ISS 2016 Statistics Paper-2 Solution: Question 33 (Type-I Error, Power, and Tailed Hypotheses)
This post continues our ongoing question-by-question walkthrough of ISS Statistics Paper-2 previous year papers, where we pick up the very next question in sequence and solve it completely from first principles. Today we are on Question 33 of the ISS 2016 Statistics Paper-2 test booklet, which tests three of the most commonly confused ideas in hypothesis testing all in one go.
Quick Summary
Topic:Testing of Hypotheses — Type-I error, Power of a test, and one-tailed vs two-tailed alternative hypotheses
Question Reference:ISS 2016, Statistics Paper-2, Question 33
Correct Answer:Option (a) — Statements 1 and 2 only
The Question, Exactly As Asked
Consider the following statements :
Size of Type-I error is P {rejecting H0 | H0 is true}
Power of a test is P {rejecting H0 | H1 is true}
H1 : μ ≠ μ0 is a right tailed hypothesis
Which of the statements given above are correct?
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
Step 1: Set Up the Hypothesis Testing Framework From Scratch
Before touching the three statements, let's write down the four possible outcomes whenever we test a null hypothesis H0 against an alternative hypothesis H1. There are exactly two things that can be true in reality (H0 is true, or H1 is true), and exactly two decisions we can make (reject H0, or do not reject H0). That gives four combinations:
H0 is true, and we do not reject it → correct decision
H0 is true, but we reject it anyway → this is called Type-I error
H0 is false (i.e. H1 is true), but we fail to reject H0 → this is called Type-II error
H0 is false, and we correctly reject it → correct decision
Every quantity in this question is just a probability attached to one of these four rows, so once this table is clear in your head, the rest of the question is bookkeeping.
Step 2: Check Statement 1 — Size of Type-I Error
By the very definition we just wrote down, a Type-I error happens when H0 is actually true but our test rejects it. In probability notation, the "size" of the test (usually written as α) is defined as the probability of this exact event:
α = P {reject H0 | H0 is true}
This is exactly what Statement 1 says, word for word. So Statement 1 is correct.
Step 3: Check Statement 2 — Power of the Test
Now look at the opposite good outcome: H0 is actually false (equivalently, H1 is true), and our test correctly rejects H0. The probability of catching a false null hypothesis is called the power of the test, and it is defined as:
Power = P {reject H0 | H1 is true} = 1 − β
where β is the probability of a Type-II error (failing to reject a false H0). Statement 2 states precisely this definition. So Statement 2 is correct as well.
Notice the clean symmetry: Type-I error size is a probability computed under H0, while power is a probability computed under H1. Keeping straight which hypothesis you are conditioning on is really the entire battle in this topic.
Step 4: Check Statement 3 — Is μ ≠ μ0 a Right-Tailed Hypothesis?
This is where the question tries to trip you up, so let's go slowly. When we write an alternative hypothesis, its exact form tells us where the rejection region sits on the number line:
H1 : μ > μ0 — we only suspect the true value is larger, so we reject H0 only for large values of the test statistic. The rejection region sits entirely in the right tail. This is a right-tailed (one-sided) test.
H1 : μ < μ0 — symmetric story, rejection region sits entirely in the left tail. This is a left-tailed (one-sided) test.
H1 : μ ≠ μ0 — here we are not claiming any direction at all. The true mean could be higher or lower than μ0, and either extreme should lead us to reject H0. So the rejection region is split between both tails.
Because H1 : μ ≠ μ0 puts rejection regions on both sides, it is called a two-tailed (two-sided) hypothesis, not a right-tailed one. Statement 3 calls it right-tailed, which is simply incorrect terminology — it has confused "not equal to" with "greater than". So Statement 3 is false.
Pro Tip: Whenever you see ≠ in an alternative hypothesis, your first instinct should be "two-tailed" — no exceptions. Save "right-tailed" strictly for a strict > sign and "left-tailed" strictly for a strict < sign. This one-line rule is drilled so often in a good classroom that it becomes automatic, and it alone is enough to eliminate options (b) and (c) in this question without doing any other work.
Step 5: Combine the Three Statements
We found:
Statement 1 — Correct
Statement 2 — Correct
Statement 3 — Incorrect
So the correct statements are 1 and 2 only, which is exactly option (a).
Step 6: Confirm Against the Official Answer Key
The official ISS 2016 Statistics Paper-2 answer key lists the answer to Question 33 as (a), which matches our step-by-step derivation exactly. No discrepancy here — both the independent working and the official key agree.
Why This Question Matters
This question is a favourite examiner trick because it mixes two things you must know cold (the definitions of Type-I error and power) with one thing that is easy to mix up under exam pressure (tail direction of a hypothesis). In the actual exam hall, a candidate who has only memorised formulas without understanding what "conditioning on H0 versus H1" really means will often flip statements 1 and 2, while a candidate who has never been explicitly warned about the ≠ sign will often mark statement 3 as correct too. Getting comfortable with this exact kind of statement-based MCQ — where each individual line looks plausible on its own — is one of the most efficient ways to build accuracy for the Testing of Hypotheses portion of the syllabus, and it's a pattern that repeats across nearly every year of this paper.
Frequently Asked Questions
What is the difference between the size of a test and the power of a test?
The size (or level of significance, α) is the probability of wrongly rejecting a true H0, so it is computed assuming H0 is true. The power is the probability of correctly rejecting a false H0, so it is computed assuming H1 is true. They are probabilities of two different events under two different assumed states of the world.
Why is H1 : μ ≠ μ0 always two-tailed and never right-tailed?
Because "not equal to" allows the true parameter to lie above or below μ0, evidence of either kind should count against H0. That forces the rejection region to be split across both tails of the sampling distribution, which is exactly the definition of a two-tailed test.
How is Type-II error related to power?
Type-II error, denoted β, is the probability of failing to reject a false H0. Power is the complementary probability of correctly rejecting a false H0, so Power = 1 − β. A test with high power has a correspondingly low Type-II error rate.
Does increasing sample size affect the size (α) of a test?
No. The size α is fixed by the analyst in advance (commonly 0.05 or 0.01) as part of choosing the critical region, and it does not automatically change with sample size. What does typically improve with a larger sample size is the power of the test, since a larger sample gives a more precise estimate and makes it easier to detect a genuinely false H0.
Is there a quick way to remember which tail a one-sided test uses?
Yes — look at the inequality sign in H1 and picture it as an arrow pointing toward the tail you reject in. A ">" sign points right, so you reject in the right tail; a "<" sign points left, so you reject in the left tail. A "≠" sign points in both directions at once, which is the visual cue for a two-tailed test.
Why do ISS papers repeatedly test this same set of definitions?
Testing of Hypotheses is a core theory block that underlies large-sample tests, SPRT, and likelihood ratio tests elsewhere in the same paper, so examiners treat these foundational definitions as must-know building blocks and test them from slightly different angles year after year.
If anything here felt unclear, or if you solved this one differently, drop a comment below — and if you know a fellow ISS aspirant who is working through the same papers, feel free to share this post with them.
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