ISS 2016 Statistics Paper-2 Solution: Question 38 (Pivotal Quantity and Confidence Intervals)
Continuing our question-by-question walk through the ISS Statistics Paper-2 archive, today we take up Question 38 from the 2016 paper. This one sits at the heart of interval estimation, and if you can explain it in your own words by the end of this post, you have understood one of the most useful ideas in the whole syllabus.
Quick Summary
Topic: Pivotal quantities and their role in constructing confidence intervals
Question Reference: ISS 2016, Statistics Paper-2, Question 38
Correct Answer: Option (c) — Statements 2 and 3 only
Here is the question exactly as it appeared in the paper.
Question 38.In usual notations, if a quantity φ(T, θ) exists, whose distribution is independent of θ, then which of the following are correct?
φ(T, θ) is called a confidence interval.
φ(T, θ) is a pivotal quantity.
φ(T, θ) can be used to define (1 − α)100% confidence interval for θ.
Select the correct answer using the code given below:
(a) 1 and 3 only
(b) 1 and 2 only
(c) 2 and 3 only
(d) 1, 2 and 3
Step 1: Read the Definition Given in the Question Very Literally
The question hands you a definition on a plate: "a quantity φ(T, θ) exists, whose distribution is independent of θ." Do not rush past this sentence. It is telling you that φ is built from two things — the statistic T (something you compute from your sample data, like the sample mean or sample variance) and the unknown parameter θ itself. Even though φ is a function of the unknown θ, its probability distribution does not depend on θ at all. That is a strange and powerful property, and it has a name.
Step 2: Recall What a Pivotal Quantity Actually Is
By definition, a pivotal quantity is exactly a function of the sample statistic(s) and the parameter, g(T, θ), whose sampling distribution does not involve the parameter θ. Compare this word for word with what the question describes: φ(T, θ) depends on T and θ, but its distribution is free of θ. That is the textbook definition of a pivotal quantity, no more and no less.
So Statement 2 — "φ(T, θ) is a pivotal quantity" — is simply restating the definition already given in the question. It is correct.
Step 3: Check Statement 1 — Is φ(T, θ) Itself Called a "Confidence Interval"?
This is where students often get tripped up, so let us be very precise about the difference between a pivotal quantity and a confidence interval.
A pivotal quantity φ(T, θ) is a single random quantity — a formula involving your data and the unknown parameter. A confidence interval, on the other hand, is a pair of numbers, say (L, U), computed purely from the data T (with no θ left in it), such that P(L ≤ θ ≤ U) = 1 − α.
These are two different objects. φ(T, θ) is the tool; the confidence interval is the outcome you build using that tool, after you isolate θ algebraically. Calling φ(T, θ) itself "a confidence interval" mixes up the raw pivotal quantity with the final interval derived from it.
So Statement 1 is incorrect.
Step 4: Check Statement 3 — Can φ(T, θ) Be Used to Build a (1 − α)100% Confidence Interval?
This is precisely why pivotal quantities are taught in the first place. Here is the standard three-step construction, worked out in full so nothing is skipped:
(i) Find two constants that trap the pivotal quantity with known probability.Because the distribution of φ(T, θ) does not depend on θ, you can find two fixed numbers a and b (read off from a standard table — normal, t, chi-square, F, whichever distribution φ follows) such that
P(a ≤ φ(T, θ) ≤ b) = 1 − α.
This step is only possible because φ's distribution is free of θ — if θ were still lurking in the distribution, a and b would themselves depend on the unknown θ, which would be circular and useless.
(ii) Algebraically rearrange the inequality to isolate θ in the middle.Since φ is a known function of T and θ, you manipulate the double inequality a ≤ φ(T, θ) ≤ b using ordinary algebra until it reads
L(T) ≤ θ ≤ U(T),
where L(T) and U(T) now involve only the observed statistic T and the known constants a, b — the unknown θ has been isolated in the middle, and everything on either side is computable from data alone.
(iii) Read off the confidence interval.Because the rearrangement was purely algebraic (an equivalent event), the probability statement is preserved:
P(L(T) ≤ θ ≤ U(T)) = 1 − α.
[L(T), U(T)] is now a genuine (1 − α)100% confidence interval for θ, built entirely from the pivotal quantity.
