Questions: Confidence Intervals for Population Means

Question 1

A researcher computes a 95% CI for a population mean and gets [3.1, 4.7]. She says: 'There is a 95% probability that μ is between 3.1 and 4.7.' What is wrong with this statement?

• A. Nothing — this is the correct interpretation of a 95% confidence interval
• B. The interval should be wider because 95% is too narrow a confidence level
• C. μ is a fixed (unknown) constant, not a random variable, so it cannot have a probability of being in any interval
• D. She should say 'certainty' rather than 'probability' since the data was collected

Answer: (C)

The critical error is treating μ as random. μ is a fixed (though unknown) parameter — it either is or is not in [3.1, 4.7], with probability 1 or 0 respectively. The 95% refers to the *procedure*: if you repeated the sampling and CI-construction many times, 95% of the resulting intervals would contain μ. The interval [3.1, 4.7] is one realized value of a random interval; once observed, the randomness is gone. 'Confidence' lives at the level of the method, not any particular interval.

Question 2

Why do we use the t-distribution instead of the standard normal when σ is unknown?

• A. The t-distribution is simpler to compute and gives the same results for large samples
• B. Using sample standard deviation s introduces additional uncertainty, so the standardized quantity follows a t-distribution with heavier tails
• C. The normal distribution cannot handle sample sizes smaller than 30
• D. The t-distribution corrects for bias in the sample mean X̄

Answer: (B)

When σ is replaced by s, the standardized quantity (X̄ − μ)/(s/√n) no longer follows N(0,1) — it follows a t-distribution with n−1 degrees of freedom. The t has heavier tails than the normal because s itself is a random variable that adds uncertainty. This produces wider intervals (more conservative) for small n, appropriately reflecting that we estimated σ from the data. As n increases, s → σ and the t approaches the normal. Option C describes a common rule-of-thumb heuristic, not the statistical reason.

Question 3

Once you have computed a specific 95% confidence interval from your data, there is a 95% probability that μ falls within it.

• T. True
• F. False

Answer: False

This is the most widespread misinterpretation of confidence intervals. Once the interval is computed, μ either is or is not inside it — the probability is 1 or 0, not 95%. The 95% describes the long-run frequency with which the *procedure* captures μ: if you drew many samples and computed a CI from each, 95% of those intervals would contain μ. The randomness is in the interval (which varies across samples), not in μ (which is fixed). Holding this picture clearly is essential preparation for hypothesis testing.

Question 4

Increasing the sample size narrows the confidence interval for a given confidence level, all else being equal.

• T. True
• F. False

Answer: True

The margin of error in a CI is proportional to σ/√n (or s/√n for the t-interval). As n increases, √n grows, so the margin of error shrinks and the interval narrows. This reflects the statistical intuition that more data provides more precise estimates of μ. Note that to halve the margin of error, you need to quadruple the sample size, since the improvement scales as √n.

Question 5

What does '95% confidence' actually mean as a statement about the procedure for constructing confidence intervals?

Model answer: A 95% confidence level means that if you were to repeat the entire procedure many times — draw a new random sample, compute X̄ and s, and construct the interval — 95% of the resulting intervals would contain the true population mean μ. The confidence is a property of the method, not of any single interval. Any specific computed interval either contains μ or it doesn't; we simply cannot know which. The 95% is a guarantee about long-run performance: out of 100 such intervals constructed under the same procedure, approximately 95 will capture μ.

This frequentist interpretation is the correct one. The common error is to assign probability to μ's location, as if μ were random. Instead, μ is fixed and the interval is random (it changes every time you sample). The confidence level quantifies how often the random interval succeeds in capturing the fixed target. This framing directly supports hypothesis testing, where the same logic applies: the p-value is about the behavior of test statistics across samples, not about the probability of a hypothesis being true.