A confidence interval around an estimated mean expresses:
Correct Answer
D) The range likely to contain the true value at a stated probability
Why this is correct: The correct answer, 'The range likely to contain the true value at a stated probability,' defines a confidence interval. It quantifies the uncertainty in a sample-based estimate (like a mean) by providing a range within which the true population parameter is expected to fall, with a specified level of confidence (e.g., 95%). This aligns with the original explanation that it 'converts sample uncertainty into a stated range' and avoids overstating precision. Why the other choices are wrong: 'The correlation between two variables' describes a measure of association, not a range around an estimate. 'The exact value the parameter must take' is incorrect because a confidence interval expresses a range of plausible values, not a single exact point. 'The total number of individual observations included in the sample' refers to the sample size (n), not a probabilistic range. Exam tip: Remember, a confidence interval is about the precision of an estimate, not the data points themselves. It's a range, not a single number or a measure of relationship.
Why This Is the Correct Answer
Option D describes exactly what the interval delivers: a range likely to contain the true value at a stated probability. Both elements matter, the range and the attached confidence level, because an interval quoted without its level is meaningless. This framing also correctly signals that the interval expresses uncertainty about the parameter rather than describing the individual data points. It is the only choice that involves both a range and a probability statement.
Why the Other Options Are Wrong
Option A: The correlation between two variables
Correlation measures the strength and direction of association between two variables and is reported as a coefficient between negative one and positive one, not as an interval around an estimate. A confidence interval can be built around a correlation coefficient, but the interval and the coefficient are different things. Candidates pick this when they group all statistical terms together instead of asking what each one outputs.
Option B: The exact value the parameter must take
An exact value is precisely what a sample cannot deliver, and claiming one would overstate the precision of the analysis. Confidence intervals exist because point estimates alone conceal sampling error. This option is tempting to anyone who wants statistics to remove uncertainty rather than quantify it.
Option C: The total number of individual observations included in the sample
Sample size is the count of observations, denoted n, and it is an input to the interval rather than the interval itself. Sample size does influence width, since a larger n reduces the standard error and narrows the interval, and that indirect relationship is what makes this distractor plausible. But an interval is expressed in the units of the variable, such as dollars per square foot, not as a count.
Net, Not Needle
A confidence interval is a net, not a needle. You cast a net of a certain width and say you are 95% confident the true value is somewhere inside it. Widen the confidence and you need a bigger net; gather more data and you can catch the same fish with a smaller one.
How to use: When a statistics option offers a single exact number, it is a needle and it is wrong. When an option describes a count, a coefficient, or a relationship, it is answering a different question. Keep the option that gives you a span of values with a probability attached.
Exam Tip
Read the units in the answer: a confidence interval is expressed in the units of the thing measured, so any option describing a count, a ratio, or a single point can be eliminated on sight.
Common Mistakes to Avoid
- -Reporting an interval without stating the confidence level that goes with it
- -Treating a confidence interval as the range containing most individual observations rather than the likely range of the parameter
- -Assuming a wide interval means the analysis was done wrong rather than that the sample is small or variable
Concept Deep Dive
Analysis
This question tests what a confidence interval measures and, just as importantly, what it does not measure. When you compute a mean from a sample of sales, that mean is an estimate of an unobservable population parameter, and it carries sampling error. A confidence interval converts that error into a stated range, built from the sample mean plus and minus a critical value times the standard error, where the standard error shrinks as the sample grows. The confidence level, commonly 90%, 95%, or 99%, describes the long-run reliability of the procedure: intervals built this way capture the true parameter that share of the time. For appraisers, the practical payoff is that a wide interval is an honest warning that the data will not support a precise point conclusion, while a narrow interval indicates the sample is tight enough to rely on.
Background Knowledge
You need to know the difference between a sample statistic and the population parameter it estimates, and that the standard error measures how much a sample mean would vary from sample to sample. You should also know the general construction of an interval as the point estimate plus and minus a margin of error, and that increasing sample size narrows the interval while raising the confidence level widens it.
Real-World Application
An appraiser analyzing 34 warehouse sales computes a mean price of $118 per square foot with a 95% confidence interval of $109 to $127. That $18 spread tells the appraiser and the client that the data support a range conclusion or a bracketed point value, and it justifies gathering additional sales before narrowing the opinion further.
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