sample Means Distribution”
Link: Khan Academy’s Statistics: “Difference of Sample Means Distribution” (YouTube)
Instructions: View the lecture above, which discusses the difference of sample means distribution.
This lecture should take less than 20 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Khan Academy’s Statistics: “Confidence Interval of Difference of Means”
Link: Khan Academy’s Statistics: “Confidence Interval of Difference of Means” (YouTube)
Instructions: Watch the above lecture, which discusses confidence interval of difference of means.
This lecture should take less than 20 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 9: Hypothesis Testing: Single Mean and Single Proportion
- 5.2 Hypothesis Testing: Two Means, Paired Data, Two Proportions
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5.2.1 Comparing Two Independent Population Means with Unknown Population Standard Deviations
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions: “Section 1: Hypothesis Testing: Two Population Means and Two Population Proportions” (PDF) and “Section 2: Comparing Two Independent Population Means with Unknown Population Standard Deviations” (PDF)
Instructions: Read sections 1 and 2. They discuss how to compare two independent population means with unknown population standard deviations.
These readings, including examples and problems, should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions
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5.2.2 Comparing Two Independent Population Means with Known Population Standard Deviations
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions: “Section 3: Comparing Two Independent Population Means with Known Population Standard Deviations” (PDF)
Instructions: Read the section linked above. Section 3 provides an overview of hypothesis testing in situations where there are two independent population means and known population standard deviations in statistics.
This reading should take you approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions
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5.2.3 Comparing Two Independent Population Proportions
- Lecture: Khan Academy’s Statistics: “Comparing Population Proportions 1,” “Comparing Population Proportions 2,” and “Hypothesis Test Comparing Population Proportions”
Link: Khan Academy’s Statistics: “Comparing Population Proportions 1” (YouTube), “Comparing Population Proportions 2” (YouTube), and “Hypothesis Test Comparing Population Proportions” (YouTube)
Instructions: View these three video lectures. These lectures will discuss comparing population proportions and hypothesis test comparing population proportions.
These lectures should take approximately 30 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions: “Section 4: Comparing Two Independent Population Proportions” (PDF)
Instructions: Read the section linked above. Section 4 discusses how to compare two population parameters.
This reading should take you approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Comparing Population Proportions 1,” “Comparing Population Proportions 2,” and “Hypothesis Test Comparing Population Proportions”
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5.2.4 Matched or Paired Samples
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions: “Section 5: Matched or Paired Samples”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions: “Section 5: Matched or Paired Samples” (PDF)
Instructions: read the section linked above in its entirety. Section 5 provides an overview of hypothesis testing for matched or paired samples.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 10: Hypothesis Testing with Two Means
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 10: Hypothesis Testing with Two Means” (YouTube)
Instructions: View lecture 10 to continue studying hypothesis testing, with a focus on testing two means and two proportions.
This lecture should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Hypothesis Test for Difference of Means”
Link: Khan Academy’s Statistics: “Hypothesis Test for Difference of Means” (YouTube)
Instructions: View the above lecture. It discusses hypothesis tests for difference of means.
This lecture should take approximately 10 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Assessment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions: “Practice 1: Hypothesis Testing for Two Proportions” (PDF) and “Practice 2: Hypothesis Testing for Two Averages” (PDF)
Instructions: Click on the above links and solve all the problems in the practice lessons. Next, click on the link titled “Homework” and solve problems 1, 5, 7, 9, 11, 15, 17, 21, 23, 27, 29, 33, 35, and 43 - 48. To check your solutions, go to the end of the document. attempt all of the assigned problems before checking your answers against the solutions.
This assessment should take approximately 3 hours to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 10: Hypothesis Testing: Two Means, Paired Data, Two Proportions: “Section 5: Matched or Paired Samples”
- 5.3 Chi-Square Distribution
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5.3.1 Notation and Facts
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution: “Section 1: The Chi-Square Distribution” (PDF), “Section 2: Notation” (PDF), and “Section 3: Facts About the Chi-Square Distribution” (PDF)
Instructions: Read sections 1 - 3. These sections provide an introduction to the Chi-square distribution.
These readings should take approximately 45 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Chi-Square Distribution Introduction,” “Pearson's Chi Square Test (Goodness of Fit),” and “Contingency Table Chi-Square Test”
Link: Khan Academy’s Statistics: “Chi-Square Distribution Introduction” (YouTube), “Pearson's Chi Square Test (Goodness of Fit)” (YouTube), and “Contingency Table Chi-Square Test” (YouTube)
Instructions: View all three video lectures. These lectures will provide an introduction to the Chi-square distribution as well as the Chi-square test.
