A machine is designed to fill automobile tires to a mean air pressure of 30 pounds per square inch (psi). The manufacturer tests the machine on a random sample of 12 tires. The mean air pressure for these 12 tires is calculated to be 30.5 psi. Fill pressures for the machine are known to follow a normal distribution with standard deviation 1.2 psi. (a) Construct a 90% confidence interval for the true mean fill pressure for this machine. (b) Provide an interpretation of the confidence interval in (a). (c) Conduct a hypothesis test at the 10% level of significance to determine whether there is significant evidence that the true mean fill pressure differs from 30 psi. Show all your steps. (d) Provide an interpretation of the P-value of the test in (c). (e) Could the confidence interval in (a) have been used to conduct the test in (c)? Why or why not? If it could be used, what would the conclusion be, and why? If it could not be used, explain why not.