Questions: Confidence Intervals: General Framework

Question 1

A researcher reports: 'The 95% confidence interval for the average daily step count is [7,200, 8,400].' Which interpretation is correct?

• A. There is a 95% probability that the true population mean is between 7,200 and 8,400
• B. 95% of individuals in the population take between 7,200 and 8,400 steps per day
• C. If this study were repeated many times, 95% of the resulting intervals would contain the true population mean
• D. The sample mean has a 95% chance of lying between 7,200 and 8,400

Answer: (C)

The correct interpretation describes the long-run performance of the procedure, not a probability about this specific interval. The parameter (true mean) is a fixed number — it either is or isn't in [7,200, 8,400]. We can't assign it a probability post-hoc. The 95% refers to what would happen across many repetitions of the study: 95% of intervals constructed this way would contain the true mean. Option A is the most common wrong interpretation and is subtly but fundamentally incorrect.

Question 2

A researcher wants a narrower confidence interval without reducing the confidence level from 95%. Which change achieves this?

• A. Use a higher critical value (e.g., z = 2.33 instead of 1.96)
• B. Increase the sample size
• C. Decrease the confidence level to 90% while keeping interpretation the same
• D. Report the interval in different units to make it appear narrower

Answer: (B)

The interval width is estimate ± critical_value × SE, where SE = σ/√n. Increasing sample size n decreases SE proportionally, directly narrowing the interval. The confidence level controls the critical value — increasing it to 99% widens the interval. The population variability σ is fixed and outside the researcher's control. Larger sample size is the only practical lever for achieving both high confidence and narrow intervals simultaneously.

Question 3

Once a specific confidence interval has been calculated — say [2.1, 3.4] — it is correct to say there is a 95% probability that the true population mean lies within that interval.

• T. True
• F. False

Answer: False

After the interval is computed, there is no randomness left to assign probability to. The true mean μ is a fixed constant; [2.1, 3.4] is a fixed interval. Either μ ∈ [2.1, 3.4] or μ ∉ [2.1, 3.4] — the probability is 1 or 0, not 0.95. The 95% describes the procedure: before sampling, the probability that the (random) interval will cover μ is 95%. Once you observe the specific interval, the probabilistic statement no longer applies to it. This distinction is subtle but fundamental to frequentist inference.

Question 4

A 99% confidence interval is always wider than a 95% confidence interval computed from the same data, all else being equal.

• T. True
• F. False

Answer: True

Higher confidence requires a larger critical value. For a 95% CI, z = 1.96; for a 99% CI, z = 2.576. The interval is estimate ± critical_value × SE, so a larger critical value produces a wider interval from the same data. This reflects the fundamental tradeoff: to be more confident of capturing the parameter, you must sacrifice precision (narrowness). There is no way to increase confidence without widening the interval, short of collecting more data.

Question 5

Explain why it is incorrect to say 'there is a 95% probability that μ lies in [2.1, 3.4]' once a specific interval has been computed from a sample.

Model answer: The parameter μ is a fixed (though unknown) constant — it does not have a probability distribution under frequentist statistics. The specific interval [2.1, 3.4] is also fixed once computed. Either μ is in the interval or it isn't. The 95% describes the sampling procedure: before collecting data, 95% of all intervals the procedure could generate would contain μ. Once a specific interval is realized, that probabilistic statement applies to the method, not to the particular interval.

The confusion arises from treating a fixed unknown parameter as random. In frequentist statistics, probability applies to the behavior of estimators and intervals across repeated sampling — not to fixed but unknown quantities. The correct framing is: 'This interval was produced by a method that works 95% of the time.' Not: 'This particular interval has a 95% chance of being correct.' Bayesian credible intervals do allow direct probability statements about parameters, but they require a prior distribution and answer a different question.