Solution
A Sixty-Forty Coin in a Hundred Flips
Show the problem again
A coin shows 60 heads in 100 tosses. For testing $H_0:p=1/2$, compute the likelihood ratio statistic $G=-2\ln\Lambda$, where $\Lambda$ is the ratio of the likelihood at $p=1/2$ to the maximised likelihood. Use natural logarithms and give $G$ rounded to three decimal places.
Worked solution
The MLE is $\hat p=0.6$. Then $\Lambda=\dfrac{0.5^{100}}{0.6^{60}0.4^{40}}$, so $G=2\left[60\ln\frac{60}{50}+40\ln\frac{40}{50}\right]=2[60(0.18232)+40(-0.22314)]\approx 4.027.$ This exceeds the $\chi^2_1$ 5% critical value 3.841, so Wilks' asymptotic test rejects, although the exact two-sided binomial test gives p about 0.057. The two approaches disagree at this borderline.
Source: Likelihood ratio (G) test for a binomial proportion, standard textbook material. Numbers written for AxiomIQ. Statement written for AxiomIQ.