ISS 2016 Statistics Paper-2 Solution: Question 40 (Simple vs Composite Hypotheses)

Continuing our question-by-question walk through the ISS Statistics Paper-2 archive, today's post picks up 2016, Question 40. This one looks like a short definitional question at first glance, but it hides a trap that catches a surprising number of students who have only half-memorised the terms "simple" and "composite" hypothesis — so we will slow right down and build the idea from scratch.

Quick Summary

• Topic: Simple vs. composite statistical hypotheses, set in a two-sample Poisson testing problem

• Question reference: ISS 2016, Statistics Paper-2, Question 40

• Correct answer (derived from first principles):(d) Composite and composite

The Question, Exactly As Asked

To compare the quality of LED TVs of Brand 'A' and Brand 'B', a statistician collects the following data: X1, X2, X3, ..., Xn are the number of dead pixels in a random sample of n Brand 'A' TVs, and Y1, Y2, Y3, ..., Ym are the number of dead pixels in a random sample of m Brand 'B' TVs.

Assuming that the Xi's and Yi's are independent Poisson random variables with mean λ1 and λ2 for Brand 'A' and Brand 'B' TVs respectively, the statistician decides to test the hypothesis H0: λ1 = λ2 versus H1: λ1 ≠ λ2.

Here H0 and H1 are, respectively:

• (a) Simple and simple

• (b) Simple and composite

• (c) Composite and simple

• (d) Composite and composite

Step 1: What Does "Simple" Actually Mean for a Hypothesis?

Before touching this specific question, we need a rock-solid definition, because the whole question is really just testing whether you know this one rule precisely.

A statistical hypothesis is a statement about the unknown parameter(s) of a distribution. We call a hypothesis simple only when it pins down every unknown parameter to one exact numerical value — in other words, it completely specifies the probability distribution with nothing left unknown. If even a single parameter is still allowed to range over more than one possible value, the hypothesis is called composite.

A few bite-sized examples to build intuition before we go further:

• For X ~ N(μ, 4): "H: μ = 10" is simple — μ is fully nailed down to one number, and σ² = 4 was already known, so the distribution N(10, 4) is completely specified.

• For the same X: "H: μ > 10" is composite — μ could be 10.1, 50, or a million different values greater than 10, so we don't know which exact distribution we have.

• For X ~ N(μ, σ²) with both unknown: "H: μ = 10" is composite, even though μ is fixed, because σ² is still free to be anything — the distribution still isn't pinned to one member of the family.

Step 2: Write Down the Parameter Space for This Problem

Here we have two independent Poisson populations, Brand A with mean λ1 and Brand B with mean λ2. Both λ1 and λ2 are unknown positive numbers. So the full parameter space — every pair of values the data could possibly have come from — is:

Θ = {(λ1, λ2) : λ1 > 0, λ2 > 0}

Picture this as the entire positive quadrant of a plane, with λ1 on one axis and λ2 on the other. Every single point in that quadrant is a distribution the data could, in principle, have been generated from. Our job is to see how much of that quadrant is "left over" once we impose H0 or H1.

Step 3: Classify H0: λ1 = λ2

This null hypothesis says the two means are equal to each other — but it never says what that common value actually is. λ1 could equal λ2 at 2, or at 7.5, or at 1000; the condition "λ1 = λ2" is satisfied by infinitely many such pairs.

Geometrically, this set of points is the diagonal line λ1 = λ2 running through the positive quadrant. A line clearly contains infinitely many points, not just one. Since H0 does not reduce us down to a single, fully-specified distribution, it fails the definition of "simple" from Step 1.

Conclusion: H0 is composite.

Pro Tip: The single most common mistake on this exact style of question is assuming that writing an equation (λ1 = λ2) automatically makes a hypothesis "simple," because it "looks like one condition." It doesn't. Simplicity is about whether the

Step 4: Classify H1: λ1 ≠ λ2

The alternative hypothesis says the two means are different from each other, but places no restriction on by how much, or on what either value individually is. This set is simply the entire positive quadrant minus the diagonal line from Step 3 — still an enormous, infinite collection of (λ1, λ2) pairs.

