Question 1181286
The sample mean is the average of the endpoints of the C.I., thus {{{X-bar = (83.9219 + 90.6782)/2 = 87.3}}}.

Now from the initial given, {{{t[0.975]*(s/sqrt(15)) =  90.6782 - 87.3 = 3.3782}}}

===> {{{2.145*(s/sqrt(15)) = 3.3782}}}  ==> s = 6.1.

Then the new 99% C.I. will be {{{(X-bar) +- t[0.995]*(s/sqrt(15)) = 87.3 +- 2.977*(6.1/sqrt(15))}}} = (82.6112, 91.9888).


The t-probabilities were gotten from https://stattrek.com/online-calculator/t-distribution.aspx.