Compute an A/B test p-value by hand

You're examining a game feature through an A/B test where the metric of interest is the conversion rate, a binary outcome.

• Control: $n_1 = 1000$ users, $x_1 = 120$ conversions
• Treatment: $n_2 = 980$ users, $x_2 = 145$ conversions

Tasks:

1. Calculate the two-sided p-value against $H_0: p_1 = p_2$ by applying a two-proportion z-test; include the formulas and the intermediate numbers.
2. Say whether the outcome reaches significance at $\alpha = 0.05$.
3. (Implementation) Explain how you would obtain this p-value in Python without SciPy, allowed to use only the math module.

Overview: This prompt tests grasp of proportion hypothesis testing—namely the two-proportion z-test, p-value calculation, and what statistical significance means for an A/B experiment with binary outcomes.