ISS 2016 Statistics Paper-2 Solution: Question 23 (SPRT Optimality and Graphical Method)

Continuing our question-by-question walk through the ISS Statistics Paper-2 previous year papers, today we take up Question 23 from the 2016 exam. This one moves away from estimation theory and into testing procedures — specifically, the Sequential Probability Ratio Test (SPRT), a topic that trips up a lot of aspirants because it mixes a theoretical optimality result with a practical, almost mechanical, procedure.

Quick Summary

• Topic: Sequential Probability Ratio Test (SPRT) — optimality property and graphical implementation

• Question Reference: ISS 2016, Statistics Paper-2, Question 23

• Correct Answer: Option (b) — Both Statement-I and Statement-II are true, but Statement-II is not the correct explanation for Statement-I

The Question, As Asked

Consider the following statements:

Statement-I:Sequential Probability Ratio Test (SPRT) requires, on an average, smaller number of observations than a non-sequential test with identical error controlling properties.

Statement-II:SPRT can be carried out graphically.

Which one of the following is correct in respect of the above two statements?

Step 1: Understand What SPRT Actually Is

Before judging the two statements, it helps to be crystal clear on what SPRT does differently from an ordinary (fixed-sample-size) test.

In a usual hypothesis test, you decide the sample size n in advance, collect all n observations, compute a test statistic, and then make one decision: accept H0 or reject it. The sample size never changes once the data collection starts.

SPRT, developed by Abraham Wald, works completely differently. You take observations one at a time, and after each new observation you compute the likelihood ratio

Lm = [Product of f1(xi) for i = 1 to m] / [Product of f0(xi) for i = 1 to m]

where f0 and f1 are the densities (or probability mass functions) under H0 and H1 respectively. You then compare Lm against two pre-fixed constants A and B (with B < 1 < A, chosen from the desired error probabilities α and β):

• If Lm ≥ A, stop and reject H0.

• If Lm ≤ B, stop and accept H0.

• If B < Lm < A, take one more observation and repeat.

So the number of observations you end up needing, N, is itself a random variable — it is not fixed in advance. That single fact is the key to understanding Statement-I.

Step 2: Checking Statement-I — The Optimality Property

Statement-I claims that SPRT needs, on average, fewer observations than a fixed-sample test that controls the same error probabilities α (Type I error) and β (Type II error).

This is not a vague claim — it is a proven theorem, usually called the Wald–Wolfowitz fundamental theorem of SPRT. It states that among all tests (sequential or fixed-sample) that have error probabilities no larger than α and β when testing a simple H0 against a simple H1, the SPRT minimises both E(N | H0) and E(N | H1) — the expected sample size under both hypotheses.

In plain words: if you fix how wrong you're willing to be (your α and β), SPRT is the most "economical" procedure in terms of expected number of observations needed to reach that same level of reliability. Empirically, this saving is often substantial — SPRT can need on average roughly half the observations of an equivalent fixed-sample test in many standard setups (e.g., testing a simple hypothesis on the mean of a normal distribution or on a binomial proportion).

So Statement-I is a genuine, well-established optimality result — it is TRUE.

Step 3: Checking Statement-II — The Graphical Procedure

Statement-II claims that SPRT can be carried out graphically. This is also true, and it's actually one of the most practical, exam-friendly features of SPRT.

Here's how the graphical version works. Taking logs of the stopping rule from Step 1 (since log is monotonic, the inequalities carry over):

log B < Σ log[f1(xi)/f0(xi)] < log A

If you plot the cumulative sum Sm = Σ log[f1(xi)/f0(xi)] on the vertical axis against the observation number m on the horizontal axis, the two stopping boundaries log A and log B become two straight, parallel lines running across the graph. For many standard distributions (normal, binomial, Poisson, exponential), this cumulative sum itself simplifies to a linear function of m, so the "path" of the test is just a sequence of points that you plot after each new observation.

