Exam Question

A large batch of electrical components contains a proportion $\theta$ which are defective and not repairable, a proportion $3\theta$ which are defective but repairable and a proportion $1-4\theta$ which are satisfactory.

• (a)

  What values of $\theta$ are admissible?

Fifty components are selected at random (with replacement) from the batch, of which 2 are defective and not repairable, 5 are defective and repairable and 43 are satisfactory.

• (b)

  Write down the likelihood function, $L(\theta )$ and make a rough sketch of it.

• (c)

  Obtain the maximum likelihood estimate of $\theta$.

• (d)

  Obtain an approximate 95% confidence interval for $\theta$. A value of $\theta$ equal to 0.02 is believed to represent acceptable quality for the batch. Do the data support the conclusion that the batch is of acceptable quality?