A student was asked to find a 95% confidence interval for the population proportion of students who take notes using data from a random sample of size n=87. Which of the following is a correct interpretation of the interval 0.15, 0.25? Check all that are correct.
With 95% confidence, a randomly selected student takes notes in a proportion of their classes that is between 0.15 and 0.25.
With 95% confidence, the proportion of all students who take notes is between 0.15 and 0.25.
There is a 95% chance that the proportion of the population is between 0.15 and 0.25.
There is a 95% chance that the proportion of notetakers in a sample of 87 students will be between 0.15 and 0.25.
The proportion of all students who take notes is between 0.15 and 0.25, 95% of the time.
The Correct Answer and Explanation is:
✅ Correct Interpretation:
- With 95% confidence, the proportion of all students who take notes is between 0.15 and 0.25.
❌ Incorrect Interpretations:
- With 95% confidence, a randomly selected student takes notes in a proportion of their classes that is between 0.15 and 0.25.
Incorrect: This interpretation confuses the population proportion with an individual’s behavior. Confidence intervals refer to population parameters, not individuals. - There is a 95% chance that the proportion of the population is between 0.15 and 0.25.
Incorrect: The population proportion is a fixed (but unknown) value. The confidence interval either contains it or it doesn’t—there’s no “chance” involved after the interval is calculated. - There is a 95% chance that the proportion of notetakers in a sample of 87 students will be between 0.15 and 0.25.
Incorrect: This misstates what the confidence interval means. It is not about the proportion in future samples but about estimating the population parameter. - The proportion of all students who take notes is between 0.15 and 0.25, 95% of the time.
Incorrect: This implies a probability applied to the fixed population parameter, which is a misunderstanding of the confidence interval concept.
Explanation
A confidence interval is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter. In this case, the parameter is the population proportion of students who take notes.
The interval given—(0.15, 0.25)—is a 95% confidence interval. This means that if we were to take many random samples of size 87, and for each sample calculate a confidence interval in the same way, then 95% of those intervals would contain the true population proportion.
The correct interpretation focuses on estimating a population characteristic, not an individual’s behavior or future samples. The statement “With 95% confidence, the proportion of all students who take notes is between 0.15 and 0.25” correctly expresses this. It reflects our degree of confidence in the procedure, not the probability of the interval after the fact.
The misconception that there is a “95% chance the population proportion is in this interval” wrongly treats the population proportion as variable. In fact, it’s fixed—we’re uncertain about it, but it’s not changing. It’s the interval that varies from sample to sample.
Understanding this helps prevent common misinterpretations and reinforces the idea that confidence intervals are about populations, not individuals or probabilities after calculation.
