You work as a Quality Control Technician for a manufacturer of kitchen countertops and your job is to randomly sample countertops to ensure product consistency. The machining time for each countertop is, on average, 20.2 seconds for a particular machine centre.

a)if it is known that the standard deviation is 4 seconds, what is the probability that the sample mean will lie between 19.2 seconds and 21.2 seconds in a sample of 4 countertops?

b)if it is known that the standard deviation is 4 seconds, what is the probability that the sample mean will lie between 19.2 seconds and 21.2 seconds in a sample of 16 countertops?

c)Based on your answers to questions a) and b), why are the confidence intervals different? Explain what happened when the sample size increased from 4 to 16. It is known that 80% of the countertops are completely defect-free

d)If a sample of 40 countertops is taken, what is the probability that less than 70% are completely defect-free?

e)If a sample of 40 countertops is taken, what is the probability that more than 36 are completely defect-free?

It is acceptable for the countertop thickness to be slightly oversized. The average amount that a countertop is oversized is 4 mm with a known standard deviation of 2 mm. You measure 16 countertops and find that the dimensions are oversized by an average of 4.8 mm.

f) Find the 95% and 99% confidence intervals for the oversize.

g)Is there a quality problem here? Justify your answer based on the results you got from the 95% and 99% confidence intervals

h) Show the 95% confidence interval assuming that the above standard deviation of 2 mm was not known, but came from a sample of 16 countertops.

i) Use your 95% confidence intervals obtained from question f) and h) to explain how the z-distribution and the t-distribution are related