Interpret a confidence level in context.
Describe how the sample size and confidence level affect the margin of error.
Explain how practical issues like nonresponse, undercoverage, and response bias can affect the interpretation of a confidence interval.
Activity: What does "95% confident" mean?
In this activity, students will be using the Confidence Intervals applet to better understand the interpretation of a confidence level and also to discover how changing the confidence level and sample size will affect the confidence interval.
Inform students that they do not have to memorize the interpretation of the confidence level. Instead, they need to be able to accurately describe what the above image is showing. Here are two possibilities for 95%:
If we took many, many samples and constructed a confidence interval for each sample, we would expect 95% of the intervals to capture the true (parameter in context).
In 95% of all possible samples, the interval computed from the sample data will capture the true (parameter in context).
Interpretations: Confidence Interval vs. Confidence Level
Really, we should interpret the confidence level first, because we can do this without taking an actual sample.
In _____% of all possible samples, the interval computed from the sample data will capture the true (parameter in context).
To interpret a confidence interval, we must take an actual sample and use that sample to compute an interval. This means we have numbers to plug into the blank spots in the following interpretation:
We are _____% confident that the interval from _____ to _____ captures the true (parameter with context).
For the AP Exam, it is much more important that students can interpret a confidence interval. Soon, we will use the 4-step process to construct and interpret a confidence interval and this interpretation will be our conclusion. The interpretation of the confidence level more often shows up on the Multiple Choice section of the AP Exam.
95% confidence or 95% probability or 95% chance
Take a look at a few AP Exam rubrics, and you will quickly realize that they don’t want students saying “95% probability” or “95% chance” when interpreting a confidence interval. For many years I didn’t really understand why (even though I still hammered this into my students’ heads). And then I was enlightened by an amazing analogy.
Stand in front of students and show them a fair coin. Tell them that in a moment you are going to toss this coin and let it land flat in the palm of your hand. Ask students if it is acceptable to say:
“There is a 50% probability that the coin will show heads”
Most will agree that this is acceptable (it is!). Now physically toss the coin, let it land flat in the palm of your hand, and then cover it up. Ask students if it is acceptable to say:
“There is a 50% probability that this coin toss is showing heads”
Some will say that this is OK, but (hopefully) someone will bring up a potential issue. The coin has already been tossed, and the outcome is already determined. The coin is either heads (in which case the probability is 100%) or tails (in which case the probability is 0%). So it is not acceptable to say there is a 50% probability that this coin toss is showing heads.
Similarly, we can’t use the word probability when interpreting a confidence interval, because the outcome is already determined. We have already constructed a confidence interval and the true parameter is either captured (in which case the probability is 100%) or is not captured (in which case the probability is 0%). We use the word confidence to avoid any suggestion of the idea of probability.