ISS 2016 Statistics Paper-2 Solution: Question 11 (Setting Up ANOVA Hypotheses)
Picking up the ISS Statistics Paper-2 PYQ series where the last post left off, this entry works through the eleventh item of the 2016 paper. It looks like a "two-second" question on paper, but it hides one of the most commonly mis-stated ideas in all of hypothesis testing, so it is worth slowing down for.
Quick Summary
Topic:Setting up the null and alternative hypothesis for one-way ANOVA
Question Reference:ISS 2016, Statistics Paper-2, Question 11
Correct Answer:(a) 1 and 3
The Question, As Asked
For the next three (03) items that follow:
A company used three different methods to train its employees. The number of units of output produced by different employees trained by the three training methods is given below:
Method A : 50, 45, 55, 44
Method B : 64, 48, 52, 56, 44
Method C : 46, 42, 48, 45, 57, 42
It has to be tested whether the three different methods are satisfactory at 5% level of significance (Given F2, 12 at 0.05 = 3.89 as the critical value).
Question 11.Which of the following is/are the null and alternate hypothesis for testing whether the mean number of units produced per hour is different for the three different training methods?
H0 : μA = μB = μC
H0 : μA ≠ μB ≠ μC
H1 : μA, μB and μC are not all equal
H1 : μA = μB = μC
Select the correct answer using the code given below:
(a) 1 and 3
(b) 1 and 4
(c) 1 and 2
(d) 2 and 3
Step 1: Recall What a Null Hypothesis Is Actually Allowed to Say
Before touching this specific question, fix the rule that governs every hypothesis test you will ever set up: the null hypothesis, written H0, must always be a single, exact, testable statement — usually a statement of "no difference," "no effect," or "equality." It can never be phrased as an inequality like "not equal to," because you cannot compute a sampling distribution, a test statistic, or a p-value under a vague claim such as "the means are different." A test statistic needs one fixed numerical scenario to test against, and only an equality gives you that.
So immediately, any option that writes H0 using "≠" is structurally wrong, no matter what it is claiming. That rules out statement 2 on sight.
Step 2: Identify the Correct Null Hypothesis for This Problem
We are comparing three training methods (A, B, C) and asking whether their true population mean outputs are the same or not. Since this is a one-way ANOVA setup with three groups, the null hypothesis is that all three population means are equal:
H0 : μA = μB = μC
This is exactly statement 1. It says nothing has changed between training methods — any differences we see in the sample means (50.83 for A, 52.8 for B, 46.67 for C, if you compute them) are assumed, under H0, to be pure sampling fluctuation rather than a real effect of the training method.
Step 3: Identify the Correct Alternative Hypothesis
This is where most students lose the mark, and it is the trickiest part of this question. The natural instinct is to write the alternative as the "opposite" of the null in the most literal sense — that is, "all three means are different from each other," i.e., μA ≠ μB ≠ μC. That instinct is wrong for ANOVA.
Pro Tip: The alternative hypothesis in ANOVA is
The correctly worded alternative is statement 3:
H1 : μA, μB and μC are not all equal
Statement 4 can be eliminated immediately too — it is word-for-word identical to the null hypothesis, and a hypothesis test cannot have H0 and H1 saying the same thing.
Step 4: Match the Statements to the Answer Codes
Putting Steps 2 and 3 together: the correct null is statement 1, and the correct alternative is statement 3. Scanning the four answer codes:
(a) 1 and 3 — correct null, correct alternative ✓
(b) 1 and 4 — correct null, but the "alternative" duplicates the null ✗
(c) 1 and 2 — correct null, but statement 2 is not even a valid null (and is being wrongly offered as if paired with 1) ✗
(d) 2 and 3 — starts from an invalid null statement ✗
So the answer is (a) 1 and 3.
Final Answer and Confirmation
Working through the logic of what a null hypothesis is permitted to state, and what "not all equal" means versus "all different," gives option (a) 1 and 3 as the correct choice. This matches the official answer key for ISS 2016 Statistics Paper-2, Question 11, which also lists the answer as A. Our independent derivation and the official key agree — no discrepancy here.
Why This Question Matters
This question tests something more fundamental than computation: whether you actually understand what a hypothesis test is claiming before you ever touch an F-statistic or a p-value. In the ISS exam, ANOVA questions almost always come bundled as a set (notice this question is the first of "the next three items that follow," with Questions 12 to 15 all built on the same training-method data). Getting the hypothesis statement wrong at the very first step means every later interpretation — "do we reject H0," "are the methods significantly different" — collapses too, even if your arithmetic for the F-statistic itself is flawless. Examiners plant this exact "all different vs. not all equal" trap repeatedly across years precisely because it separates candidates who memorized ANOVA mechanically from those who understand it.
Frequently Asked Questions
Why can't the null hypothesis be written with a "not equal to" sign?
A null hypothesis must describe one exact, calculable scenario so that a sampling distribution and test statistic can be derived under it. "Not equal to" describes infinitely many possible scenarios, not one, so no single test statistic distribution could ever be computed from it. This is a structural rule that applies to every hypothesis test, not just ANOVA.
Is "all means are different" ever a valid alternative hypothesis in any test?
For a two-sample test comparing only two means, "μ1 ≠ μ2" is perfectly valid because there is no middle ground — either they are equal or they aren't. The trouble starts with three or more groups, where "different from each other" and "not all equal" stop meaning the same thing, and only the second phrasing is correct for ANOVA.
What if only two of the three training methods actually differ — does H0 still get rejected?
Yes. As long as even one population mean differs from the rest, the statement "μA = μB = μC" is false, so H0 is false and H1 ("not all equal") is true. The ANOVA F-test itself only tells you that at least one difference exists somewhere; identifying which specific pair differs requires a separate post-hoc test, not the ANOVA F-test.
Does rejecting H0 in ANOVA tell us which method is best?
No. Rejecting H0 only tells you that the three population means are not all equal — it does not say which one is highest, lowest, or which pairs differ. Ranking the methods or isolating the differing pair needs a follow-up procedure such as Tukey's HSD test, run only after the ANOVA F-test has already rejected H0.
Why does this question mention "at 5% level of significance" and a critical F value if it's only asking about the hypotheses?
That information belongs to the shared "next three items" block and is needed for the sibling questions (12 to 15) in the same set, which ask you to actually compute variance estimates and the F-statistic. Question 11 itself only needs the correctly worded H0 and H1 — the significance level and critical value become relevant once you reach the decision step in the later questions.
Is this the same hypothesis structure used for regression F-tests too?
Conceptually yes — any F-test built on comparing "restricted vs. unrestricted" models follows the same logic: H0 is a single equality (or set of equalities) describing the restricted case, and H1 is simply "H0 is not true," never a stronger claim than that. Once this pattern is internalized here, it carries over directly to the joint-significance F-tests you'll see later in the regression portion of this same paper.
If any step above felt shaky, drop a comment below — doubts on exactly where the "not all equal" wording trips people up are common, and working through them in the comments helps the next aspirant reading this too. Feel free to share this with a fellow ISS aspirant who is revising ANOVA basics.
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