What is the interpretation of a 96% confidence level?
The interval contains 96% of all sample means.
There’s a 96% chance that the given interval includes the true value of the population parameter.
Approximately 96 out of 100 such intervals would include the true value of the population parameter.
There’s a 4% chance that the given interval does not include the true value of the population parameter.
The correct answer and explanation is :
The correct answer is:
“Approximately 96 out of 100 such intervals would include the true value of the population parameter.”
Explanation:
A 96% confidence level refers to the long-term proportion of confidence intervals, constructed using repeated random samples, that will contain the true population parameter. This means that if we were to construct 100 different confidence intervals from 100 different samples, we would expect about 96 of them to contain the true parameter.
Why is this the correct interpretation?
Confidence intervals are based on repeated sampling and probability theory. The confidence level indicates the reliability of the method used to construct the interval, not the probability of any single interval containing the true parameter. The true population parameter is a fixed value, while the confidence interval varies from sample to sample.
Why are the other options incorrect?
- “The interval contains 96% of all sample means.”
- Incorrect because the confidence interval does not describe the distribution of sample means; it estimates the population parameter based on a single sample.
- “There’s a 96% chance that the given interval includes the true value of the population parameter.”
- Incorrect because probability statements apply to random variables, not fixed values. The true parameter is either inside or outside the interval, but we do not assign a probability to this specific interval.
- “There’s a 4% chance that the given interval does not include the true value of the population parameter.”
- While this seems reasonable, it incorrectly implies a probability for the specific interval. Instead, it’s better to say that 4% of similarly constructed intervals will not contain the true value over repeated sampling.
Thus, the best interpretation of a 96% confidence level is that approximately 96 out of 100 such intervals will contain the true population parameter if the sampling process is repeated many times.