Step 1 :In a two-tailed test, the rejection regions are at both ends of the distribution. The level of significance, \(\alpha\), is split between these two tails. So, each tail will contain \(\alpha/2 = 0.10/2 = 0.05\) of the total area under the curve.
Step 2 :The critical value is the t-score that corresponds to this area in the tail. We can find this value from the t-distribution table. However, the t-distribution depends on the degrees of freedom, which is \(n-1 = 7-1 = 6\) in this case.
Step 3 :Using a t-distribution table with \(\alpha = 0.05\) and degrees of freedom \(df = 6\), we find the critical value to be approximately 1.943.
Step 4 :The rejection regions are the t-scores less than -1.943 or greater than 1.943.
Step 5 :Final Answer: The critical value is approximately \(\boxed{1.943}\). The rejection regions are the t-scores less than \(\boxed{-1.943}\) or greater than \(\boxed{1.943}\).