Final answer:
The critical t-value for a degrees of freedom of 3 and an alpha of 0.01 is 6.64, based on a t-distribution table. This value is the same for either tail in a two-tailed test scenario, due to the symmetric nature of the t-distribution.
Step-by-step explanation:
To find the critical t-value when the degrees of freedom (df) is 3 and the alpha (α) is 0.01, we refer to the t-distribution table which is typically used in statistical hypothesis testing. Since the t-distribution is symmetric around zero, the critical values at a given α level are the same but with opposite signs (+ and -), depending on the tail(s) of interest.
Looking at the provided table, for df of 3, the critical t-value at a significance level of 0.01 is 6.64 (Table 19.6). This is a one-sided threshold, which means if we were conducting a two-tailed test, we would use the same value to determine the rejection regions on both ends of the t-distribution.
If you are conducting a one-tailed test, you would use the same value of 6.64 to identify the critical region. However, for a two-tailed test with α = 0.01, the value is split between both tails, and since the table is symmetric, the critical value remains the same for either tail.