For each of the following situations, calculate a \(95 \%\) confidence interval for the mean ( \(\sigma\) not known), beginning with the step, "Calculate the degrees of freedom ( \(d f\) ) and identify the critical value of \(t . "\)

a. \(\quad N=11, \bar{X}=3.00, s_{\bar{X}}=.13\)

b. \(\quad N=20, \bar{X}=17.83, s_{\bar{X}}=2.37\)

c. \(\quad N=17, \bar{X}=1.56, s_{\bar{X}}=.14\)

d. \(N=30, \bar{X}=69.71, s_{\bar{X}}=4.76\)