Trial for Likelihood ratio test for multinomial distribution Promblem

Question

So the promblem is: For two factors—starchy or sugary, and green base leaf or white base leaf—the following counts for the progeny of self-fertilized heterozygotes were observed (Fisher 1958):

Starchy green $1997$

Starchy white $906$

Sugary green $904$

Sugary white $32$

According to genetic theory, the cell probabilities are $9/16$, $3/16$, $3/16$, $1/16$.

3. Conduct a likelihood ratio test.

My trial:

Does this look right? If so, is this routine a general solution for all likelihood ratio test for multinomial distribution problems?

Answer 1

Yes, your work is correct; specifically, your calculation of the likelihood ratio test statistic is correct, and the critical value is also correct. However, I would state clearly that your choice of critical value corresponds to a test at the $\alpha = 0.05$ level, since the decision of whether there is sufficient evidence to reject $H_0$ depends on the selected significance level $\alpha$.

Alternatively, you may elect to compute a $p$-value; in this case, the test statistic is so enormous that the $p$-value is approximately $1.1260393 \times 10^{-83}$.