Question

Let $X$ be a uniformly distributed random variable in $[0, b]$. If the critical region for testing the null hypothesis $H_0: b = 2$ against the alternative $H_A: b \ne 2$ is $\{x \le 0.1 \text{ or } x \ge 1.9\}$, where $x$ is the value of a single draw of $X$, then the probability of Type-I error is

• A. 0.2
• B. 0.1
• C. 0.05
• D. 0.01

Hint:

In a uniform distribution, tail probabilities are directly proportional to the length of the critical region. Always divide by total range to get the probability.

Answer 1

The Correct Option is B

Step 1: Under $H_0$, $X \sim U(0,2)$. Probability density = $\frac{1}{2}$ over $[0,2]$.
Step 2: Type-I error probability.
\[ P(X \le 0.1 \text{ or } X \ge 1.9) = P(X \le 0.1) + P(X \ge 1.9) \] \[ = \frac{0.1}{2} + \frac{2 - 1.9}{2} = \frac{0.1}{2} + \frac{0.1}{2} = 0.1. \] Wait — this gives $0.1$. But each side area is 0.05 of the range (since 0.1/2 = 0.05). Thus, total = 0.05 + 0.05 = 0.1. Correction: The total area = $0.1$, but half on each tail means 0.05 each side if total probability = 0.1. Hence, Type-I error = 0.1. The question’s intended correct value = **0.1**, matching Option (B).