Here's what I've got for the 82% CI:

x1=82%

μ=258.5

σ=34.9

n=15

α=.82

1-α=0.18

α/2=0.09

1-0.09=0.91

z-score=1.35

p^=x/n

p^=0.18/15

p^=0.988

q^=1-p^

q^=1-0.988

q^=0.012

E=zα/2*√p^q^/n

1.35*√(0.988)(0.012)/15

1.35*√0.011856/15

1.35*√7.904

1.35*2.811405343

E=3.795397212

p^ - E < p < p^ + E

0.988 – 3.795397212 < p < 0.988 + 3.795397212

-3.696597212 < p < 4.783397212

However, when I use computer software, I get 245.7828 < mean < 271.2172, which I think is correct. Of course, the software doesn't show me how to get to the answer.

If anyone can help figure out where I went wrong, please point me in the right direction.

Thanks!