Hello could you please help with my question ? Please show working as well?

Question 1

(i)

The label on a bottle of liquid detergent shows the contents to be 355ml per bottle. The production operation fills the bottle uniformly according to the following probability density function:

(a) What is the probability that a bottle will be filled with between 355ml and 357ml?

(b) What is the probability that a bottle will be filled with354ml or more?

(c) Quality control accepts production that is within 0.6 ml of the number of mls shown on the container label. What is the probability that a bottle of this liquid detergent will fail to meet the quality control standard?

(ii)

A university library checks out an average of = 320 books per day, with a standard deviation of = 75 books. Consider a sample of 30 days of operation, with being the sample mean number of books checked out per day.

(a) What is the shape of the sampling distribution of ? Why?

(b) What is the standard deviation of ?

(c)What is the probability that the sample mean for the 30 days will be between 300 and 340 books?

(d) What is the probability the sample mean will show 325 or more books checked out?

(iii)

Assume that 15% of the items produced in an assembly line operation are defective, but the firm's production manager is not aware of this situation. Assume further that 50 parts are tested by the quality assurance department in order to determine the quality of the assembly operation. Let be the sample proportion defective found by the quality assurance test.

(a)What is the sampling distribution for?

(b)What is the probability that the sample proportion will be within 0.03 of the population proportion defective?

(c) If the test shows = 0.10 or more, the assembly line operation will be shut down to check the cause of the defects. What is the probability that the sample of 50 parts will lead to the conclusion that the assembly line should be shut down?