Which statement is true about confidence intervals?

Which statement is true about confidence intervals?

• A. They guarantee the true parameter is within the interval for every sample.
• B. They are fixed once computed.
• C. They provide a range that would contain the true parameter with a specified probability over many samples.
• D. They replace hypothesis testing.

Answer: (C)

The main idea here is how a confidence interval reflects uncertainty about a population parameter. A confidence interval is built from sample data to estimate a range where the true parameter would lie if we repeated the study many times with the same method. The best statement says that the interval would contain the true parameter with a specified probability over many samples. In other words, if we could repeat the whole process lots of times and construct an interval each time at a chosen confidence level (say 95%), about 95% of those intervals would include the true parameter. This describes the long-run performance of the method, not guarantees for any single obtained interval. Why the others don’t fit: for any given sample, the true parameter doesn’t have to be inside the single interval we computed; it can be outside due to sampling variation. The interval we get from one sample is a fixed outcome for that data, but the process that generates intervals is random across repeated samples, so saying they are fixed once computed misrepresents that variability. Finally, confidence intervals and hypothesis testing address different questions: a CI provides a range of plausible parameter values, while hypothesis testing evaluates evidence against a specific null hypothesis; they’re related but do not replace one another.

The main idea here is how a confidence interval reflects uncertainty about a population parameter. A confidence interval is built from sample data to estimate a range where the true parameter would lie if we repeated the study many times with the same method.

The best statement says that the interval would contain the true parameter with a specified probability over many samples. In other words, if we could repeat the whole process lots of times and construct an interval each time at a chosen confidence level (say 95%), about 95% of those intervals would include the true parameter. This describes the long-run performance of the method, not guarantees for any single obtained interval.

Why the others don’t fit: for any given sample, the true parameter doesn’t have to be inside the single interval we computed; it can be outside due to sampling variation. The interval we get from one sample is a fixed outcome for that data, but the process that generates intervals is random across repeated samples, so saying they are fixed once computed misrepresents that variability. Finally, confidence intervals and hypothesis testing address different questions: a CI provides a range of plausible parameter values, while hypothesis testing evaluates evidence against a specific null hypothesis; they’re related but do not replace one another.