Learn by Doing – Proportions (Step 4)

Published: October 22nd, 2013

Category: Activity 1: Learn By Doing

The purpose of this activity is to give you a better sense of the p-value, and in particular, help strengthen your intuition about how it measures the evidence against the null hypothesis.


In this activity we will use example 2. Click here for the associated questions. Recall that we’ve just completed this example, and summarized it using the following figure:

The entire population is represented as a large circle, representing Students at the college. We are interested in p for this population, which is the proportion which use marijuana. H_0: p = .157, H_a: p > .157 . From the population we generate a sample of 100 students, and find that 19 use marijuana. Our conditions are met, and we find that p-hat = 19/100 = .19, z = .91, and p-value = .182 . We conclude that H_0 cannot be rejected.

We’ve seen that the evidence the data provided—19 marijuana users out of a sample of 100—was not enough for us to conclude that the proportion of marijuana users in the college is higher than the national figure (.157). An interesting question, therefore is: how many marijuana users out of 100 should we have found for it to be enough evidence to reject Ho? 19 was not enough, but what would have been enough? 21? 25?

To help us answer this question more accurately, let’s look at a table that lists various sample counts/proportions of users, the corresponding z statistic, and the associated p-value.


Note that we highlighted in red the result we found in our sample—19 users.

This is a great opportunity to see how the p-value “works” as a measure of evidence against Ho; the smaller it is the more evidence is “stored” in the data against Ho. Obviously, if finding 19 marijuana users was not enough evidence to reject Ho. We therefore conclude that p, the proportion of marijuana users in the college is higher than 0.157 (the national figure), then anything below 19 would not be enough either, since it provides even less evidence against Ho.

See in the green section of the table how this is depicted by the values of the p-value, which get larger as the number of marijuana users gets smaller. On the other hand, it is pretty clear that the more marijuana users we see in our sample, the more evidence we have to reject Ho and conclude that p > 0.157. Indeed, note that the p-values get smaller as the number of marijuana users increases.

This document is linked from Proportions (Step 4 & Summary).