A researcher computes a 95% CI for mean sleep hours and gets [6.8, 7.4]. Which interpretation is correct?
AThere is a 95% probability that the true mean lies between 6.8 and 7.4 hours
BIf this procedure were repeated many times, 95% of such intervals would contain the true mean
C95% of people in the sample sleep between 6.8 and 7.4 hours
DThe true mean is 95% likely to be close to 7.1 hours
The true mean μ is a fixed constant — it either lies in [6.8, 7.4] or it doesn't. Probability doesn't apply to a specific computed interval. What 95% describes is the long-run procedure: if you repeated this sampling and interval-construction process many times, 95% of the resulting intervals would contain μ. Option A is the most common misconception and is incorrect for exactly this reason.
Question 2 Multiple Choice
A 95% CI is computed from a sample of n = 50. If the sample size is increased to n = 200 with the same confidence level, what happens to the interval width?
AIt stays the same — confidence level determines width, not sample size
BIt decreases by a factor of 2 — the margin of error is proportional to 1/√n
CIt increases — more data introduces more sources of variability
DIt doubles — larger samples cover more of the population
The margin of error is z*(s/√n), so it shrinks like 1/√n. Quadrupling n (from 50 to 200) halves √n's denominator effect, cutting the margin of error in half. The common misconception is that more data means more uncertainty; in fact, more data means better precision and a narrower interval.
Question 3 True / False
A 99% confidence interval computed from the same data is wider than a 95% confidence interval.
TTrue
FFalse
Answer: True
Higher confidence requires a larger multiplier (z* = 2.576 for 99% vs. 1.96 for 95%), which widens the margin of error. The intuition: to be more confident of capturing μ, you must cast a wider net. More confidence always means a wider interval, all else equal.
Question 4 True / False
A 95% CI guarantees that 95% of future sample means drawn from the same population will fall inside it.
TTrue
FFalse
Answer: False
A CI makes a claim about the population parameter μ, not about future sample means. The 95% refers to the proportion of confidence intervals (computed by this procedure) that would contain μ — a statement about the interval-generating process, not about the distribution of x̄ values. Future sample means are covered by the sampling distribution, which is a separate concept.
Question 5 Short Answer
Why is it incorrect to say 'there is a 95% probability that the true mean is in this interval'?
Think about your answer, then reveal below.
Model answer: The true mean μ is a fixed (though unknown) constant — it either is or isn't in the computed interval. Since μ is not random, probability doesn't apply to it. The randomness is in the interval itself, which varies from sample to sample. '95%' describes what fraction of intervals would contain μ across many repetitions of the procedure, not a probability about where μ sits relative to one particular interval.
This is the central interpretive challenge of confidence intervals. Frequentist probability only applies to random events. Once an interval is computed, μ's location relative to it is a fixed fact — we just don't know which. The correct statement shifts the randomness to the procedure: 'this method produces intervals that capture μ 95% of the time,' not 'this interval has a 95% chance of being right.'