(a) Find the $t$-value such that the area in the right tail is 0.25 with 25 degrees of freedom. 0.684 (Round to three decimal places as needed.)
(b) Find the t-value such that the area in the right tail is 0.15 with 19 degrees of freedom.
$\square$ (Round to three decimal places as needed.)
(c) Find the $t$-value such that the area left of the $t$-value is 0.01 with 30 degrees of freedom. [Hint: Use $\square$ (Round to three decimal places as needed.)
(d) Find the critical t-value that corresponds to $60 \%$ confidence. Assume 22 degrees of freedom. $\square$ (Round to three decimal places as needed.)

Step 1 ：The problem is asking for the t-value for a given area in the right tail and degrees of freedom. The t-distribution is a type of probability distribution that is symmetric and bell-shaped, like the standard normal distribution, but has heavier tails, which means it is more prone to producing values that fall far from its mean. The t-distribution is used in hypothesis testing and in constructing confidence intervals.

Step 2 ：To find the t-value, we can use the Percent Point Function (PPF), which is the inverse of the Cumulative Distribution Function (CDF). The CDF gives the probability that a random variable is less than a certain value, and the PPF gives the value associated with a certain probability.

Step 3 ：The probability is 1 minus the area in the right tail, because the PPF function gives the value associated with the probability to the left of that value.

Step 4 ：Let's calculate the t-value for an area of 0.25 in the right tail and 25 degrees of freedom.

Step 5 ：Given that the area in the right tail is 0.25 and the degrees of freedom is 25, the t-value is calculated to be approximately 0.6844299675118172.

Step 6 ：Final Answer: The $t$-value such that the area in the right tail is 0.25 with 25 degrees of freedom is \( \boxed{0.684} \).