""" We can estimate the probability of the unfair coin by looking at the average value of the Bernoulli variables corresponding to each flip - 1 if heads, 0 if tails. If we observe 525 heads out of 1000 flips, then we estimate p equals 0.525. How confident can we be about this estimate? """ from probability_intervals import normal_two_sided_bounds p_hat = 525 / 1000 mu = p_hat sigma = math.sqrt(p_hat * (1 - p_hat) / 1000) # 0.0158 normal_two_sided_bounds(0.95, mu, sigma) # [0.4940, 0.5560] """ Using the normal approximation, we conclude that we are "95% confident" that the above interval contains the true parameter p We do not conclude that the coin is unfair because 0.5 falls within our confidence interval. """
from probability_intervals import normal_two_sided_bounds # 95% bounds based on assumption p is 0.5 lo, hi = normal_two_sided_bounds(0.95, mu_0, sigma_0) # actual mu and sigma based on p = 0.55 mu_1, sigma_1 = normal_approximation_to_binomial(1000, 0.55) # a type 2 error means we fail to reject the null hypothesis # which will happen when X is still in our original interval type_2_probability = normal_probability_between(lo, hi, mu_1, sigma_1) power = 1 - type_2_probability # 0.887
""" We can estimate the probability of the unfair coin by looking at the average value of the Bernoulli variables corresponding to each flip - 1 if heads, 0 if tails. If we observe 525 heads out of 1000 flips, then we estimate p equals 0.525. How confident can we be about this estimate? """ from probability_intervals import normal_two_sided_bounds p_hat = 525 / 1000 mu = p_hat sigma = math.sqrt(p_hat * (1 - p_hat) / 1000) # 0.0158 normal_two_sided_bounds(0.95, mu, sigma) # [0.4940, 0.5560] """ Using the normal approximation, we conclude that we are "95% confident" that the above interval contains the true parameter p We do not conclude that the coin is unfair because 0.5 falls within our confidence interval. """
