Use simulation to decide whether data are consistent with a proposed model.
Test a proposed population proportion by simulation
Problem
Under a model proportion of \(0.50\), a sample of \(40\) has \(27~\text{successes}\). In \(2,000~\text{simulations}\), \(91\) produced \(\text{at}~\text{least}~27~\text{successes}\). State the simulation model, statistic, and tail; compute the empirical \(p\text{-}\text{value}\); and make the decision at \(\text{alpha}~=~0.05\).
Big Picture
What this problem is really about
A valid model test simulates the original sample size repeatedly under the proposed success proportion and records the same count statistic each time. Translate the stated direction into an extremeness inequality before counting simulations. Divide the qualifying count by all simulations, compare that empirical p-value with alpha, and interpret the decision as evidence about the model rather than certainty.
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