Pro Tip: The step students most often fumble under exam pressure is Step (i) — picking a and b. A very common mistake is to assume a and b must be symmetric percentile points (like −zα/2 and zα/2) even when the pivotal quantity's distribution is not symmetric, for example a chi-square pivotal quantity built from sample variance. The way this is usually drilled in classroom practice — the habit that eventually becomes automatic for a well-prepared candidate — is to always sketch the actual density of φ(T, θ) first and mark off the correct lower and upper α/2 tail areas from *that* shape, rather than assuming symmetry by default. Get into that habit early and this entire question type stops being a source of silly mistakes.
So Statement 3 is correct — this is exactly the mechanism by which pivotal quantities are used in practice.
Step 5: Combine the Findings and Match to the Options
We have determined:
Statement 1 — Incorrect (confuses the pivotal quantity with the interval itself)
Statement 2 — Correct (this is the definition of a pivotal quantity)
Statement 3 — Correct (this describes the standard use of a pivotal quantity)
So the correct statements are 2 and 3 only, which is option (c).
Checking against the official ISS 2016 Statistics Paper-2 answer key: Question 38 is listed as (c). Our independent derivation matches the official key exactly.
Why This Question Matters
Interval estimation questions built around pivotal quantities appear almost every year in one form or another, sometimes as a direct definitional question like this one, and sometimes hidden inside a numerical problem asking you to construct an actual confidence interval for a mean, variance, or proportion. If you are shaky on the distinction between a pivotal quantity, an estimator, and a confidence interval, every one of those numerical problems becomes shakier too. Getting the vocabulary exactly right here — as we did above, question by question — pays off across the entire inference portion of the syllabus, not just on this one MCQ.
Frequently Asked Questions
What is the difference between a pivotal quantity and a statistic?
A statistic, like T, is a function of the sample data alone and does not involve the unknown parameter θ. A pivotal quantity φ(T, θ) explicitly involves both T and θ, but its probability distribution is free of θ. So every pivotal quantity uses a statistic as a building block, but a pivotal quantity itself is not a statistic in the strict sense, because it still contains the unknown parameter.
Why does the distribution of the pivotal quantity need to be free of θ?
If the distribution still depended on θ, you would not be able to find fixed constants a and b from a standard table to trap the pivotal quantity with a known probability 1 − α. Those constants would themselves become functions of the unknown θ, which defeats the entire purpose of building an interval you can actually compute from data.
Is the sample mean a pivotal quantity for the population mean?
Not by itself, because the distribution of the sample mean generally still depends on the unknown population mean and variance. However, a standardized version, such as (sample mean − θ) divided by the standard error, often has a distribution (like the standard normal or Student's t) that is completely free of θ. That standardized quantity is the actual pivotal quantity used to build the confidence interval for the mean.
Can a pivotal quantity be used for hypothesis testing as well as confidence intervals?
Yes. The same pivotal quantity that gives you a confidence interval can be inverted to test a hypothesis about θ: you simply check whether the hypothesized value θ0 falls inside the confidence interval, or equivalently whether the observed pivotal value falls in the rejection region for that test. This is why confidence intervals and two-sided hypothesis tests are so closely linked.
Do all parameters have a pivotal quantity available?
Not always in closed form. For many standard distributions (normal, exponential, gamma) neat pivotal quantities exist and are tabulated in every textbook. For more complicated models, an exact pivotal quantity may not exist, and practitioners fall back on approximate or asymptotic pivotal quantities, often derived using the asymptotic normality of maximum likelihood estimators.
Why is option (a), "1 and 3 only," a tempting wrong answer here?
Because Statement 3 is genuinely correct and closely related in wording to Statement 1, a hurried reader can conflate "used to define a confidence interval" (Statement 3, true) with "is called a confidence interval" (Statement 1, false) and mistakenly tick both as correct together. Reading each statement slowly and separately, exactly as we did in Step 3 and Step 4 above, is the only reliable way to avoid this trap.
That covers Question 38 in full. If any step above still feels unclear, or if you spot something you would solve differently, leave a comment below — and if this walkthrough helped, do share it with a fellow ISS aspirant who is working through the same paper.
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