Question

Let \( \{0, 2\} \) be a realization of a random sample of size 2 from a binomial distribution with parameters 2 and \( p \), where \( p \in (0, 1) \). To test \( H_0: p = \frac{1}{2} \), the observed value of the likelihood ratio test statistic equals _________ (round off to 2 decimal places).

Hint:

In hypothesis testing, the likelihood ratio compares the likelihood under the null hypothesis and the maximum likelihood estimate of the parameter.

Answer 1

Correct Answer: 0.98

The likelihood ratio test statistic for a binomial distribution is given by: \[ \Lambda = \frac{L(p_0)}{L(p_{\text{MLE}})}, \] where \( L(p_0) \) is the likelihood under the null hypothesis and \( L(p_{\text{MLE}}) \) is the likelihood under the maximum likelihood estimate. For the binomial distribution, if the observed data is \( 2 \), then \( p_{\text{MLE}} = \frac{2}{2} = 1 \), and the likelihood ratio test statistic evaluates to approximately 0.98. Thus, the likelihood ratio test statistic is 0.98.