Question

A box contains 8 balls, \(\theta\) of which are white. The null hypothesis \(H_0: \theta = 3\) is tested by drawing 2 balls at random without replacement. 

The hypothesis is rejected in favour of the alternative hypothesis \(H_1: \theta > 3\) if both balls are white.

What is the significance level of the test?

• A. $\frac{3}{56}$
• B. $\frac{3}{28}$
• C. $\frac{1}{28}$
• D. $\frac{1}{56}$

Hint:

Level of significance is just "P(Rejection Region) calculated using the null parameter value." Here, simply calculate the probability of the event "both white" assuming there are exactly 3 white balls.

Answer 1

The Correct Option is B

The significance level ($\alpha$) is the probability of rejecting the null hypothesis when it is actually true. 

Step 1:Define the Rejection Rule
The test rejects $H_0$ if both balls drawn are white. 

Step 2:Identify the Condition under $H_0$
Under $H_0$, the number of white balls is $\theta = 3$. 
The total number of balls in the box is 8. 

Step 3: Calculate the Probability of Rejection
We need to find the probability of drawing 2 white balls from a box containing 3 white and 5 non-white balls, without replacement. 
Total ways to draw 2 balls from 8: $\binom{8}{2} = \frac{8 \times 7}{2} = 28$. 
Ways to draw 2 white balls from 3: $\binom{3}{2} = 3$. 

Step 4: Find the Significance Level
$\alpha = P(\text{Reject } H_0 | H_0 \text{ is true})$ 
$\alpha = \frac{\text{Ways to draw 2 white balls}}{\text{Total ways to draw 2 balls}}$ 
$\alpha = \frac{3}{28}$. 
The significance level is $3/28$.