2 marks, −0.667StatisticsHypothesis testingType I and type II errors and powerModerateabout 1 min

IIT JAM 2021 Question 15: Hypothesis testing

Let $X$ be uniform on $[0, b]$ . Testing $H_0: b = 2$ against $H_A: b \neq 2$ uses the critical region $\{x \le 0.1 \text{ or } x \ge 1.9\}$ from a single draw $x$ . The probability of Type-I error is

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Answer and worked solution

Answer: (B)

Under $H_0$ , $X \sim U[0,2]$ with density $\tfrac12$ . Type-I error is the probability the draw lands in the rejection region when $H_0$ is true.

$P(X \le 0.1) = \tfrac{0.1}{2} = 0.05$ and $P(X \ge 1.9) = \tfrac{0.1}{2} = 0.05$ , summing to $0.10$ . The answer is (B).

The trap: (C) 0.05

Only one tail (0.05) is easy to compute; the rejection region has two tails, so both must be added to reach 0.10.

Plain-text version

Let X be uniform on [0, b]. Testing H₀: b = 2 against H_A: b ≠ 2 uses the critical region {x ≤ 0.1 or x ≥ 1.9} from a single draw x. The probability of Type-I error is (A) 0.2 (B) 0.1 (C) 0.05 (D) 0.01

Also written as: Let X be uniform on [0, b]. Testing H_0: b = 2 against H_A: b != 2 uses the critical region {x <= 0.1 or x >= 1.9} from a single draw x. The probability of Type-I error is

Hypothesis testing has come up 47 times in 7 exams since 2007.

What this question tests

This question is on Type I and type II errors and power, part of Hypothesis testing in Statistics.

The two ways a test can be wrong and the probability of detecting a real effect.

TestStop has 15 past-paper questions on this subtopic; the nearest in year are below.

IIT JAM 2022 · Q8In hypothesis testing, which of the following defines the size of the power of the test?

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