The area under the t-distribution with 18 degrees of freedom to the right of t = 1.56 is 0.0681. What is the area under the t-distribution with 18 degrees of freedom to the left of t = –1.56? Why?

Verified step by step guidance

1. Understand that the t-distribution is symmetric about zero, meaning the shape on the left side mirrors the shape on the right side.
2. Given the area to the right of t = 1.56 with 18 degrees of freedom is 0.0681, recognize that this area corresponds to the tail probability on the positive side.
3. Because of symmetry, the area to the left of t = -1.56 is equal to the area to the right of t = 1.56.
4. Therefore, the area to the left of t = -1.56 is also 0.0681.
5. Summarize that the symmetry property of the t-distribution allows us to conclude that \(P(T < -1.56) = P(T > 1.56) = 0.0681\) for \(T\) with 18 degrees of freedom.

Key Concepts

t-Distribution

The t-distribution is a probability distribution used in statistics when estimating population parameters with small sample sizes. It is symmetric and bell-shaped like the normal distribution but has heavier tails, which account for extra uncertainty. The shape depends on degrees of freedom, affecting the spread and tail thickness.

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 t-distribution, df typically equals the sample size minus one. The value of df influences the shape of the t-distribution, with higher df making it closer to the normal distribution.

Symmetry of the t-Distribution

The t-distribution is symmetric about zero, meaning the area to the right of a positive t-value equals the area to the left of its negative counterpart. Therefore, the area to the left of t = –1.56 is the same as the area to the right of t = 1.56, which is 0.0681 in this case.