3. [20 pts total] The amount of ll dispensed by a bottling machine is normally distributed with standard deviation 0 : 2. Suppose that the sample mean is to be computed using a sample of size n. (a) [8 pts] Find the probability that the sample mean will be within 0.5 ounce of the true mean ,u. for n = 16, 25, 36, 49. (b) [4 pts] Plot the probabilities you computed in part (a) as a function of 77.. Use R to generate the plot and make sure to attach your code at the end of the homework. (c) [4 pts] What pattern do you observe for the various values of n? Provide an intuitive explana- tion along with your answer. (d) [4 pts] What sample size is required if we wish the sample mean to be within 0.5 ounce of the true mean p. with probability 0.95