ISS 2016 Statistics Paper-2 Solution: Question 49 (True Statements on Hypothesis Testing)
Continuing our question-by-question walk through the ISS Statistics Paper-II previous year papers, today we take up Question 49 from the 2016 paper. This one isn't a calculation at all — it's a conceptual statement-matching question on the logic of hypothesis testing, and it rewards reading each statement slowly rather than reacting to familiar-sounding phrases.
Quick Summary
Topic:Hypothesis testing — conceptual understanding (Statistical Inference)
Question reference:ISS 2016, Statistics Paper-II, Question 49
Correct answer (by independent derivation):Option (c) — Statements 1, 2 and 4
Official answer key (Series A):Option (a) — Statements 1, 2 and 3 — this post explains why we disagree and flags the discrepancy
The Question As Asked
49. Consider the following statements:
The purpose of hypothesis testing is to make a judgement about the difference between the sample statistic and the hypothesized population parameter.
When the Null hypothesis is rejected the alternate hypothesis is accepted.
The purpose of hypothesis testing is to test the correctness of the computed value of the sample statistic.
If the Null hypothesis is not rejected, this does not ensure that the Null hypothesis is true.
Which of the above statements are correct?
(a) 1, 2 and 3
(b) 1, 3 and 4
(c) 1, 2 and 4
(d) 2, 3 and 4
Step 1: Set Up The Framework We're Testing Each Statement Against
Before judging any of the four statements, it helps to write down, in plain words, what hypothesis testing actually is. In a hypothesis test:
We start with a null hypothesis H₀, a specific claim about an unknown population parameter (for example, a population mean μ equals some value μ₀).
We collect a random sample and compute a sample statistic (for example, the sample mean x̄).
We ask: is the gap between what we observed (x̄) and what H₀ claims (μ₀) small enough to be explained by ordinary sampling variability, or is it large enough that we conclude H₀ itself is probably wrong?
Based on that comparison, we either reject H₀(in favour of the alternative hypothesis H₁) or we fail to reject H₀.
With that skeleton in place, we examine each of the four statements on its own merits.
Step 2: Statement 1 — The Purpose Of The Test
"The purpose of hypothesis testing is to make a judgement about the difference between the sample statistic and the hypothesized population parameter."
This is, almost word for word, the standard textbook description of what a hypothesis test does. The entire test is built around one question: is the observed difference (x̄ − μ₀, or its standardized version) large enough to be statistically significant, or small enough to be dismissed as chance variation? That is exactly "making a judgement about the difference between the sample statistic and the hypothesized parameter."
Statement 1 is TRUE.
Step 3: Statement 2 — Rejecting H₀ And Accepting H₁
"When the Null hypothesis is rejected the alternate hypothesis is accepted."
In the classical (Neyman–Pearson) testing framework used throughout this syllabus, H₀ and H₁ are set up to exhaust the parameter space between them — together they cover every possibility the test is designed to distinguish. The sample space is correspondingly split into an acceptance region and a critical (rejection) region. If the test statistic falls in the critical region, we reject H₀, and by the very construction of the test, that is equivalent to accepting H₁. This is the standard operational meaning of "reject H₀" in this framework.
Statement 2 is TRUE.
Step 4: Statement 3 — The Decoy Restatement
"The purpose of hypothesis testing is to test the correctness of the computed value of the sample statistic."
Read this one slowly against Statement 1 — they look similar but they are not saying the same thing. The sample statistic (say, x̄ = 52.3 from your data) is simply a number you computed from your sample. There is nothing to "test" about whether that number is "correct" — it is exactly what the arithmetic on your sample produced; it isn't a claim that can be true or false. What hypothesis testing actually evaluates is whether the hypothesized population parameter is consistent with that observed statistic — the object under test is the parameter's hypothesized value, not the arithmetic correctness of the statistic itself. Statement 3 quietly swaps "population parameter" for "sample statistic," which changes the meaning and makes it incorrect.
Statement 3 is FALSE.
Pro Tip: Whenever two statements in a "which are correct" question look like near-paraphrases of each other (as 1 and 3 do here), assume the examiner has deliberately planted one genuine definition and one subtly broken mirror of it. The habit worth drilling until it becomes automatic — the way it's taught when students work through these statement-based MCQs systematically — is to ask "what noun did they swap?" rather than just pattern-matching on familiar vocabulary like "purpose of hypothesis testing."
Step 5: Statement 4 — Failing To Reject Is Not Proof
"If the Null hypothesis is not rejected, this does not ensure that the Null hypothesis is true."
This is one of the most heavily emphasized cautions in all of hypothesis testing, and for good reason. Not rejecting H₀ only tells you that your sample did not provide strong enough evidence against it — it does not prove H₀ is actually true. There are at least two reasons this caution is necessary:
Your sample size might simply be too small to detect a real difference, so you fail to reject H₀ purely due to low statistical power.
