I need help with c, as I have attempted a and b already, but believe they help with context.
Suppose you took a random sample of 100 accounts in a large department-store chain, and found that the mean balance due was \$74 and the standard deviation was \$86.
(a) Find the $95%$ confidence interval for the mean balance due. (Since the sample is large, the population standard deviation σ can be safely approximated with the sample standard deviation s = $86$.)
For this, my interval was $57.144-90.856$
(b) If there were 243,000 accounts altogether, find a 95% confidence interval for the total balance due. Briefly explain to the vice president the meaning of your answer.
For this, I just replaced the square root of 100 with the square root of 243,000 and that interval was $178,989.083-180,650.917$
(c) Suppose that the skeptical vice president undertook a complete accounting of the whole population of balances due, and found a total of $19,714,00. What would you say?
So you know with 95% confidence that the answer he should be getting is your answer from part b. Note that for part b you want the "total balance," so unless I'm misreading, you need to multiply by $243000.$
What the question is looking for is whether or not the estimate in part C is in the confidence interval from part b. If it is, then good. If it isn't then he is wrong, and possibly trying to embezzle.