Inferring Confidence Interval from Hypothesis Test

Question

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.

Answer 1

I managed to find a fairly good explanation of this here, which is paraphrased below.

Rejection regions, confidence intervals, and indeed p-values will always agree about statistical significance.

The rejection region defines the distance the sample mean must be from the mean under the null in order to be considered statistically significant. The boundaries of the confidence interval are also defined by a distance from the sample mean. The distance in both cases is exactly the same, typically $$z^{*} \times \sigma_{\hat{\mu}}$$

For this reason, hypothesis testing may be performed using rejection regions, confidence intervals or p-values with no change in the results.