Let X subscript 1 comma space X subscript 2 comma space X subscript 3 comma space horizontal ellipsis comma space X subscript n space be iid random variables with X subscript i space tilde thin space N left parenthesis mu comma sigma squared right parenthesis , and define the usual sample mean ie top enclose X space equals space 1 over n space sum from i equals 1 to n of X subscript i

We know that if Z space equals space fraction numerator top enclose X space minus space mu over denominator begin display style bevelled fraction numerator sigma over denominator square root of n end fraction end style end fraction then Z space tilde thin space N left parenthesis 0 comma 1 right parenthesis .



If we now define W space equals space fraction numerator top enclose X space minus space mu over denominator begin display style bevelled fraction numerator S over denominator square root of n end fraction end style end fraction where S comes from the usual estimator of sigma , ie S squared space equals space fraction numerator 1 over denominator n minus 1 end fraction space space sum from i space equals space 1 to n of left parenthesis X subscript i space minus space top enclose X right parenthesis squared , then what distribution does W follow?

a.
t subscript n (ie a t-distribution with n degrees of freedom)

b.
N(0,1)

c.
t subscript n minus 1 end subscript (ie a t-distribution with n-1 degrees of freedom)

d.
N(0,1.5)