Question

Let \( X_1, X_2, \dots, X_{10} \) be a random sample from a probability density function

\[ f_\theta(x) = f(x - \theta), \quad -\infty<x<\infty, \]

where \( -\infty<\theta<\infty \) and \( f(-x) = f(x) \) for \( -\infty<x<\infty \). For testing

\[ H_0: \theta = 1.2 \quad \text{against} \quad H_1: \theta \neq 1.2, \]

let \( T^+ \) denote the Wilcoxon Signed-rank test statistic. If \( \eta \) denotes the probability of the event \( \{T^+<50\} \) under \( H_0 \), then \( 32\eta \) equals

\[ \underline{\hspace{2cm}} \]

(round off to 2 decimal places).

Hint:

For the Wilcoxon Signed-rank test, calculate the ranks and their signs to compute the test statistic, then use the CDF to find the probability.

Answer 1

Correct Answer: 31.6

The Wilcoxon Signed-rank test statistic \( T^+ \) is based on the ranks of the sample values and their signs. To find the value of \( \eta \), we calculate the cumulative distribution function (CDF) under \( H_0 \) and use it to find the probability: \[ 32 \eta = 31.70. \] Thus, the value is \( 31.70 \).