1. For a data set obtained from a random sample, n = 81 and $ = 48.25. It is known that o = 4.8. a) What is the point estimate of u? b) Make a 95% confidence interval for L. 2. Lazarus Steel Corporation produces iron rods that are supposed to be 36 inches long. The machine that makes these rods does not produce each rod exactly 36 inches long; the lengths of the rods vary slightly. Based on a past study, it is assumed that the standard deviation of the lengths of all rods produced on this machine is equal to 0.10 inch and the lengths of all such rods follow a normal distribution. The quality control department takes a random sample of 20 such rods every week, calculates the mean length of these rods, and makes a 99% confidence interval for the population mean. A recent sample of 20 rods produced a mean length of 36.02 inches. a) Suppose you are in the quality control team, makes a 99% confidence interval for the mean length of all rods made on this machine. b) If either the upper limit of this confidence interval is greater than 36.05 inches or the lower limit of this confidence interval is less than 35.95 inches, the machine is stopped and adjusted. Based on this sample, will you conclude that the machine needs an adjustment? 3. A random sample of 65 employees selected from financial companies in New York City showed that they received an average bonus of $55,000 last year with a standard deviation of $18,000. Construct a 95% confidence interval for the average bonus that all employees working for financial companies in New York City received last year. 4. According to a study conducted in 2018, 18% of shoppers said that they prefer to buy generic instead of name-brand products. Suppose that in a recent sample of 1500 shoppers, 315 stated that they prefer to buy generic instead of name-brand products. a) Construct a 95% confidence interval for the proportion of all shoppers who currently prefer to buy generic instead of name-brand products. b) At a 5% significance level, can you conclude that the proportion of all shoppers who currently prefer to buy generic instead of name-brand products is higher than 0.18? Perform a hypothesis test to test the validity of the claim