The amount of fill dispensed by a bottling machine is normally distributed with standard deviation =2. Suppose that the sample mean is to be computed using a sample of size n.

(a)Find the probability that the sample mean will be within 0.5 ounce of the true mean for n=16, 25, 36, 49.

(b)Plot the probabilities you computed in part (a) as a function of n. Use R to generate the plot and make sure to attach your code at the end of the homework.

(c)What pattern do you observe for the various values of n? Provide an intuitive explanation along with your answer.

(d) What sample size is required if we wish the sample mean to be within 0.5 ounce of the true mean with probability 0.95.