Question

A survey organization conducted telephone interviews in December 2018 in which 2,000 randomly selected adults in the United States responded to the following question:

"Did you vote in the 2016 Presidential election?"

Do the data provide convincing evidence that the proportion of adults in the United States who would respond "yes" to the question changed from December 2015 to December 2018?

Of the 2,000 adults surveyed in 2018, 1340 responded "yes." On December of 2015, 915 of 1,500 randomly selected adults in the United States had responded "yes" to the question: "Do you plan to vote in the 2016 election?"

Answer 1

To determine if there is convincing evidence that the proportion of adults in the United States who would respond "yes" to the question changed from December 2015 to December 2018, we need to perform a hypothesis test.

Let p1 be the proportion of adults who responded "yes" in December 2015, and p2 be the proportion of adults who responded "yes" in December 2018. We want to test the null hypothesis that the two proportions are equal: H0: p1 = p2.

The alternative hypothesis is that the proportions are not equal: Ha: p1 ≠ p2.

We can use a two-sample proportion test to test this hypothesis. Using the given data, the sample proportions are:

p1 = 915/1500 = 0.61 (December 2015)

p2 = 1340/2000 = 0.67 (December 2018)

Using a significance level of α = 0.05, we can use a z-test to determine if there is evidence to reject the null hypothesis. The test statistic is:

z = (p1 - p2) / sqrt{ (p1*(1-p1)/n1) + (p2*(1-p2)/n2) }

= (0.61 - 0.67) / sqrt{ (0.61*(1-0.61)/1500) + (0.67*(1-0.67)/2000) }

= -4.58

The critical value for a two-tailed z-test with α = 0.05 is ±1.96. Since the calculated test statistic of -4.58 is outside the critical region, we reject the null hypothesis and conclude that there is convincing evidence that the proportion of adults in the United States who would respond "yes" to the question changed from December 2015 to December 2018