91Ó°ÊÓ

Given:

Let

\({d_{ij}} = {x_i} - {y_i}\)

where\({x_1},{x_2}, \ldots ,{x_m}\)and\({y_1},{y_2}, \ldots ,{y_n}\)are observed value of continuous distributions that differ only in location but not in shape.

The general form of a\(100(1 - \alpha )\)confidence interval for\({\mu _1} - {\mu _2}\)is

\(\left( {{d_{ij(mn - c + 1)}},{d_{ij(c)}}} \right),\)

Where\({d_{ij(1)}},{d_{ij(2)}} \ldots ,{d_{ij(mn)}}\) are the ordered differences.

Use Appendix Table A.16 to determine values of\(c\).
For\(m = n = 5\)and the Table A.16, for\(90.5\% \)confidence interval,\(c = 21\), the confidence interval is in the form of

\(\left( {{d_{ij(mn - c + 1)}},{d_{ij(c)}}} \right) = \left( {{d_{ij(5 \cdot 5 - 21 + 1)}},{d_{ij(21)}}} \right) = \left( {{d_{ij(5)}},{d_{ij(21)}}} \right)\)

Compute all differences, find\({5^{{\rm{th }}}}\)and\({21^{{\rm{th }}}}\)ordered difference values.

The five smallest ordered differences are

\( - 18, - 2,3,4,16\)

and the five largest ordered differences are

\(136,123,120,107,87.\)

The\({5^{th}}\)and\({21^{th}}\)ordered difference are\(16\)and\(87\), respectively; thus, a\(90.5\% \)confidence interval is

\(\left( {{d_{ij(5)}},{d_{ij(21)}}} \right) = (16,87).\)