Confidence Intervals: every key term you need (+ practice quiz)
25 flashcard terms for Probability & Statistical Inference Topic 7, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A range of parameter values reported together with a stated reliability, rather than a single number. It communicates precision, which a bare point estimate hides entirely.
Confidence level
The long-run proportion of intervals built by a given procedure that would contain the true parameter across repeated sampling. It is a property of the method, not of any particular interval already computed.
Coverage probability
The actual probability that a procedure's interval captures the parameter, which may differ from the nominal level when assumptions fail. Comparing actual to nominal coverage is how interval methods are judged.
Frequentist interpretation trap
A computed interval either contains the parameter or it does not, so saying there is a ninety-five percent chance the parameter lies inside it misstates the guarantee. The probability describes the procedure over repetitions.
Margin of error
The half-width of a symmetric interval, equal to a critical value times the standard error. Reporting it separately makes clear how much of the width comes from confidence level and how much from sample variability.
Critical value
The quantile of a reference distribution that sets how many standard errors the interval extends. Higher confidence demands a larger critical value and therefore a wider interval, with no free lunch.
Pivotal quantity
A function of the data and the parameter whose distribution does not depend on any unknown parameter. Inverting a probability statement about a pivot is the standard route to an exact interval.
One-sample z interval
An interval for a mean using normal critical values, valid when the population standard deviation is genuinely known or the sample is large enough for the estimate to be treated as fixed.
One-sample t interval
An interval for a mean using t critical values because the standard deviation is estimated from the same data. It is the default in practice and is moderately robust to nonnormality at reasonable sample sizes.
Wald interval for a proportion
The estimate plus and minus a normal critical value times the estimated standard error. It undercovers badly for proportions near zero or one and can produce endpoints outside the unit interval.
Wilson score interval
An interval for a proportion obtained by inverting the score test rather than using the plug-in standard error. Its coverage is far better near the boundaries and its endpoints always stay inside the valid range.
Interval for a variance
Built from the chi-square distribution of the scaled sample variance, so it is asymmetric about the point estimate. It is highly sensitive to departures from normality, more so than intervals for a mean.
Two-sample interval for a mean difference
Centred on the difference of sample means with a standard error combining both groups. Whether variances are pooled or kept separate changes the degrees of freedom and, in unbalanced designs, the conclusion.
Paired versus independent designs
Paired data are analysed as a single sample of differences, which removes between-subject variation. Treating paired observations as independent inflates the standard error and wastes real precision.
Welch adjustment
A method for comparing two means without assuming equal variances, using an approximate fractional degrees of freedom. It is the safer default because equality of variances is rarely known in advance.
Width determinants of an interval
Width grows with the confidence level and the population spread and shrinks with the square root of the sample size. Only the sample size is usually under the analyst's control.
Sample size planning
Choosing how much data to collect by fixing a target margin of error and solving for the sample size, using a planning value for the spread. Conservative planning values guard against underpowered studies.
Bootstrap percentile interval
An interval read off the quantiles of a resampled distribution of the statistic. It needs no closed-form standard error but can be biased for skewed statistics unless a correction is applied.
Prediction interval
A range for a single future observation rather than for a parameter, so it must include individual variability as well as estimation error. It is always wider than the corresponding interval for a mean.
Tolerance interval
A range claimed to contain a specified proportion of the population with stated confidence. It answers a different question from either a parameter interval or a prediction interval and is common in quality work.
Credible interval
A Bayesian range holding a stated share of the posterior probability, so the direct probability statement about the parameter is legitimate. It generally differs from the frequentist interval unless the prior is uninformative.
Duality of intervals and tests
A value lies inside a confidence interval exactly when the corresponding two-sided test would not reject it at the complementary level. Intervals therefore carry all the information of a family of tests plus a sense of magnitude.
Overlapping intervals fallacy
Two intervals that overlap do not imply a nonsignificant difference between the groups. The correct procedure builds a single interval for the difference, which is narrower than naive comparison of two separate ranges suggests.
Interval reporting practice
Reporting the estimate, the interval and the sample size together lets a reader judge both direction and precision. An interval that includes trivially small and practically large values signals an inconclusive study.
Effect of nonnormality on coverage
Strong skewness or heavy tails degrade the coverage of small-sample intervals for means and severely damage variance intervals at any size. Transformation or resampling methods are the usual remedies.