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
ANSWERED
asked 2021-08-03
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 \(\displaystyle\%\)

Expert Answers (1)

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 \(\displaystyle={28}\%\)
\(\displaystyle\text{Confidence Interval}=\text{Sample Statistic}\pm\text{Margin of Error}\)
\(\displaystyle\text{Lower Limit}=\text{Sample Statistic}-\text{Margin of Error}\)
\(\displaystyle\text{Lower Limit}={25}-{3}={22}\%\)
\(\displaystyle\text{Upper Limit}=\text{Sample Static}+\text{Margin of error}\)
\(\displaystyle\text{Upper Limit}={25}+{3}={28}\%\)
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