Step 1 :Calculate the degrees of freedom (df) by subtracting 1 from the sample size. For the first problem, df = 57 - 1 = 56.
Step 2 :Look up the $t$-value in the $t$-distribution table or use a $t$-distribution calculator with df = 56 and left tail area = 0.025. The $t$-value is approximately -2.0047.
Step 3 :\(\boxed{t = -2.0047}\)
Step 4 :Calculate the degrees of freedom (df) for the second problem. df = 74 - 1 = 73.
Step 5 :Look up the $t$-value in the $t$-distribution table or use a $t$-distribution calculator with df = 73 and right tail area = 0.05. The $t$-value is approximately 1.6672.
Step 6 :\(\boxed{t = 1.6672}\)
Step 7 :Calculate the degrees of freedom (df) for the third problem. df = 60 - 1 = 59.
Step 8 :Look up the $t$-value in the $t$-distribution table or use a $t$-distribution calculator with df = 59 and right tail area = 0.01. The $t$-value is approximately 2.3901.
Step 9 :\(\boxed{t = 2.3901}\)
Step 10 :Calculate the degrees of freedom (df) for the fourth problem. df = 16 - 1 = 15.
Step 11 :Look up the $t$-values in the $t$-distribution table or use a $t$-distribution calculator with df = 15 and the area in the two tails = 1 - 0.95 = 0.05. The $t$-values are approximately -2.1314 and 2.1314.
Step 12 :\(\boxed{t = -2.1314, 2.1314}\)