(20 points) A bottling company uses a machine to fill bottles with a tasty beverage. The bottles are advertised to contain 300 milliliters (ml), but in reality, the amount varies according to a normal distribution with mean u=298 ml and standard deviation o=3. For this problem, let us assume 6 is known. a What is the probability that a randomly chosen bottle contains less than 296 .5 m1? 1). Given a simple random sample of size n=5 bottles, what is the probability that the sample mean is less than 296.5 ml? c. What is the probability that a single bottle is lled within 4 ml of the true mean u=298 ml? Hint: Add +4 units and -4 units to the mean to calculate. d. What is the probability that the mean of 9 randomly selected bottles is within 4 ml of the mean? What about the mean of a sample of n=30 bottles? (10 points) A social worker is interested in estimating the average length of time spent outside of prison for rst offenders who later commit a second crime and are sent to prison again. A random sample of n=35 prison records in the county courthouse indicates that the average length of prison-free life between rst and second offenses is 4.2 years, with a sample standard deviation of 1.1 years. Use this information to construct a 90% condence interval for u, the average time between rst and second offenses for all prisoners on record in the county courthouse. Incorporate the values you obtained in a sentence that explains what the condence interval is telling you