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 \( \boxed{0.684} \).