Question:

(a) Packs of food have weights that are distributed with standard deviation 2.3 g. A random sample of 200 packets is taken. The mean weight of this sample is found to be 99.2 g. Calculate a 99% confidence interval for the population mean weight.

(b) Insects stick to a wall at random points. The mean number of insects is 2 per square metre. A wall with area 25 m² is painted with a new type of paint which the manufacturer claims is insect repellent. It is found that 29 insects stick to this wall.

Use a suitable approximation to test the manufacturer's claim at the 1% significance level. Take the null hypothesis to be µ = 50, where µ is the population mean.

(c) Bottles of liquor are stacked in racks of 12. The weights of these bottles are normally distributed with mean 1.2 Kg and standard deviation 0.055 Kg. The weights of empty racks are normally distributed with mean 2 Kg and standard deviation 0.295 Kg.

(i) Show that the total weight of a full rack of 12 bottles of liquor follow a normal distribution with mean 16.4 Kg and variance 0.123.

(ii) Find the probability that the total weight of a full rack of 12 bottles of liquor is between 16.5 Kg and 17 Kg.

(iii) Two bottles of liquor are chosen at random. Find the probability that they differ in weight by less than 0.04 Kg.