How to find the exact and likelihood ratio p-value for a hypothesis test that a population sex ratio is 1:1? 
Skulls were excavated from a dig. The sex of each skull was determined
  by anatomical appreciation.  For Sites A, the data was Male=162,
  Female=110.
a) Find the exact p-value for a hypothesis test about whether the sex
  ratio of the population buried at this sites was 1:1.
b) Instead use a likelihood ratio test.  Find the likelihood ratio
  test statistic, and resulting p-value.  Compare with (a).

Any help would be appreciated.
 A: This is a binomial test. The hypothesis is that if you choose a skull you have a 50% chance it is female. Kind of like a coin flip, except instead of testing heads and tails you're testing male and female.
$X = 110$
$N = 110 + 162 = 272$
$p = .5$
$\alpha = .05$
Formally stated the hypotheses are:
$H_0: p = .5$
$H_1: p \ne .5$
To find the test statistic assuming the above alpha value and given that we have a two tailed test we consult this z table: http://lilt.ilstu.edu/dasacke/eco148/ztable.htm
Because it's a two-tailed test we need to find the z-value of alpha/2 which is 1.96. So that's our test statistic.
Then we set up a binomial test and evaluate it like so (refer to http://www.elderlab.yorku.ca/~aaron/Stats2022/BinomialTest.htm):
$$z = \frac{\frac{X}{N} - p}{\sqrt{\frac{pq}{N}}}=\frac{\frac{110}{272} - .5}{\sqrt{\frac{.5*.5}{272}}} = -\frac{.0956}{.0303} = -3.1551$$
Since the absolute value of the statistic we calculated is greater than the test statistic of 1.96 we reject the null hypothesis that the proportion of female skulls is .5.
As far as likelihood goes, we can find the probability of heads given the number of female heads found in the population of 272 total heads found.
$$L(p | n, y) = {n \choose y}p^y{(1-p)}^{n-y}$$
Where n is the total number of heads found, y is the number of female heads found, p is the anticipated ratio that $y \over n$ may represent.
$$L(.5 | 272, 110) = {272 \choose 110}{.5}^{110}{(1-.5)}^{272-110}= 0.00033$$
I computed this in Wolfram Alpha here:
http://www.wolframalpha.com/input/?i=choose%28272%2C110%29*%28.5%5E110%29%281-.5%29%5E%28272-110%29
So both results agree! We can conclude it is very unlikely the ratio of male and female skulls is in fact 1:1.
A: You only know the number of the females and males in site A? Do you want to test all the sites have the same ratio or site A has ratio 1:1?
I guess this is a problem wrt to Contingency table, for example, you may write it down in the form:
\begin{align*}
  &  A\ B\ C\ sum\\
F\\
M\\
sum\\
\end{align*}
Then you may use some formulas to calculate p values,etc.
