#### Answer

$s_{1} = \sqrt {\frac{(93 - 89.857) + .... + (86 - 89.857)} {7 - 1}} = 11.625$
$s_{2} = \sqrt {\frac{(118 - 99.625) + .... + (98 - 99.625)} {8 - 1}} = 27.3806$
$s_{p} =\sqrt \frac{(7 - 1)(11.625)^{2} + (8 - 1)(27.3806)^{2}}{7 + 8 - 2}$
$ = \sqrt \frac{(6)(135.1406) + (7)(749.697)}{13}$
$ = \sqrt \frac{(810.84375 +5247.8808}{13}$
= 21.588
Using students t distribution table for df = 13, we have:
$t_{α/2} = 1.771$
$E = t_{α/2} . s_{p}. \sqrt {1/n_{1} + 1/n_{2}}$
$= 1.771 . 21.588 . \sqrt 15/56 = 19.787$
$(x̅_{1} - x̅_{2}) - E = (89.857 - 99.625) - 19.787 = -29.555$
$(x̅_{1} - x̅_{2}) + E = (89.857 - 99.625) + 19.787 = 10.0195$

#### Work Step by Step

As above