A company produces bottled water. The volume of water the bottling machine puts into each bottle is distributed normally with a known standard deviation of 10 ml. Each bottle needs to be filled to 500 ml, but small variations to either side are acceptable. In a random sample of 5 bottles, the following volumes are observed: 502, 529, 454, 510, 545. (a) Does it appear that the machine is working as expected? Use 0.10 significance. (b) Give the p-value for your test in (a). (c) Give the 90% confidence interval for the mean fill amount. Explain how it matches your answer to (a). (d) The company starts a campaign, putting rewards under some bottle caps: a motivational message (with probability 10%), a discount code (5%), or a code to redeem a small prize (1%). In a random sample of 500 bottles, they observe 41 motivational messages, 18 discount codes, and 9 prize codes. Do the rewards appear distributed as expected at 0.10 significance? At 0.05 significance? Give the p-value