1. A company sells toothpaste in a tube advertised to contain 8 ounces. The tube filling process is set with a mean of 8.1764 ounces. In this continuous production process, the amount of toothpaste put in a tube is normally distributed with a mean of 8.1764 ounces and a standard deviation of 0.0900 ounces.

(a) If the actual capacity of the tubes used is 8.3764 ounces, what proportion of the tubes do we expect to be filled beyond capacity?

(b) What proportion of the tubes do we expect to contain less than the advertised 8 ounces?

(c) In the normal distribution the probability that X > μ + 2σ is the same as the probability that X < μ - 2σ where μ and σ are the population mean and standard deviation respectively.  What characteristic of the normal distribution guarantees this is true?

2. A production engineer is checking the volume of paint in cans as the cans leave the filling line.  The machine is set to fill the cans with 4 litres of paint.  However, the engineer wishes to determine if the cans are being filled with less than 4 litres of paint.  A sample of n=36 cans is taken from the production line.  The sample shows a mean of X=3.97 litres and a standard deviation of S=0.14 litres.

Explaining each step, answer the following:

(a) Test the hypothesis at the 1 per cent level of significance that the paint cans are on average underweight (i.e. test if the sample mean is less than 4 litres).

(b) Construct the 95 % confidence interval for μ= the mean quantity of paint in each can.

need full answer of those 2 question