Module 7 Lesson: Sampling Variation and Confidence Intervals
Classroom Explanation
Every estimate from a sample would have come out differently with a different sample. A confidence interval is the range of values consistent with the data, and its width is the honest measure of how much you know.
In this module, the goal is to interpret estimates with uncertainty ranges instead of false precision. The important part is not only the calculation. The important part is the thinking before and after the calculation.
Why This Matters
The common mistake is saying there is a 95% chance this exact interval contains the true value. The beginner-safe wording is that the method gives a reasonable uncertainty range. This mistake can make a report sound more confident than the evidence deserves.
Key Ideas
1. A statistic describes the sample.
A statistic is calculated from the sample in front of us. It is useful, but it is still only an estimate of something broader.
2. A parameter is the unknown value in the wider population.
A parameter is the wider value we wish we knew. We rarely see it directly, so we use sample evidence carefully.
3. A confidence interval gives a reasonable range for the unknown value under the method used.
A sample estimate is not the final truth. The interval reminds us that another reasonable sample could have produced a nearby but different result.
Worked Example
A course completion estimate of 68% with an interval from 61% to 75% should be written as an uncertain estimate, not as exactly 68%.
Here is the dataset used in this module.
| estimate_id | metric | estimate | lower_ci | upper_ci | sample_size |
|---|---|---|---|---|---|
| CI01 | average study hours | 5.2 | 4.6 | 5.8 | 120 |
| CI02 | course completion rate | 0.68 | 0.61 | 0.75 | 180 |
| CI03 | average delivery time | 33.5 | 30.2 | 36.8 | 75 |
| CI04 | support ticket rate | 0.052 | 0.044 | 0.060 | 900 |
| CI05 | proportion preferring practice tasks | 0.57 | 0.49 | 0.65 | 150 |
Read the width before the centre. An estimate of forty-two per cent with an interval from thirty-five to forty-nine is a different message from the same estimate with an interval two points wide, and only the width tells you whether a target inside the range has been met.
How To Think Through It
- Report the estimate.
- Report the interval around it.
- Read the width and relate it to the sample size.
- Compare the interval with any threshold that matters.
- Word the finding so the uncertainty travels with it.
Common Mistake
The common mistake is saying there is a 95% chance this exact interval contains the true value. The beginner-safe wording is that the method gives a reasonable uncertainty range.
How To Write The Result
The sample completion rate is 68%, and the uncertainty range suggests the wider completion rate may plausibly be around 61% to 75%.
Practice Prompt
Interpret three intervals and write one common misinterpretation to avoid.
Takeaway
The interval is the result; the point estimate is only its midpoint.