Hypothesis Test for a Proportion: COVID-19 and Capitalism

Pew Research Center Data:  Impressions of Socialism and Capitalism

In early 2019, the Pew Research Center surveyed 10,170 Americans about their impressions of "socialism" and "capitalism."   4271 reported a positive view of “socialism”  and  5594 reported a negative view.   Also, 6611 reported a positive view of "capitalism" and 3390 reported a negative view.  (Note that some of those surveyed did not respond to each question).

In 2011 (according to Pew) 31% of Americans had a positive view of socialism and 60% had a negative view.  Also at that time, 50% had a positive view of capitalism and 40% had a negative view of capitalism.  These figures were little different from the previous year.

Question 1:   Do the 2019 sample data support the claim that the percentage of Americans with a positive view of socialism has increased since 2011?  Use a 1% level of significance.

Solution:  This is a "hypothesis test" problem using a single proportion.  If necessary, review this procedure using the e-text or Hypothesis Tests for a Population Proportion.

On the face of it, the sample proportion with a positive view of socialism is 4271/10170 or about 42%.   This is certainly greater than the 31% baseline figure.  But it is just a SAMPLE proportion.  If Pew had sampled a different 10,170 people they would almost certainly have gotten a different number with a positive view of socialism.  How do we know this wasn't an atypical (even if randomly chosen) sample?

We don't.  But we can use what we know about sampling distributions for proportions to calculate the probability of getting a sample with this many positive responses if the true population proportion of positives were still 31%.  If this probability is "small enough" then we conclude that our result was not due to sampling error.  It (probably) represents a real change (the alternative hypothesis).

How small is "small enough"?  Less than the level of significance we have chosen (or that has been given to us) -- in this problem, 1%.

Your instructor may have a preferred way for you to organize your hypothesis test, but it will include the following elements:

The null hypothesis H₀:  the 2019 population proportion = 31%

The alternative hypothesis H₁: the 2019 population proportion > 31% ("has increased")

The level of significance = 0.01 

The distribution to be used:  We have n = 10171,  p (the hypothesized population proportion) = .31 and 1-p = .69.  Since npq >>10, the normal approximation to the binomial distribution is appropriate so we use the Z-distribution.

Your instructor will tell you whether they prefer for you to use the classical (critical-value) approach or the p-value approach, and whether you are to use computational formulas or technology.  In either case, you will find that  Z= 23.98 (definitely in the critical region!) and  (definitely less than = 0.01).

Either result tells us to REJECT the null hypothesis at the 1% level of significance.

Conclusion:  The 2019 sample data support the claim (at a 1% level of significance) that the percentage of Americans with a positive view of socialism has increased since 2011.

Question 2:  Do the 2019 sample data support the claim that the percentage of Americans with a positive view of capitalism has changed since 2011?  Use a 1% level of significance.  (Notice that the alternative hypothesis will now be a "two-tailed" ($\ne$) test.)

Has Covid-19 changed how people in the US view our society?

JustCapital is an organization dedicated to reforming -- not abolishing! -- corporate capitalism.  Since 2015 it has conducted polls on how people in the US think businesses could become more responsive to their needs. (JustCapital.com) 

A JustCapital/Harris poll in June 2020 asked whether American capitalism "produces the kind of society I want for the next generation."  Out of 807 respondents, 161 said that "no form of capitalism can achieve this" and another 412 said that "we need a  more evolved form of capitalism to achieve this."

Remember that, in the 2019 Pew poll, 3390/10170  $\approx$ 0.33 or 33% of the respondents had a negative view of capitalism.  Which response(s) to JustCapital/Harris should count as a "negative view of capitalism"?

Let's try it both ways.

Question 3a:  Do the JustCapital/Harris sample data support the claim that the percentage of Americans with a negative view of capitalism has changed since 2019?  Use a 1% level of significance.  Only count as "negative" the 161 people who said that "no form of capitalism can achieve this."

Question 3b:  Do the JustCapital/Harris sample data support the claim that the percentage of Americans with a negative view of capitalism has changed since 2019?  Use a 1% level of significance.  Count as "negative" the 412 people who said that "we need a more evolved form of capitalism" AND the 161 people who said that "no form of capitalism can achieve this.

Now post your answers in the DISCUSSION:  Capitalism, Socialism and Other Ways of Organizing Society