Question

Let X be a random sample of size 1 from a population with cumulative distribution function \[ F(x) = \begin{cases} 0 & \text{if } x < 0, \\ 1 - (1 - x)^\theta & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x \geq 1 \end{cases} \] where \( \theta > 0 \) is an unknown parameter. To test \( H_0 : \theta = 1 \) against \( H_1 : \theta = 2 \), consider using the critical region \( \{ x \in \mathbb{R} : x < 0.5 \} \). If \( \alpha \) and \( \beta \) denote the level and power of the test, respectively, then \( \alpha + \beta \) (rounded off to two decimal places) equals:

Hint:

- The level \( \alpha \) is the probability of rejecting \( H_0 \) when \( H_0 \) is true, and the power \( \beta \) is the probability of rejecting \( H_0 \) when \( H_1 \) is true.
- The cumulative distribution function (CDF) gives the probability of the random variable being less than or equal to a specific value.

Answer 1

The critical region is \( \{ x \in \mathbb{R} : x<0.5 \} \), meaning we reject \( H_0 \) if \( x<0.5 \). - The level \( \alpha \) is the probability of rejecting \( H_0 \) when \( H_0 \) is true. This is \( P(X<0.5 | \theta = 1) \). For \( \theta = 1 \), the CDF is: \[ F(x) = 1 - (1 - x)^\theta \] Substituting \( \theta = 1 \): \[ F(x) = 1 - (1 - x) \] Thus, \[ P(X<0.5 | \theta = 1) = F(0.5) = 1 - (1 - 0.5) = 0.5 \] Therefore, \( \alpha = 0.5 \). - The power \( \beta \) is the probability of rejecting \( H_0 \) when \( H_1 \) is true. This is \( P(X<0.5 | \theta = 2) \). For \( \theta = 2 \), the CDF is: \[ F(x) = 1 - (1 - x)^2 \] Thus, \[ P(X<0.5 | \theta = 2) = F(0.5) = 1 - (1 - 0.5)^2 = 1 - 0.25 = 0.75 \] Therefore, the power \( \beta = 0.75 \). Thus, \( \alpha + \beta = 0.5 + 0.75 = 1.20 \).