Lesson 3.2: Confidence Intervals
Lesson Learning Objectives
- Construct and interpret a confidence interval for a proportion.
- Check the conditions required for the validity of a confidence interval for a proportion.
- Distinguish between the standard error and the margin of error of a sample proportion.
Lesson 3.2 Checklist
| Learning activity | Graded? | Estimated time |
|---|---|---|
| Read OpenIntro Statistics section 5.2 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 5.2: Confidence Intervals for a Proportion in OpenIntro Statistics (Diez et al., 2019) CC BY-SA 3.0. In the last lesson, we considered the problem of estimating a population proportion with a point estimate, the sample proportion. Since the sample proportion has some uncertainty as a result of sampling variability, we can provide a plausible range of values for the population proportion to quantify this uncertainty. This plausible range of values is called a confidence interval. As you read, look-up new terminology in the Glossary and self-assess your understanding by attempting the guided practice exercises.
Watch the video, Intro to Confidence Intervals via Proportions (Diez, 2019), on this topic (duration 00:09:39).
Confidence Interval for a Proportion
Read Supplementary Notes 3.2, which begins by developing a confidence interval for a proportion intuitively using the 68–95–99.7 rule (empirical rule) from Lesson 2.3. It then fills-in details of constructing and interpreting a confidence interval for a proportion by working through a couple of examples.
Lesson Check-in Questions ✍
Virtual Statistical Software Lab 💻
Work through the virtual statistical software lab: Software Lab 3.2: Confidence Intervals. In this lab you will investigate the construction of a confidence interval for a proportion based on a random sample of data from a known population (so you can learn about the properties of the confidence interval). 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 3.2 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 5.7, 5.9, 5.11, and 5.13, and Chapter Exercises 5.27 and 5.33 (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 🌎
- To better understand how confidence intervals for the proportion work and how to interpret them, check-out the Proportion Confidence Interval Simulator [Application] (CPM Educational Program, 2023). You can use it to mimic the simulations used to create Figure 5.6 in Section 5.2.2 of the textbook. Set “sample size” to 1000, “population proportion of successes” to 0.88, “confidence level” to 95, and use “normal model with standard error (from p-hat)” as the “method for calculating.” Also set “width of the x-axis” to 0.12 (this setting just affects the visual appearance of the figure).
- To practice proportion confidence interval calculations, use this Confidence Interval For Proportions Calculator [Application] (Green, 2021). For example, consider the examples in Supplementary Notes 3.2:
- For the aquarium expansion example, set n = 600, x = 510 (since 85% of 600 is 510), and “CL” to 0.95.
- For the Canada’s troops in Afghanistan examples, set n = 550, x = 291.5 (since 53% of 550 is 291.5), and “CL” to 0.90 or 0.95.
Media Attributions
References
CPM Educational Program. (2023). Proportion confidence interval simulator [Application]. https://stats.cpm.org/propCIs/
Diez, D. (2019, Sep. 5). Intro to confidence intervals via proportions [Video]. YouTube. https://www.youtube.com/watch?v=A6_W8qY8zJo
Diez, D. M., Çetinkaya-Rundel, M., Barr, C. D. (2019). OpenIntro Statistics (4th ed.). OpenIntro. https://www.openintro.org/book/os/
Green, L. (2021, May 21). 25: Confidence interval for proportions calculator [Application]. LibreTexts Statistics. https://stats.libretexts.org/Learning_Objects/02%3A_Interactive_Statistics/25%3A_Confidence_Interval_For_Proportions_Calculator
Undermedia. (2019). Opinion polling during the campaign period of 2019 Canadian federal election [Chart]. Wikimedia. https://commons.wikimedia.org/wiki/File:Opinion_polling_during_the_campaign_period_of_2019_Canadian_federal_election.svg
