Find the value of \( t \) for a \( t \)-distribution with 15 degrees of freedom such that the area to the left of \( -t \) plus the area to the right of \( t \) equals 0.20 . Round your answer to three decimal places, if necessary.

To find the value of \( t \) for a \( t \)-distribution with 15 degrees of freedom such that the area to the left of \( -t \) plus the area to the right of \( t \) equals 0.20, we need to determine the tail areas. Since the total area under the curve is 1, if the area to the left of \( -t \) plus the area to the right of \( t \) equals 0.20, then the area between \(-t\) and \(t\) must equal \(0.80\) (since \(1 - 0.20 = 0.80\)). This means that: - The area to the left of \( -t \) (which is the same as the area to the right of \( t \)) is \(0.20 / 2 = 0.10\). Thus, we need to find \( t \) such that the cumulative probability \( P(T \leq -t) = 0.10 \) or \( P(T \geq t) = 0.10 \). Using a \( t \)-distribution table or calculator: 1. Look for the value of \( t \) where the cumulative probability is 0.10. 2. For 15 degrees of freedom, this value is approximately \( t \approx -1.341 \) for the left tail (and \( 1.341 \) for the right tail). Thus, rounding to three decimal places, we have: \[ t \approx 1.341 \]