P-value calculations and interpretations depend on the alternative hypothesis of a test, a description of the difference from expectation that we are interested in.

For example, let’s return to the 10-coin-flip example from earlier. Suppose that we flipped a coin 10 times and observed only 2 heads. We might run a hypothesis test with the following null and alternative hypotheses:

• Null: the probability of heads is 0.5
• Alternative: the probability of heads is less than 0.5

This hypothesis test asks the question: IF the probability of heads is 0.5, what’s the probability of observing 2 or fewer heads among a single sample of 10 coin flips?

Earlier, we used a for-loop to repeatedly (10000 times!) flip a fair coin 10 times, and store the number of heads (for each set of 10 flips) in a list named outcomes. The probability of observing 2 or fewer heads among 10 coin flips is approximately equal to the proportion of those 10000 experiments where we observed 0, 1, or 2 heads:

import numpy as np
outcomes = np.array(outcomes)
p_value = np.sum(outcomes <= 2)/len(outcomes) 
print(p_value) #output: 0.059

This calculation is equivalent to calculating the proportion of this histogram that is colored in red:

We estimated that the probability of observing 2 or fewer heads is about 0.059 (5.9%). This probability (0.059) is referred to as a one-sided p-value.