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: Understand what the test is checking.
We have five values from a continuous distribution with median \(\theta_{1/2}\). We test \[ H_0:\theta_{1/2}=2.5 \quad \text{against} \quad H_1:\theta_{1/2}>2.5. \]

Step 2: Link order statistics to a count.
Let \(S\) be the number of observations bigger than \(2.5\). \(X_{(r)}>2.5\) iff \(S\ge 5-r+1\).

Step 3: Find the distribution of \(S\) under \(H_0\).
Under \(H_0\), \(S\sim\text{Binomial}(5,0.5)\).

Step 4: Work out the size of each region.
\(X_{(1)}>2.5\): \(S\ge5\), \(P=1/32=0.03125\). \(X_{(2)}>2.5\): \(S\ge4\), \(P=6/32=0.1875\). \(X_{(3)}>2.5\): \(S\ge3\), \(P=16/32=0.5\). \(X_{(4)}>2.5\): \(S\ge2\), \(P=0.8125\).

Step 5: Pick the region with size at most \(0.05\).
Only \(X_{(1)}>2.5\) gives size below \(0.05\).

Final Answer: \[ \boxed{X_{(1)}>2.5} \]