These lectures should take approximately 40 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution
-
5.3.2 Goodness-of-Fit Test
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution: “Section 4: Goodness-of-Fit Test”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution: “Section 4: Goodness-of-Fit Test” (PDF)
Instructions: Read the section linked above. Section 4 describes how the Chi-square distribution is used to conduct the goodness-of-fit test.
This reading should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution: “Section 4: Goodness-of-Fit Test”
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5.3.3 Test of Independence
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution: “Section 5: Test of Independence”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution: “Section 5: Test of Independence” (PDF)
Instructions: Read section 5, which describes how the Chi-square distribution can be used to test for independence.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution: “Section 5: Test of Independence”
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5.3.4 Test of a Single Variance (Optional)
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution: “Section 6: Test of a Single Variance (Optional)”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution: “Section 6: Test of a Single Variance (Optional)” (PDF)
Instructions: Read section 6, which provides an overview on Chi-square distribution tests of variance.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 11: The Chi-Square Distribution”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 11: The Chi-Square Distribution” (YouTube)
Instructions: View lecture 11 to study the Chi-square distribution and its properties.
This lecture should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Assessment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution: “Practice 1: Goodness-of-Fit Test” (PDF), “Practice 2: Contingency Tables” (PDF), and “Practice 3: Test of a Single Variance” (PDF)
Instructions: Click on the above links to solve all the problems in each practice lesson. Next, click on the link titled “Homework” and solve the odd numbered problems from 3 - 37. To check your solutions, go to the end of the document. Solve all of the problems before checking the solutions.
This assessment should take approximately 3 hours to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 11: The Chi-Square Distribution: “Section 6: Test of a Single Variance (Optional)”
-
Unit 6: Correlation, Regression, and ANOVA
One of the main reasons you will conduct analysis is in order to understand how two variables are related to one another. The most common type of relationship is a linear relationship. For example, you may want to know what happens to one variable when you increase or decrease the other variable. You want to answer questions such as, “Does one increase as the other increases, or does it decrease?” For example, how does drinking soda relate to weight gain for teenagers? Does drinking more soda really relate to more weight gain? In this unit, you will learn to measure the degree of a relationship between two or more variables. Both correlation and regression are measures for comparing variables. However, they are quite different from one another. Correlation quantifies the strength of a relationship between two variables and is a measure of existing data. Regression, on the other hand, is the study of the strength of a linear relationship between an independent and dependent variable, and can be used to predict the value of the dependent variable when the value of the independent variable is unknown. A note of caution: Be careful to not automatically interpret correlation and regression as establishing cause-and-effect relationships!
Unit 6 Time Advisory show close
Also, you will learn about a method called Analysis of Variance (abbreviated ANOVA), which is used for hypothesis tests involving more than two averages. ANOVA is about examining the amount of variability in the Y variable and trying to see where that variability is coming from. You will study the simplest form of ANOVA, called single factor or one-way ANOVA. Finally, you will briefly study the F-distribution, used for ANOVA, and the test of two variances.
Unit 6 Learning Outcomes show close
- 6.1 Linear Regression
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6.1.1 Linear Equations, Slope, and Y-intercept
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 1: Linear Regression and Correlation”(PDF), “Section 2: Linear Equations” (PDF), and “Section 3: Slope and Y-Intercept of a Linear Equation” (PDF)
Instructions: Read sections 1 - 3. Section 1 introduces the idea of bivariate and multivariate data. Section 2 discusses linear equations, and Section 3 discusses the slope and Y-intercept of a linear equation.
These readings should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation
-
6.1.2 Scatter Plots
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 4: Scatter Plots”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 4: Scatter Plots” (PDF)
Instructions: Read section 4. It introduces scatter plots, which are a way to display the relation between two variables, X and Y.
This reading should take less than 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 4: Scatter Plots”
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6.1.3 The Regression Equation
- Lecture: Khan Academy’s Statistics: “Covariance and the Regression Line”
Link: Khan Academy’s Statistics: “Covariance and the Regression Line” (YouTube)
Instructions: View the video lecture linked above. This lecture discusses covariance, variance, and the slope of the regression.
