Final answer:
The critical t-value for a left-tailed test with a significance level of 0.005 and a sample size of 12 (df=11) is approximately -3.106. The rejection region consists of all t-scores smaller than this critical value.
Step-by-step explanation:
To find the critical value(s) and rejection region(s) for a left-tailed t-test with a level of significance α of 0.005 and a sample size n of 12, we must first calculate the degrees of freedom (df), which is df = n - 1. In this case, df = 12 - 1 = 11.
Since we are dealing with a t-distribution, we cannot use the z-table directly for our answer. We need to refer to the t-distribution table. The critical t-value is the value that separates the rejection region on the left tail of the t-distribution. Because the test is left-tailed, we are only concerned with the negative side of the t-distribution.
Consulting a t-distribution table or using statistical software, we find that the critical t-value for a df of 11 and α = 0.005 is approximately -3.106. This number might vary slightly depending on the t-table version you are referencing.
The rejection region is, therefore, all t-scores less than -3.106. If the calculated t-score from the t-test is smaller than -3.106, we reject the null hypothesis.