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My task was to flip my penny $11$ times and record the number of successes (heads) and then determine with $99\%$ confidence whether the coin is fair or foul.

my $\hat{p}$ is $0.45$, thus $\hat{q}$ is $0.55$.

What I calculated: $99\%$ confidence gave me a $Z_{\alpha/2}$ of $2.58$.

The margin of Error $E = 2.58 \times \sqrt{\hat{p}\hat{q}/11} = 38.73\%$ [seems huge]

Then I construct the Confidence interval as

$\hat{p}-E < p < \hat{p} + E$

$.4545-.3873 < p < .4545 + .3873$

$.0672 < p < .8418$

I get stuck here because this seems too large to be even reasonable.

What am I doing wrong here?

ant11
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user163862
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    When you are designing a CI interval as you've done above you are doing assuming that the distribution of of our average coin flips over 11 flips is approximately normal, but with only 11 flips this may not be a reasonable assumption – Kamster Jul 23 '14 at 04:07
  • I don't know, but I wouldn't say that is the proper term for it. You are certainly not 99% confident that the coin is fair, only that "fair" is in the 99% confidence interval. :) – Thomas Andrews Jul 23 '14 at 04:07
  • Also remember the way a confidence interval is designed, it is designed that so if you where to construct interval with procedure as you did over many other 11 flip samples about 99% of intervals would contain the true p, thus if there is a lot variability in amount of heads you can get you will want a wide interval to ensure it will capture the true p which will be true for if you are only flipping 10 times – Kamster Jul 23 '14 at 04:22
  • The confidence interval should be quite large because with only 11 flips to test your coin, you would have to have either an overwhelming number of heads or an overwhelming number of tails to be 99% confident the coin was not fair. You can compute a better confidence interval, but it will still be large. – David K Jul 23 '14 at 13:48

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As noted by the comments, your sample size is too small for using a normal approximation. I would recommend you use one of the improved intervals, as listed on Wikipedia (hopefully one of these has been covered or referenced in your class).