dsfs 07 hypothesis and inference

e.g. flipping a coin

1. normal distirbution

        def normal_approximation_to_binomial(n, p):
            """find mu and sigma corresponding to a binomial(n, p)"""
            mu = p * n
            sigma = math.sqrt(p * (1 - p) * n)
            return mu, sigma
   

   1. whenever a random variable follows a normal distribution

       # we can use `normal_cdf`
       # the normal cdf is the probability
       # the variable is below a threshold
       normal_probability_below = normal_cdf
      
       # it's above the threshold if it's not below the threshold
       def normal_probability_above(lo, mu=0, sigma=1):
           return 1 - normal_cdf(lo, mu, sigma)
      
       # it's between if it's less than hi, but not less than lo
       def normal_probability_between(lo, hi, mu=0, sigma=1):
           return normal_cdf(hi, mu, sigma) - normal_cdf(lo, mu, sigma)
      
       # it's outside if it's not between
       def normal_probability_outside(lo, hi, mu=0, sigma=1):
           return 1 - normal_probability_between(lo, hi, mu, sigma)
      

   2. we can do the reverse

       def normal_uper_bound(probability, mu=0, sigma=1):
           """returns the z for which P(Z <= z) = probability"""
           return inverse_normal_cdf(probability, mu, sigma)
      
       def normal_lower_bound(probability, mu=0, sigma=1):
           """returns the z for which P(Z >= z) = probability"""
           return inverse_normal_cdf(1 - probability, mu, sigma)
      
       def normal_two_sided_bounds(probability, mu=0, sigma=1):
           """returns the symmetric (about the mean) bounds
           that contain the specified probability"""
           tail_probability = (1 - probability) / 2
      
           # upper bound should have tail_probability above it
           upper_bound = normal_lower_bound(tail_probability, mu, sigma)
      
           # lower bound should have tail_probability below it
           lower_bound = normal_upper_bound(tail_probability, mu, sigma)
      
           return lower_bound, upper_bound
      

   3. filp coin n = 1000 times

       mu_0, sigma_0 = normal_approximation_to_binomial(1000, 0.5)
      
       normal_two_sided_bounds(0.95, mu_0, sigma_0)  # (469, 531)
      

   4. calculate the power of the test

       # 95% bounds based an 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