Question

# A poll asked the question, "What do you think is the most important problem facing this country today?" Twenty-five percent of the respondents answered "crime and violence." The margin of sampling error was plus or minus 3 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.

Confidence intervals
A poll asked the question, "What do you think is the most important problem facing this country today?" Twenty-five percent of the respondents answered "crime and violence." The margin of sampling error was plus or minus 3 percentage points. Following the convention that the margin of error is based on a $$\displaystyle{95}\%$$ confidence interval, find a $$\displaystyle{95}\%$$ confidence interval for the percentage of the population that would respond "crime and violence" to the question asked by the pollsters.
Lower limit $$\displaystyle\%$$
Upper limit $$\displaystyle\%$$

2021-08-08
Step 1
The percentage of responndents answered crime and violence $$\displaystyle={25}\%$$
$$\displaystyle{95}\%$$ confidence interval margin of error $$\displaystyle={3}\%$$
Theformulato calculate the confidence interval is shown below
$$\displaystyle\text{Confidence Interval}=\text{Sample Statistic}\pm\text{Margin of Error}$$
Step 2
We know sample statistic is given to us as $$\displaystyle{25}\%$$. So substituting the values we get the lower limit and upper limit of $$\displaystyle{95}\%$$ confidence interval as shown below.
Lower Limit $$\displaystyle={22}\%$$
Upper Limit $$\displaystyle={28}\%$$
$$\displaystyle\text{Confidence Interval}=\text{Sample Statistic}\pm\text{Margin of Error}$$
$$\displaystyle\text{Lower Limit}=\text{Sample Statistic}-\text{Margin of Error}$$
$$\displaystyle\text{Lower Limit}={25}-{3}={22}\%$$
$$\displaystyle\text{Upper Limit}=\text{Sample Static}+\text{Margin of error}$$
$$\displaystyle\text{Upper Limit}={25}+{3}={28}\%$$