Since this region clearly contains far more than one point — indeed almost the whole parameter space — H1 is, just as obviously, composite.

Conclusion: H1 is composite.

Step 5: Final Answer and a Flagged Discrepancy With the Official Key

Putting Steps 3 and 4 together: H0 is composite, and H1 is composite. That matches option (d) Composite and composite.

When we checked the official answer key for ISS 2016 Statistics Paper-2 (Series A), it lists option (a) "Simple and simple" against Question 40. We rechecked our derivation carefully — re-examined the parameter space, re-verified that neither hypothesis fixes λ1 or λ2 to a specific number — and the reasoning still points firmly to (d). Testing the equality of two unspecified means (or rates) against their inequality is one of the standard textbook examples of a composite-versus-composite test precisely because no numeric value is ever pinned down on either side of the hypothesis. We are flagging this openly as a discrepancy between our fully-worked derivation and the published key, rather than silently following either source, so that readers can follow the mathematics themselves and judge it on its merits.

Why This Question Matters

Simple-versus-composite classification shows up again and again once you move into the Neyman-Pearson framework, likelihood ratio tests, and power function analysis — all heavily tested topics in ISS Paper-2. Misjudging whether a hypothesis is simple or composite early on can quietly wreck an entire downstream derivation, such as deciding whether the Neyman-Pearson Lemma even applies in its basic form (it is stated for simple-versus-simple testing) or whether you need a generalized/uniformly most powerful test framework instead. Getting this one classification step right, every time, is what lets everything built on top of it stand up.

Frequently Asked Questions

Why isn't H0: λ1 = λ2 considered simple, since it's just one equation?

Writing one equation does cut the parameter space down from two dimensions to one, but "simple" requires cutting it all the way down to a single point. Since λ can still be any positive number while satisfying λ1 = λ2, infinitely many distributions remain consistent with H0, so it stays composite.

Is there any version of this question where H0 would actually be simple?

Yes — if the question had instead said "H0: λ1 = λ2 = 5," that fixes both parameters to one exact numeric value, giving a single fully-specified Poisson(5) versus Poisson(5) pair. That version would be simple. The presence or absence of an actual number is the deciding factor.

Does the sample size n or m affect whether a hypothesis is simple or composite?

No. Simplicity and compositeness are properties of the parameter space defined by the hypothesis statement itself, not of the data or the sample sizes. You could have n = m = 1000 or n = m = 2, and the classification of H0 and H1 here would be unchanged.

How does this concept connect to the Neyman-Pearson Lemma?

The classical Neyman-Pearson Lemma is proved for testing one simple hypothesis against another simple hypothesis, giving the most powerful test via a likelihood ratio. When both hypotheses are composite, as in this question, you instead need extensions such as uniformly most powerful (UMP) tests or likelihood ratio tests, which is exactly why examiners like testing this classification early.

What's a quick way to double-check my classification during the exam?

Ask yourself: "If this hypothesis were true, could I write down one specific, fully numerical distribution, or do I still have unknown numbers left over?" If you can write exact numbers for every parameter, it's simple; if any parameter is still free to vary, it's composite. Apply this test to both H0 and H1 separately, since they don't have to match.

Are equality-type hypotheses between two unknown parameters always composite?

In almost every practical testing scenario, yes — whenever the common value being tested for equality is itself unspecified (as with λ1 = λ2 here, or the more familiar μ1 = μ2 in two-sample mean tests), the hypothesis remains composite because that shared value can still range freely.

If anything in this derivation felt shaky, especially the simple-versus-composite distinction, work through the small examples in Step 1 again with your own numbers until they feel automatic — and feel free to drop your doubts in the comments below. If this walkthrough helped clear things up, consider sharing it with a fellow ISS aspirant who might be stuck on the same question.