Question

Let \(X_1, X_2, \ldots, X_5\) be random observations from a continuous distribution. Let \(\theta_p\) be the \(p\)-th population quantile. Consider the following hypotheses
\[ H_0:\theta_{1/2}=2.5 \quad \text{against} \quad H_1:\theta_{1/2}>2.5. \]
Let \(X_{(r)}\) denote the \(r\)-th order statistic of the given observations. Then which of the following is a critical region of a level \(0.05\) test?

• A. \(X_{(1)}>2.5\)
• B. \(X_{(2)}>2.5\)
• C. \(X_{(3)}>2.5\)
• D. \(X_{(4)}>2.5\)

Hint:

Convert the order statistic event into a count of observations exceeding 2.5, then use that this count is Binomial(5, 0.5) under H0.

Answer 1

The Correct Option is A

Step 1: Think of this as a sign test on the median.
Under $H_0$, each of the five observations has a $50\%$ chance of exceeding $2.5$. Let $S$ count how many exceed it; $S\sim$Binomial$(5,0.5)$.

Step 2: Read each option as a statement about $S$.
$X_{(1)}>2.5$ needs $S=5$; $X_{(2)}>2.5$ needs $S\ge4$; $X_{(3)}>2.5$ needs $S\ge3$; $X_{(4)}>2.5$ needs $S\ge2$.

Step 3: Compute the false alarm rate for option (A).
$P(S=5)=(0.5)^5=1/32\approx0.031$, below $0.05$.

Step 4: Show the others break the level.
$X_{(2)}>2.5$: $P(S\ge4)=6/32\approx0.188$, too big; $X_{(3)},X_{(4)}$ even bigger.

Step 5: Conclude. \[ \boxed{X_{(1)}>2.5} \]