Question

A poll finds that 54% of the 600 people polled favor the incumbent. Shortly after the poll is taken, it is disclosed that the incumbent had an extramarital affair. A new poll finds that 50% of the 1030 polled now favor the incumbent. We want to know if support for the incumbent has decreased. What is the test statistic

Answer 1

There is not enough evidence to support the claim that the proportion that polled in favor of the incumbent has decreased significantly (P-value=0.06).

Test statistic z=1.56.

Explanation:

This is a hypothesis test for the difference between proportions.

The claim is that the proportion that polled in favor of the incumbent has decreased significantly.

Then, the null and alternative hypothesis are:

`H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2> 0`

The significance level is 0.05.

The sample 1 (old poll), of size n1=600 has a proportion of p1=0.54.

The sample 2 (new poll), of size n2=1030 has a proportion of p2=0.5.

The difference between proportions is (p1-p2)=0.04.

`p_d=p_1-p_2=0.54-0.5=0.04`

The pooled proportion, needed to calculate the standard error, is:

`p=(X_1+X_2)/(n_1+n_2)=(324+515)/(600+1030)=(839)/(1630)=0.5147`

The estimated standard error of the difference between means is computed using the formula:

`s_(p1-p2)=\sqrt{(p(1-p))/(n_1)+(p(1-p))/(n_2)}=\sqrt{(0.5147*0.4853)/(600)+(0.5147*0.4853)/(1030)}\\\\\\s_(p1-p2)=√(0.0004+0.0002)=√(0.0007)=0.0257`

Then, we can calculate the z-statistic as:

`z=(p_d-(\pi_1-\pi_2))/(s_(p1-p2))=(0.04-0)/(0.0257)=(0.04)/(0.0257)=1.56`

This test is a right-tailed test, so the P-value for this test is calculated as (using a z-table):

`\text{P-value}=P(z>1.56)=0.06`

As the P-value (0.06) is bigger than the significance level (0.05), the effect is not significant.

The null hypothesis failed to be rejected.

There is not enough evidence to support the claim that the proportion that polled in favor of the incumbent has decreased significantly.