You keep plotting points as data comes in. The very first time your plotted path touches or crosses the upper line, you stop and reject H0; the first time it touches or crosses the lower line, you stop and accept H0. As long as the path stays strictly between the two lines, you take another observation. This is literally how SPRT was taught and even used in quality-control settings before computers made likelihood-ratio computation trivial — you could hand a factory inspector a pre-drawn chart with the two boundary lines already marked, and they would just plot a point after each item inspected.

So Statement-II is also TRUE.

Pro Tip: The trap in this question is not figuring out whether each statement is true — both clearly are, from standard theory. The trap is deciding whether Statement-II "explains" Statement-I. The graphical method is just a computational/visual convenience for applying the stopping rule; it says nothing about

Step 4: Putting It Together

We have established:

• Statement-I: TRUE (Wald–Wolfowitz optimality theorem)

• Statement-II: TRUE (the log-likelihood-ratio path against two parallel boundary lines)

• Statement-II does NOT explain Statement-I — one is about expected sample size efficiency, the other is about a plotting technique for implementing the stopping rule.

This matches option (b):Both Statement-I and Statement-II are true, but Statement-II is not the correct explanation for Statement-I.

Checking against the official ISS 2016 Statistics Paper-2 answer key: the key lists option (b) for Question 23 — so our independently derived answer matches the official key exactly.

Why This Question Matters

SPRT questions show up repeatedly across ISS and similar statistical service exams because the topic sits at the intersection of three examinable ideas: hypothesis testing theory, the idea of a random stopping time, and a genuinely elegant practical implementation. Examiners like the Assertion-Reason format here precisely because it's easy to construct two independently-true statements about SPRT and test whether a candidate actually understands the logical structure of the theorem versus just memorising "SPRT is efficient" and "SPRT has a graph" as isolated facts. Being able to cleanly separate "what is true" from "what explains what" is a skill that pays off across the whole testing-theory section, not just this one question.

Frequently Asked Questions

What exactly does "on an average" mean in Statement-I?

It refers to the expected value of the sample size N, which is a random variable in SPRT since testing stops as soon as the likelihood ratio crosses a boundary. E(N) is computed under a specific hypothesis (either H0 or H1 being true), and the Wald–Wolfowitz theorem shows this expectation is minimised by SPRT compared to any fixed-sample test with the same α and β.

Does SPRT always require fewer observations than a fixed-sample test, for every single sample drawn?

No — this is a common misreading. The result is about the expected (average) sample size, not a guarantee for every individual run. In some unlucky sequences, SPRT could in principle run longer than the fixed-sample alternative; the optimality is a statement about long-run average behaviour, not a case-by-case guarantee.

Why are A and B chosen based on α and β specifically?

Wald showed that setting A ≈ (1−β)/α and B ≈ β/(1−α) keeps the actual Type I and Type II error probabilities of the SPRT close to the target α and β you specified in advance. This approximation (Wald's approximation) is what lets you fix the boundaries before seeing any data.

Is the graphical method still relevant, or is it just historical?

It's mostly a teaching and conceptual tool today, since likelihood ratios are trivial to compute on any calculator or computer now. But it remains extremely useful for building intuition about how SPRT behaves — seeing the random walk-like path move between two boundary lines makes the "random stopping time" idea concrete in a way that formulas alone don't.

Could Statement-II ever be the correct explanation for Statement-I in a different phrasing?

Only if the graphical method itself were the reason for the efficiency gain, which it isn't — the graph is just a visualisation of the same likelihood-ratio stopping rule. If a future question rephrased Statement-II to say something like "the stopping rule adapts to accumulating evidence about which hypothesis is true," that actually would explain the efficiency, and the answer would shift to option (a).

How is this different from a simple two-sided fixed-sample Z-test?

A fixed-sample Z-test commits to a sample size n in advance regardless of how clear-cut the data turns out to be. SPRT, in contrast, can stop early if the evidence strongly favours one hypothesis, or run longer if the evidence is ambiguous — and it is precisely this adaptivity that produces the lower expected sample size referenced in Statement-I.

If anything here felt unclear, or you spotted a step you'd solve differently, drop a comment below — working through these disagreements is often where the real learning happens. And if you know a fellow ISS aspirant grinding through the same previous-year papers, feel free to pass this one along.