4.28 (Examination question, University of Glasgow.)

In an industrial bottling plant, labels are glued to the bottles at the final stage of the process. Unfortunately, the shape of the bottles used in the plant sometimes leads to bottles not being held completely straight by the labelling machine, with the result that the label is sometimes glued at a slight angle. A quality control inspector has taken 20 samples, each of 25 bottles, at various times throughout the day. The numbers of poorly labelled bottles are shown below.

No. of poorly labelled bottles 0 1 2 3 4 5 6 Frequency 1 3 3 6 5 1 1 Assume that the number of poorly labelled bottles in each batch of 25 can be modelled by a Bin(25, θ) random variable.

1. Write down the likelihood and obtain the maximum likelihood estimate ˆθ.

2. Explain how to construct an approximate 95% likelihood-ratio-based confidence interval for θ, and describe a Newton-Raphson procedure to determine the end-points of this interval.