Question

The observed value of a random sample of size \(9\) from a distribution having continuous and strictly increasing cumulative distribution function is as below:

Hint:

In a one-sided sign test for \(H_1:M>0\), count the number of positive signs and calculate the upper-tail binomial probability under \(H_0\).

Answer 1

Correct Answer: 1.5

Step 1: Count the positive signs.
There are $5$ positive and $4$ negative observations, so the sign statistic is $S=5$.

Step 2: Null distribution.
Under $H_0:M=0$, $S\sim$ Binomial$(9,\frac12)$, and large $S$ favours $H_1:M>0$.

Step 3: $p$-value.
$p=P(S\ge5)$. By the symmetry of Binomial$(9,\frac12)$, $P(S\ge5)=\frac12$, so $p=0.50$.

Step 4: Decision.
Since $0.50>0.05$, do not reject $H_0$, so $\eta=1$.

Step 5: Add.
$p+\eta=0.50+1=1.50$.
\[ \boxed{1.50} \]