I want to know the detailed derivation process of this problem, the formulas used, and how the obtained number is calculated. I should look up the table

Pie. 3. A personnel manager has found that historically the scores on aptitude test given to applicants for entry-level positions follow a normal distribution with a standard deviation of 32.4 points. A random sample of nine test scores from the current group of applicants had a mean score of 187.9 points. a) Find an 80% confidence interval for the population mean score of the current group of applicants. b) Based on these sample results, a statistician found for the population mean a confidence interval extending from 165.8 to 210.0 points. Find the confidence level of this interval.6. A clinic offers a weight-loss program. A review of is records found the following amounts of weight loss, in pounds, for a random sample of 24 of its clients at the conclusion of a 4-month program. 18 25 16 11 15 20 16 19 28 25 26 31 45 40 36 19 28 25 36 16 35 20 16 19 a) Assume the customer base of this program is very large (>1000). Find a 99% confidence interval for the population mean. b) Without doing the calculations, explain whether a 90% confidence interval for the population mean would be wider than, narrower than, or the same as that found in part a). c) Now assume the total number of customers who have participated in this weight loss program is 400. What is the 99% confidence interval now