Question

In May 2006, the Gallup Poll (www.galluppoll.com) asked 510 randomly sampled adults the question "Generally speaking, do you believe the death penalty is applied fairly or unfairly in this country today?" Of these, 60% answered "Fairly," 35% said "Unfairly," and 4% said they didn't know. First, please answer this question: From this survey, what can we conclude about the opinions of all adults? (To answer this question, please build a 95% confidence interval for the proportion of all U.S. adults who believe the death penalty is applied fairly.) Please use all steps: Check requirements. Find the critical z-value. Find the margin of error. Find the confidence interval. Interpret the result (conclusion). Then, please answer the following questions (same source): Think some more about the 95% confidence interval for the proportion of US adults who think the death penalty is applied fairly. 

.1. If we wanted to be 98% confident, would our confidence interval need to be narrower or wider? 

2. Our margin of error was about ±4%. If we wanted to reduce it to ±3%, would our level of confidence be higher or lower? 

3. If the Gallup organization had polled more people, would the interval's margin of error have been larger or smaller?

Answer 1

Ans:

sample proportion=0.60

n=510

alpha=1-0.95=0.05

alpha/2=0.025

critical z value=normsinv(0.025) or normsinv(0.975)=1.960

Margin of error=1.96*sqrt(0.6*(1-0.6)/510)=0.0425

95% confidence interval is

=0.6+/-0.0425

=(0.5575, 0.6425)

We are 95% confident that true proportion of all U.S. adults who believe the death penalty is applied fairly lies between 0.5575 and 0.6425.

1. If we wanted to be 98% confident, it will be wider than 95% CI(as margin of error will be larger.)

2. Our margin of error was about ±4%. If we wanted to reduce it to ±3%, confidence level will be lower.

3. If the Gallup organization had polled more people, the interval's margin of error will be smaller.