You could be committing a Type II error — failing to reject a null hypothesis that is, in reality, false.
Because of this, standard practice is to describe a non-significant result as "failing to reject H₀," deliberately avoiding the stronger phrase "accepting H₀ as true," precisely to keep this distinction visible. This statement is a textbook-correct description of that caution.
Statement 4 is TRUE.
Step 6: Combine The Four Verdicts
Putting the four verdicts together:
Statement 1 — TRUE
Statement 2 — TRUE
Statement 3 — FALSE
Statement 4 — TRUE
That gives us Statements 1, 2 and 4 as correct, which is Option (c).
Step 7: Checking Against The Official Answer Key — A Flagged Discrepancy
The official answer key published for ISS 2016 Statistics Paper-II (Test Booklet Series A) lists the answer to Question 49 as Option (a) — Statements 1, 2 and 3, which would mean the key treats Statement 3 as correct and Statement 4 as incorrect.
We rechecked our derivation carefully before writing this up, and we are not able to reconcile that with standard hypothesis-testing theory. Statement 4 — that a non-rejected null hypothesis has not been proven true — is one of the most consistently taught, non-controversial facts in all of inferential statistics; essentially every standard statistics text (and every treatment of Type II error) states this explicitly. Conversely, Statement 3 conflates "testing a hypothesis about a population parameter" with "testing the correctness of an already-computed sample statistic," which is not a coherent description of what a hypothesis test does — a computed statistic doesn't have a truth value to test.
Given this, we are publishing our own fully worked derivation —Option (c): Statements 1, 2 and 4 — as the statistically justified answer, while explicitly flagging that it disagrees with the official key's Option (a). If you have access to an alternate, verified version of the key, we'd genuinely like to compare notes in the comments.
Why This Question Matters
Questions like this one test something calculation-heavy questions can't: whether you actually understand the logic of hypothesis testing rather than just its formulas. ISS Paper-II leans on this kind of statement-based MCQ repeatedly, precisely because it's easy to recite the steps of a t-test or a chi-square test without ever being able to say, in one clean sentence, what a "rejection" or "non-rejection" decision does and does not tell you. Being able to spot a decoy restatement like Statement 3, and being confident enough in the Type II error caution to hold your ground against a key that looks inconsistent with it, is exactly the kind of conceptual sturdiness this exam is probing for.
Frequently Asked Questions
Is "accepting H₀" the same as "proving H₀ is true"?
No. Not rejecting H₀ only means the sample evidence wasn't strong enough to reject it; it does not establish that H₀ is actually true. This is precisely the distinction Statement 4 in this question is testing, and it's one of the most important caveats in all of hypothesis testing.
Why is rejecting H₀ treated as equivalent to accepting H₁?
In the classical Neyman–Pearson framework, H₀ and H₁ are set up so that together they cover the full range of possibilities the test is designed to distinguish between, and the sample space is split into an acceptance region and a rejection region accordingly. Landing in the rejection region for H₀ is therefore, by construction, the same decision as accepting H₁.
What's the real difference between a "population parameter" and a "sample statistic" in this context?
A population parameter (like the true population mean μ) is an unknown, fixed quantity we are making a claim about. A sample statistic (like the sample mean x̄) is a number we actually compute from our data. Hypothesis testing evaluates whether the hypothesized value of the parameter is plausible given the observed statistic — it is not a test of whether the statistic itself was computed "correctly."
What is a Type II error, and how does it relate to this question?
A Type II error happens when you fail to reject a null hypothesis that is, in reality, false. It's the direct reason Statement 4 is worded so carefully: a non-significant result could simply mean your test lacked the power to detect a real effect, not that no effect exists.
How should I approach "which statements are correct" questions under exam time pressure?
Evaluate each numbered statement completely independently first, writing TRUE or FALSE next to each one based purely on its own content, before you even look at the answer options. Only after you've judged all the statements should you match your TRUE/FALSE pattern against options (a)–(d). Looking at the options first tends to anchor your judgement and makes decoy statements like Statement 3 here much easier to miss.
Should I always trust the official answer key over my own derivation?
Treat the key as strong evidence, not as infallible — keys do occasionally contain transcription or printing errors. The right habit is to rederive the answer independently first, compare it to the key, and if there's still a disagreement after careful rechecking, trust well-established theory over a key that contradicts a near-universal statistical principle, while clearly noting the discrepancy rather than silently picking one side.
If you worked through this one differently, or you've seen another verified copy of the 2016 Paper-II answer key, drop a comment below — and if a fellow ISS aspirant is grinding through the same papers, send this one their way.
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