Question

A poll asked the question, "What do you think is the most important problem facing this country today?" Twenty percent of the respondents answered "crime and violence." The margin of sampling error was plus or minus 4 percentage points. Following the convention that the margin of error is based on a 95% confidence interval, find a 95% confidence interval for the percentage of the population that would respond "crime and violence" to the question asked by the pollsters.

Answer 1

Answer:  (16%, 24%)

Step-by-step explanation:

The confidence interval for proportion is given by :-

$p\pm E$

Given : Significance level : $\alpha=1-0.95=0.05$

The proportion of the respondents answered "crime and violence." : p=0.20

Margin of sampling error : $E=\pm0.04$

Now, the 95% confidence interval for the proportion of the population that would respond "crime and violence" to the question asked by the pollsters is given by :-

$0.20\pm 0.04\\\\=0.20-0.04,\ 0.20+0.04\approx(0.16,0.24)$

In percentage, $(16\%,24\%)$

Hence, the 95% confidence interval for the percentage of the population that would respond "crime and violence" to the question asked by the pollsters =(16%, 24%)