Question

Bloomberg Media Group reported the result of an internet survey of 1000 New Yorkers.
The key question asked was: "Should a wealth tax be levied to pay for humanitarian relief efforts?". The results of the survey were as follows:

	Urban Response	Rural Response
Sample Size	550	450
Number of those who voted in favour	150	300

Is this enough evidence to draw the conclusion that the two groups i.e. Urban and Rural differ significantly with respect to their opinions? (Test at 5% level of significance)

Answer 1

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:p1​=p2​

Ha: Ha:p1​̸​=p2​

This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a two-tailed test is zc=1.96.

The rejection region for this two-tailed test is R={z:∣z∣>1.96}

(3) Test Statistics

The z-statistic is computed as follows:

4) Decision about the null hypothesis

Since it is observed that ∣z∣=12.457>zc​=1.96, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.000, and since p=0.000<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1 is different than p2, at the 0.05 significance level.