MLE and method of moments

• A.Correct

  True.

  The Beta prior is conjugate to the binomial likelihood: with prior Beta(a, b) and x successes in n trials, the posterior is Beta(a + x, b + n − x) = Beta(5 + 10, 5 + 15) = Beta(15, 20).

  Under squared error loss the Bayes estimate is the posterior mean, 15/(15 + 20) = 3/7 (verified).

  The trigger: squared error loss always means posterior mean — add successes to the first parameter, failures to the second, then take the mean.

• B.Wrong

  False.

  3/5 does not arise from either the prior or the data.

  It would be the posterior mean if the counts were reversed — 15 heads and 10 tails on top of the prior — so it is what a transposed update produces.

• C.WrongExecution slip

  False — 1/2 is the prior mean, 5/(5 + 5) (verified).

  Choosing it means never updating: the data pull the estimate down from 1/2 towards the observed proportion

  landing at 3/7.

  A Bayes estimate equal to the prior mean would require the data to say nothing at all.

• D.WrongHypothesis dropped

  False — 2/5 is the maximum likelihood estimate, 10/25 (verified)

  which is the answer with no prior at all.

  The Bayes estimate always lies strictly between the prior mean 1/2 and the MLE 2/5, and 3/7 ≈ 0.4286 does exactly that.

  Seeing your answer sit outside that interval is a quick sanity check.

2 of these 3 wrong options correspond to a named reasoning error — the same errors recur across subjects.