[NW]
a. Find the t-value such that the area in the right tail is 0.10 with 25 degrees of freedom.
b. Find the t-value such that the area in the right tail is 0.05 with 30 degrees of freedom.

Verified step by step guidance

1. Understand that the problem asks for t-values corresponding to specific right-tail areas in a t-distribution with given degrees of freedom (df). These t-values are critical values used in hypothesis testing or confidence intervals.
2. Recall that the t-distribution is symmetric about zero, and the t-value for a right-tail area \( \alpha \) with \( df \) degrees of freedom satisfies \( P(T > t) = \alpha \), where \( T \) follows a t-distribution with \( df \) degrees of freedom.
3. For part (a), identify the parameters: right-tail area \( \alpha = 0.10 \) and degrees of freedom \( df = 25 \). Use a t-distribution table or statistical software to find the t-value \( t_{0.10, 25} \) such that \( P(T > t_{0.10, 25}) = 0.10 \).
4. For part (b), identify the parameters: right-tail area \( \alpha = 0.05 \) and degrees of freedom \( df = 30 \). Similarly, find the t-value \( t_{0.05, 30} \) such that \( P(T > t_{0.05, 30}) = 0.05 \) using a t-table or software.
5. Summarize the results by stating the t-values found for each part, which represent the critical values for the given right-tail probabilities and degrees of freedom.

Key Concepts

t-Distribution

The t-distribution is a probability distribution used when estimating population parameters when the sample size is small and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which account for the extra uncertainty. The shape depends on degrees of freedom, becoming closer to normal as degrees of freedom increase.

Degrees of Freedom

Degrees of freedom (df) refer to the number of independent values that can vary in an analysis without breaking any constraints. In the context of the t-distribution, df typically equals the sample size minus one. It affects the shape of the t-distribution, with fewer degrees of freedom resulting in a wider, more spread-out distribution.

Critical t-Value and Tail Area

A critical t-value corresponds to a specific area in the tail(s) of the t-distribution, representing a cutoff point for hypothesis testing or confidence intervals. The right tail area indicates the probability of observing a value greater than the critical t-value. Finding this value involves using t-distribution tables or software based on the given tail area and degrees of freedom.