Step 1

Solution

Given:

Confidence interval in the trilinear inequality form:

Lower limit \(\displaystyle{<}{p}{<}\) Upper limit

\(\displaystyle{53.5}\%{<}{p}{<}{69.7}\%\)

Hence,

Lower limit \(\displaystyle={53.5}\%\)

Upper limit \(\displaystyle={69.7}\%\)

Step 2

Sample proportion \(\displaystyle{\left(\hat{{{p}}}\right)}={\frac{{\text{Upper limit+Lower limit}}}{{{2}}}}\)

Plug in all the values in the formula, we get

Simple proportion \(\displaystyle{\left(\hat{{{p}}}\right)}={\frac{{{69.7}\%+{53.5}\%}}{{{2}}}}\)

Sample proportion \(\displaystyle{\left(\hat{{{p}}}\right)}={61.6}\%\)

Step 3

Margin of error \(\displaystyle{\left({M}{E}\right)}={\frac{{\text{Upper limit+Lower limit}}}{{{2}}}}\)

Plug in all the values in the formula, we get

Margin of error \(\displaystyle{\left({M}{E}\right)}={\frac{{{69.7}\%+{53.5}\%}}{{{2}}}}\)

Margin of error \(\displaystyle{\left({M}{E}\right)}={8.1}\%\)

Step 4

Express the confidence interval in the form \(\displaystyle{\left(\hat{{{p}}}\right)}\pm{E}\)

\(\displaystyle{\left(\hat{{{p}}}\right)}\pm{E}-{61.6}\%\pm{8.1}\%\)