This lecture should take approximately 20 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 5: The Regression Equation”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 5: The Regression Equation” (PDF)
Instructions: Read section 5, which explains how to find and graph the regression equation.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Covariance and the Regression Line”
- 6.2 Correlation
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6.2.1 The Correlation Coefficient
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 6: The Correlation Coefficient”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 6: The Correlation Coefficient” (PDF)
Instructions: Read section 6. It introduces the correlation coefficient, which is a numerical measure of the strength of association between the independent variable, X, and the dependent variable, Y.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “Correlation and Causality”
Link: Khan Academy’s Statistics: “Correlation and Causality” (YouTube)
Instructions: View the video lecture, which discusses correlation and causality.
Viewing this lecture and taking notes should take approximately 15 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 6: The Correlation Coefficient”
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6.2.2 Prediction
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 8: Prediction”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 8: Prediction” (PDF)
Instructions: Read section 8, which explains how to predict using the regression equation.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 8: Prediction”
-
6.2.3 Outliers
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 9: Outliers”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 9: Outliers” (PDF)
Instructions: Read section 9, which discusses how to identify outliers.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 12: Linear Regression and Correlation”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 12: Linear Regression and Correlation” (YouTube)
Instructions: View lecture 12 to study linear regression and bivariate data.
This lecture should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Assessment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation “Practice: Linear Regression”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Practice: Linear Regression” (PDF)
Instructions: Click on the above link and solve the problems. Next, click on the link titled “Homework” and solve problems 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, and 21 - 25. To check your solutions, go to the end of the document. Solve all of the problems before checking the solutions.
This assessment should take approximately 2 hours and 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 12: Linear Regression and Correlation: “Section 9: Outliers”
- 6.3 F-Distribution and ANOVA
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6.3.1 ANOVA
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 13: F Distribution and ANOVA: “Section 1: F Distribution and ANOVA” and “Section 2: ANOVA”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 13: F Distribution and ANOVA: “Section 1: F Distribution and ANOVA” (PDF) and “Section 2: ANOVA” (PDF)
Instructions: Read sections 1 and 2. These sections describe the assumptions needed for implementing an ANOVA and how to set up the hypothesis test for the ANOVA.
These reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 13: F Distribution and ANOVA: “Section 1: F Distribution and ANOVA” and “Section 2: ANOVA”
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6.3.2 The F Distribution and the F Ratio
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 13: F Distribution and ANOVA
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 13: F Distribution and ANOVA: “Section 3: The F Distribution and the F Ratio” (PDF) and “Section 4: Facts about the F Distribution” (PDF)
Instructions: Read sections 3 and 4. Section 3 describes how to calculate the F-ratio and F-distribution based on the hypothesis test for the ANOVA, and Section 4 states the factors associated with F-distributions.
These readings should take approximately 1 hour to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Lecture: Khan Academy’s Statistics: “ANOVA 1: Calculating SST,” “ANOVA 2: Calculating SSW and SSB,” and “ANOVA 3: Hypothesis Test with F-Statistic”
Link: Khan Academy’s Statistics: “ANOVA 1: Calculating SST (Total Sum of Squares)” (YouTube), “ANOVA 2: Calculating SSW and SSB (Total Sum of Squares Within and Between)” (YouTube), and “ANOVA 3: Hypothesis Test with F-Statistic” (YouTube)
Instructions: View the three lectures, which discuss ANOVA.
These lectures should take approximately 30 minutes to complete.
Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike United States License 3.0. It is attributed to the Khan Academy.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 13: F Distribution and ANOVA
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6.3.3 Test of Two Variances
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 13: F Distribution and ANOVA: “Section 5: Test of Two Variances”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 13: F Distribution and ANOVA: “Section 5: Test of Two Variances” (PDF)
Instructions: Read section 5, which provides the assumptions to be considered in order to calculate a test of two variances and how to execute the test of two variances.
This reading should take approximately 30 minutes to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Assessment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 13: F Distribution and ANOVA: “Practice: ANOVA”
Link: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 13: F Distribution and ANOVA: “Practice: ANOVA” (PDF)
Instructions: Click on the above link and solve the problems. Next, click on the link titled “Homework” and solve problems 1, 3, 5, and 7. To check your solutions, go to the end of the document. Solve all of the problems before checking the solutions.
This assessment should take approximately two hours to complete.
Terms of Use: This work is licensed under a Creative Commons Attribution 2.0 Generic License. It is attributed to Barbara Illowsky and Susan Dean, and the original version can be found here.See a broken link? Please let us know!
- Reading: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 13: F Distribution and ANOVA: “Section 5: Test of Two Variances”
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Final Exam
- Final Exam: The Saylor Foundation’s MA121 Final Exam
Link: The Saylor Foundation’s MA121 Final Exam
Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.See a broken link? Please let us know!
- Final Exam: The Saylor Foundation’s MA121 Final Exam
