Question

Two polls were taken for the approval of Donald Trump. The first was a random selection taken before the events at Charlottesville of 1000 people and 440 indicated support for Trump. The second was taken immediately after Charlottesville where 1000 people were randomly selected and 380 expressed support for Trump. Test against a 95% probability Zα/2 = 1.96 that the difference was due to random variation.

Answer 1

Let be the true proportion supporting Trump before the events at Charlottesville and be the true proportion supporting Trump after the events at Charlottesville.

We want to test if the difference between the 2 proportions was due to random variation. That is we want to test if

The hypotheses are

We have the following from the 2 samples that were collected

The combined proportion supporting Trump is

The estimated standard error of difference between the proportions is

The hypothesized difference between the proportions is

We can use normal distribution as the sampling distribution of difference in sample proportions

The test statistic is

This is a 2 tailed test (The alternative hypothesis has "not equal to")

The right tail critical value for 95% confidence level or significance level is given as

The critical values are -1.96, +1.96

We will reject the null hypothesis if the test statistic does not lie within the interval -1.96 and +1.96.

Here, the test statistic is 2.728 and it lies outside the interval 1.96 and +1.96. Hence we reject the null hypothesis.

ans: We conclude that there is sufficient evidence to reject the claim that the difference was due to random variation.

That is, we can say that the proportion supporting Trump before the events at Charlottesville is different from the proportion supporting Trump after Charlottesville.