Part 2: Hypothesis Testing of proportion-based samples

In part one of this series, I introduced the concept of hypothesis testing, and described the different elements that go into using the various tests. It ended with a cheat-sheet to help you choose which test to use based on the kind of data you’re testing.

In this second post I will go into more detail on proportion-based samples.

If any of the terms Null Hypothesis, Alternative Hypothesis, p-value are new to you, I’d suggest reviewing the first part of this series before moving on.

What is a proportion-based sample?

In these cases we’re interested in checking proportions. For example 17% of a sample matches some profile, and the rest does not. This could be a test comparing a single sample against some expected value, or comparing two different samples.

Note: These tests are only valid when there are only two possible options; and if the probability of one option is p, then the probability of the other must be (1 – p).

Requirements for the quality of the sample

For these tests the following sampling rules are required:

Random	The sample must be a random sample from the entire population
Normal	The sample must reflect the distribution of the underlying population. For these tests a good rule of thumb is that: Given a sample size of n Given a sample proportion of p Then both np and n(1-p) must be at least 10 For example: if a sample finds that 80% of issues were resolved in 5 days, and 20% were not, then that sample must have at least 10 issues resolved within 5 days, and at least 10 issues resolved in more than 5 days.
Independent	The sample must be independent – for these tests, a good rule of thumb is that the sample size is less than 10% of the total population.

Code Samples for Proportion-based Tests

Note that all of these code samples are available on Github. They use the popular statsmodels library to perform the tests.

1-sample z-test

Compare the proportion in a sample to an expected value

Here we have a sample and we want to see if some proportion of that sample is greater than/less than/different to some expected test value.

In this example:

• We expect more than 80% of the tests to pass, so our null hypothesis is: 80% of the tests pass
• Our alternative hypothesis is: more than 80% of the tests pass
• We sampled 500 tests, and found 410 passed
• We use a 1-sample z-test to check if the sample allows us to accept or reject the null hypothesis

To calculate the p-value in Python:

from statsmodels.stats.proportion import proportions_ztest

# can we assume anything from our sample
significance = 0.05

# our sample - 82% are good
sample_success = 410
sample_size = 500

# our Ho is  80%
null_hypothesis = 0.80

# check our sample against Ho for Ha > Ho
# for Ha < Ho use alternative='smaller'
# for Ha != Ho use alternative='two-sided'
stat, p_value = proportions_ztest(count=sample_success, nobs=sample_size, value=null_hypothesis, alternative='larger')

# report
print('z_stat: %0.3f, p_value: %0.3f' % (stat, p_value))

if p_value > significance:
   print ("Fail to reject the null hypothesis - we have nothing else to say")
else:
   print ("Reject the null hypothesis - suggest the alternative hypothesis is true")

2-sample z-test

Compare the proportions between 2 samples

Here we have two samples, defined by a proportion, and we want to see if we can make an assertion about whether the overall proportions of one of the underlying populations is greater than / less than / different to the other.

In this example, we want to compare two different populations to see how their tests relate to each other:

• We have two samples – A and B. Our null hypothesis is that the proportions from the two populations are the same
• Our alternative hypothesis is that the proportions from the two populations are different
• From one population we sampled 500 tests and found 410 passed
• From the other population, we sampled 400 tests and found 379 passed
• We use a 2-sample z-test to check if the sample allows us to accept or reject the null hypothesis

To calculate the p-value in Python:

from statsmodels.stats.proportion import proportions_ztest
import numpy as np

# can we assume anything from our sample
significance = 0.025

# our samples - 82% are good in one, and ~79% are good in the other
# note - the samples do not need to be the same size
sample_success_a, sample_size_a = (410, 500)
sample_success_b, sample_size_b = (379, 400)

# check our sample against Ho for Ha != Ho
successes = np.array([sample_success_a, sample_success_b])
samples = np.array([sample_size_, sample_size_b])

# note, no need for a Ho value here - it's derived from the other parameters
stat, p_value = proportions_ztest(count=successes, nobs=samples,  alternative='two-sided')

# report
print('z_stat: %0.3f, p_value: %0.3f' % (stat, p_value))

if p_value > significance:
   print ("Fail to reject the null hypothesis - we have nothing else to say")
else:
   print ("Reject the null hypothesis - suggest the alternative hypothesis is true")

In the next post I will focus on hypothesis testing mean-based samples.

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• PART I: An Introduction to Hypothesis Testing
• PART III: Hypothesis Testing of mean-based samples
• PART IV: Hypothesis Testing of frequency-based samples

Comments

[Matt Curcio]

I love Stats.
There is a line of thinking in Math education that Stats should be taught much earlier, for example, ages 10-18. I have seen Stats taught so easily to young adults.