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 0.05 . Round your answer three decimal places, if necessary. Answer

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 0.05, we first recognize that this signifies a two-tailed test. Therefore, the area to the left of \( t \) must equal \( 0.95 \) (since \( 1 - 0.05 = 0.95 \)) for the critical values. Using a \( t \)-table or an inverse \( t \)-distribution calculator for 15 degrees of freedom, we can find the value of \( t \) for the 0.025 tail (as it is split between both tails for a total of 0.05). Looking up \( t_{0.025, 15} \), we find: \( t_{0.025, 15} \approx 2.131 \). Therefore, the value of \( t \) that satisfies the condition is approximately \( 2.131 \). So, the final rounded answer is: \[ t \approx 2.131 \]