I am having some trouble understanding the exact relationship between these two concepts. The following problem is from my homework:
A spinach producer is testing a new packaging line. They want the mean weight of spinach in each package to be 8 ounces. They run their new packing machine for a few days to get a large population of packages, then select 12 packages at random and weigh the spinach in each. They want to determine if there is strong evidence that they should re-calibrate their packing machine. The sample weights (in ounces) are:
7.7, 6.8, 8.0, 7.4, 7.1, 7.4, 7.2, 7.3, 8.3, 7.7, 7.6, 7.0
I stated my hypotheses:
H0 : µ = 8 and
HA : µ ≠ 8
I then constructed a hypothesis test & used a rejection region to determine that I should reject the null hypothesis since my test statistic
(xbar - µ)/(s/√n) = -4.391 was in the rejection region (T < -3.106 and T > 3.106, based on a t-distribution with 11 DoF and α = 0.01)
I think I understand all of this fine. However the last part of the question is confusing me:
(f) If you calculated a 99% confidence interval for the population mean weight, would you expect it to contain 8? Why or why not?
It is not hard to construct the confidence interval & see that it does not contain 8. It also makes intuitive sense that since I made a rejection region using 99% confidence & was able to reject
µ0 = 8 that a 99% CI for the mean would also not contain 8.
I think that in essence the CI for the mean and the rejection region for the hypothesis test are equivalent, however I am not really sure how to formalize this understanding or put it into words.
Can anybody offer a little insight into what is the exact connection between these two concepts and how I might approach this question?
P.S. I have read the question here & it doesn't seem to answer my specific question (although I am only in introductory-level statistics so it could be that I just failed to understand it completely). I think that this question might be asking the same thing that I am asking but I don't really understand the answer.