To find the t-value such that 0.05 of the area under the curve is to the left of $t$, with 12 degrees of freedom, follow these steps:

1. Identify the cumulative probability: Since 0.05 of the area is to the left of $t$, this corresponds to the cumulative probability $P(T \leq t) = 0.05$.

2. Locate the corresponding t-value in the t-distribution table: With 12 degrees of freedom, we look up the cumulative probability of 0.05 in the t-distribution table.

Typically, t-distribution tables provide the value for the upper tail (right tail). For a left tail probability of 0.05, the corresponding upper tail probability is 1 - 0.05 = 0.95.

3. Find the critical t-value: Look for the t-value in the table that corresponds to 0.95 cumulative probability with 12 degrees of freedom.

Using the t-distribution table, the critical t-value for 12 degrees of freedom and an upper tail probability of 0.95 is:

$t_{0.95, 12} = 1.782$

Since we are interested in the left tail with 0.05 probability, the t-value will be negative. Therefore, the t-value such that 0.05 of the area under the curve is to the left of $t$ is:

$t = -1.782$

Summary

For 12 degrees of freedom, the t-value such that 0.05 of the area under the curve is to the left of $t$ is:

$t = -1.782$

Would you like details or have any questions about this process?

Here are 8 related questions you might find interesting:

1. How is the t-distribution different from the standard normal distribution?
2. Why do we use degrees of freedom in the t-distribution?
3. What is the t-value for a cumulative probability of 0.95 with 15 degrees of freedom?
4. How do we calculate t-values for different levels of confidence?
5. What role does the t-distribution play in hypothesis testing?
6. How does sample size affect the t-distribution?
7. What is the relationship between the t-distribution and the standard error of the mean?
8. How would you find the t-value if the cumulative probability was 0.01 instead of 0.05?

Tip: Always ensure to understand whether you are dealing with a left-tailed or right-tailed test when using t-distribution tables, as this will affect the cumulative probability you are looking up.