Under a model proportion of \(0.40\), a sample of \(25\) has \(13~\text{successes}\). In \(3,000~\text{simulations}\), \(463\) produced \(\text{at}~\text{least}~13~\text{successes}\). State the simulation model, statistic, and tail; compute the empirical \(p\text{-}\text{value}\); and make the decision at \(\text{alpha}~=~0.05\).

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Use X = successes in 25 trials and count X ≥ 13. The empirical p-value is 463/3000 ≈ 0.1543, so do not reject the p = 0.40 model at alpha = 0.05.

Divide the 463 extreme simulations by n = 25 to obtain the p-value.

Use X ≥ 13 and p ≈ 0.1543, but reject at alpha = 0.05.

Use X = successes in 25 trials but count X ≤ 13; 463/3000 still gives the p-value.

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Curriculum context

Course: Math III
Standard: S-IC.2
Category: Statistics and Probability
Domain: Making Inferences and Justifying Conclusions
Objective: Use simulation to decide whether data are consistent with a proposed model.
Problem type: Test a proposed population proportion by simulation