Lesson 3.1: Sampling Variability
Lesson Learning Objectives
- Understand the distinction between sampling variability and bias.
- Understand the concept of a sampling distribution for a sample proportion.
- Know and apply the Central Limit Theorem for a sample proportion.
- Check the conditions needed for the normal model.
- Calculate probabilities using the normal model for the sampling distribution of a sample proportion.
Lesson 3.1 Checklist
| Learning activity | Graded? | Estimated time |
|---|---|---|
| Read OpenIntro Statistics section 5.1 and supplementary notes | No | 30 mins |
| Watch instructional video | No | 20 mins |
| Answer two lesson check-in questions | Yes | 15 mins |
| Work through virtual statistical software lab | No | 45 mins |
| Answer two virtual statistical software lab questions | Yes | 15 mins |
| Work on practice exercises | No | 1.5 hours |
| Explore suggested websites | No | 15 mins |
Learning Activities
Readings 📖 and Instructional Video 🎦
Sampling Variability
Read Section 5.1: Point Estimates and Sampling Variability in OpenIntro Statistics (Diez et al., 2019) CC BY-SA 3.0.
For the remainder of this course, we’ll use statistical inference to make decisions about a population based on a sample of data. Statistical inference works by quantifying the uncertainty we have about the true value of a population parameter. This uncertainty results from the variability associated with sampling from the population. There is some random variation inherent to this process that we need to account for. This section provides a foundation for this idea by considering the problem of estimating a population proportion. As you read, look-up new terminology in the Glossary and self-assess your understanding by attempting the guided practice exercises.
Watch the following video, Foundations for Inference: Point Estimates (Diez, 2019) on this topic (duration 00:18:57).
Normal Model for the Sampling Distribution of a Proportion
Read Supplementary Notes 3.1, which works through three examples of the normal model for the sampling distribution of a proportion in detail. Note that what the supplementary notes refers to as “technical conditions for the normal model for the sampling distribution of a proportion,” the textbook refers to as “technical conditions for the Central Limit Theorem.”
Lesson Check-in Questions ✍
Virtual Statistical Software Lab 💻
Work through the virtual statistical software lab: Software Lab 3.1: Sampling Distributions. This lab investigates how to use a statistic from a random sample of data as a point estimate for a population parameter. We’re interested in formulating a sampling distribution of our estimate in order to learn about its properties. As you work through the lab, answer the exercises in the shaded boxes. These exercises are not graded but the solutions are available: Software Lab 3.1 Solutions. The lab should take you no more than 45 minutes to complete.
Virtual Statistical Software Lab Questions ✍
Practice Exercises 🖊
Work on the following exercises in OpenIntro Statistics: Exercises 5.1, 5.3, and 5.5 (Diez et al., 2019) CC BY-SA 3.0. Check your answers using these solutions (Diez et al., 2019) CC BY-SA 3.0. You’ll deepen your understanding much more effectively if you genuinely attempt the questions by yourself before checking the solutions.
Work on the WeBWorK exercises, which are linked from your Moodle course. Check your answers using the solutions provided.
Suggested Websites 🌎
- If you want to play around with simulations of the sampling distribution of a proportion, check-out the interactive Proportion Sampling Distribution Simulator [Application] (CPM Educational Program, 2023). You can use it to mimic the simulations in Section 5.1.5 of the textbook.
- Another way to practice probability calculations based on the sampling distribution of a proportion is to use the StatPowers [Application] (Powers, n.d.) online calculator. For example, consider the examples in Supplementary Notes 3.1:
- For Example 1 (about the quality of the environmental), set “Sample Successes” to 175, “Sample Size” to 350, and “Probability” between 0.448 and 0.552.
- For Example 2 (about die rolls), set “Sample Successes” to 50, “Sample Size” to 300, and “Probability” above 0.2.
- For Example 3 (about airline overselling), set “Sample Successes” to 22.5, “Sample Size” to 225, and “Probability” below 0.0622.
Media Attributions
The Dachshund Diversity, by Tony Alter (2010), on Flickr, CC BY 2.0
References
Alter, T. [Tobyotter]. (2010). The Dachshund Diversity [Photograph]. Flickr. https://flic.kr/p/7Ja1be
CPM Educational Program. (2023). Proportion sampling distribution simulator [Application]. https://stats.cpm.org/propsamples/
Diez, D. (2019, Sep. 2). Foundations for inference: point estimates [Video]. YouTube. https://www.youtube.com/watch?v=oLW_uzkPZGA
Diez, D. M., Çetinkaya-Rundel, M., Barr, C. D. (2019). OpenIntro Statistics (4th ed.). OpenIntro. https://www.openintro.org/book/os/
Powers, B. (n.d.). StatPowers [Application]. https://www.statpowers.com/samplingDistributionProportion.html
