How to calculate p-value for coin toss?

Question

I was trying to calculate the p-value for the following coin toss example:

n = 5
h = 4  # out of 5 toss 4 are head

# calculate pvalue using equal or more extreme cases
pval = P(4H1T) + P(4T1H) + P(5H) + P(5T)
     = 5/32    + 5/32    + 1/32  + 1/32
     = 12/32
     = 0.375

But when I tried standard method:

from scipy import stats

phat = h / n
p0 = 0.5  # for unbiased coin
q0 = 1 - p0
z = (phat - p0) / sqrt(p0q0/n)      

pval = 1 - stats.norm.cdf(z)

I got:

0.08985624743949994.

Question 1

How to get the similar result using python scipy stats (answer = 0.375) ?

References

In this statquest video the author explains how to get pvalue using equal or more extreme values and we get 0.375

https://www.youtube.com/watch?v=JQc3yx0-Q9E&t=728s

But here,

https://www.csus.edu/indiv/j/jgehrman/courses/stat50/hypthesistests/9hyptest.htm

using the formula given we get another answer.

Question 2

Which method is good so that we can compare pvalue with alpha?

user_11897007 · Answer

The z value is the standardized value of a normal distribution and it is NOT a probability. Also, the probability for an exact value in a continuous distribution is a bit more tricky. This scenario sounds optimal for binomial distribution.

from scipy.stats import binom
binom.pmf(4,5,0.5)

Which outputs:

0.15625

from scipy.stats import binom
binom.pmf(4,5,0.5)

Which outputs:

0.15625

user_5200329 · Answer

This is a binomial distribution case. Follow the official example in scipy.stats.binom_test:

https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.binom_test.html#scipy.stats.binom_test

import numpy as np
from scipy import stats

alpha = 0.05
pval = stats.binom_test(4, n=5, p=0.5, alternative='two-sided')

print('pvalue = ', pval)
if pval< alpha:
    print("We Reject the Null Hypothesis.")
else:
    print("We Accept the Null Hypothesis.")

Output

pvalue =  0.375
We Accept the Null Hypothesis.

How to calculate p-value for coin toss?

Question 1

References

Question 2

2 Answers

Output