For a confidence level of $99 \%$ with a sample size of 11 , find the critical $t$ value.

Step 1 ：We are given a confidence level of 99%, which means the significance level (alpha) is 1% or 0.01.

Step 2 ：Since we are likely dealing with a two-tailed test, we need to divide alpha by 2 to get 0.005.

Step 3 ：The sample size is 11, so the degrees of freedom is the sample size minus 1, which is 11 - 1 = 10.

Step 4 ：We use the t-distribution table or a function that can calculate the critical t value. The critical t value is the value such that the area under the t-distribution curve to the right of it is equal to alpha.

Step 5 ：By doing this, we find that the critical t value is approximately 3.169.

Step 6 ：Final Answer: The critical t value for a confidence level of 99% with a sample size of 11 is \(\boxed{3.169}\).